A new modified Lehmann type – II G class of distributions: exponential distribution with theory, simulation, and applications to engineering sector

Oluwafemi Samson Balogun; Muhammad Zeshan Arshad; Muhammad Zafar Iqbal; Madiha Ghamkhar

doi:10.12688/f1000research.52494.1

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Research Article

A new modified Lehmann type – II G class of distributions: exponential distribution with theory, simulation, and applications to engineering sector

[version 1; peer review: 1 approved with reservations, 1 not approved]

Oluwafemi Samson Balogun¹, Muhammad Zeshan Arshad ², Muhammad Zafar Iqbal³, Madiha Ghamkhar⁴

PUBLISHED 17 Jun 2021

Author details Author details

¹ School of Computing, University of Eastern Finland, Kuopio, Northern Europe, 70211, Finland
² Department of Mathematics and Statistics, University of Agriculture, Faisalabad, Faisalabad, Pakistan, Pakistan
³ Department of Mathematics and Statistics, University of Agriculture, Faisalabad, Pakistan, University of Agriculture, Faisalabad, Faisalabad, Faisalabad, Pakistan
⁴ Department of Mathematics and Statistics, University of Agriculture, Faisalabad, Faisalabad, Faisalabad, Pakistan

Oluwafemi Samson Balogun
Roles: Funding Acquisition, Supervision, Writing – Original Draft Preparation, Writing – Review & Editing

Muhammad Zeshan Arshad
Roles: Conceptualization, Data Curation, Methodology, Software, Writing – Original Draft Preparation, Writing – Review & Editing

Muhammad Zafar Iqbal
Roles: Conceptualization, Supervision, Writing – Original Draft Preparation, Writing – Review & Editing

Madiha Ghamkhar
Roles: Methodology, Supervision, Writing – Original Draft Preparation, Writing – Review & Editing

OPEN PEER REVIEW

REVIEWER STATUS

Abstract

Background: Modeling against non-normal data a challenge for theoretical and applied scientists to choose a lifetime model and expect to perform optimally against experimental, reliability engineering, hydrology, ecology, and agriculture sciences, phenomena.
Method: We have introduced a new G class that generates relatively more flexible models to its baseline and we refer to it as the new modified Lehmann Type – II (ML–II) G class of distributions. A list of new members of ML–II-G class is developed and as a sub-model the exponential distribution, known as the ML-II-Exp distribution is considered for further discussion. Several mathematical and reliability characters along with explicit expressions for moments, quantile function, and order statistics are derived and discussed in detail. Furthermore, plots of density and hazard rate functions are sketched out over the certain choices of the parametric values. For the estimation of the model parameters, we utilized the method of maximum likelihood estimation.
Results: The applicability of the ML–II–G class is evaluated via ML–II–Exp distribution. ML–II–Exp distribution is modeled to four suitable lifetime datasets and the results are compared with the well-known competing models. Some well recognized goodness–of–fit including -Log-likelihood (-LL), Anderson-Darling (A*), Cramer-Von Mises (W*), and Kolmogorov-Smirnov (K-S) test statistics are considered for the selection of a better fit model.
Conclusion: The minimum value of the goodness–of–fit is the criteria of a better fit model that the ML–II–Exp distribution perfectly satisfies. Hence, we affirm that the ML–II–Exp distribution is a better fit model than its competitors.

Keywords

Lehmann Type – II Distribution; Exponential Distribution; Hazard Rate Function; Moments; Rényi Entropy; Maximum Likelihood Estimation, Simulation.

Corresponding author: Muhammad Zeshan Arshad

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2021 Balogun OS et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Balogun OS, Arshad MZ, Iqbal MZ and Ghamkhar M. A new modified Lehmann type – II G class of distributions: exponential distribution with theory, simulation, and applications to engineering sector [version 1; peer review: 1 approved with reservations, 1 not approved]. F1000Research 2021, 10:483 (https://doi.org/10.12688/f1000research.52494.1) First published: 17 Jun 2021, 10:483 (https://doi.org/10.12688/f1000research.52494.1) Latest published: 17 Jun 2021, 10:483 (https://doi.org/10.12688/f1000research.52494.1)

Introduction

Over the past two decades, the growing attention of researchers towards the development of new G families has explored the remarkable characteristics of baseline models. New models open new horizons for theoretical and applied researchers to address real-world problems, to proficiently and adequately fit them to asymmetric and complex random phenomena. Accordingly, several classes of distributions have been developed and discussed in the literature. For more details, we encourage the reader to see the credible work of some notable scientists including Marshall and Olkin¹ $[G (x) / (G (x) + α \bar{G} (x))]$ , who developed a new technique to generate models; Eugene et al.,² who proposed beta generated-G class $[G_{beta} (x)]$ ; the quadratic rank transmutation map technique introduced by Shaw and Buckley³ $[(1 + λ) G (x) - λ G^{2} (x)]$ ; the Kumaraswamy generalized-G class proposed by Cordeiro and Castro⁴ $[1 - {[1 - G^{α} (x)]}^{β}]$ ; the gamma-G class proposed by Ristic and Balakrishnan⁵ $[- lo g G (x)]$ ; the exponentiated generalized (EG) class proposed by Cordeiro et al.⁶ $[{(1 - {[1 - G (x)]}^{α})}^{β}]$ ; the T–X family proposed by Alzaatreh et al.⁷ $[1 - R (W [G (x)])]$ ; the Weibull–G family proposed by Bourguignon et al.⁸ $[1 - exp (- α {[G (x) / \bar{G} (x)]}^{β})]$ ; the beta Marshall–Olkin–G class proposed by Alizadeh et al.⁹ $[I_{Marshall - Olkin} (a, b)]$ ; the logistic-X family proposed by Tahir et al.¹⁰ $[{[1 + {[- lo g [\bar{G} (x)]]}^{- α}]}^{- 1}]$ ; Kumar et al.¹¹ proposed DUS transformation $[(e^{G (x)} - 1) / (e - 1)]$ ; Mahdavi and Kundu¹² proposed Alpha transformation $[(α^{G (x)} - 1) / (α - 1)]$ ; Elbatal et al.¹³ proposed a new Alpha power transformed family of distributions $[G (x) α^{G (x)} / α]$ ; and recently, Ijaz et al.¹⁴ proposed a Gull alpha power Weibull family $[αG (x) / α^{G (x)}]$ ; among others.

This paper is organised into the following sections. A new modified Lehmann Type–II (ML–II) G class of distributions accompanied by a table of some special models is proposed and developed in A new modified Lehmann Type – II–G Class of distributions. A special model of ML–II–G class, known as a modified Lehmann Type–II exponential (ML–II–Exp) distribution, along with its mathematical and reliability measures are derived and discussed in Mathematical properties. The method of maximum likelihood estimation is used to estimate the unknown model parameters and develop some simulation results to assess the performance of maximum likelihood estimations (MLEs) in Inference. Applications are discussed in four real data applications and finally, the conclusion is reported in Conclusions.

A new modified Lehmann Type – II–G Class of distributions

Lehmann¹⁵ proposed the Lehmann Type–I (L - I) distribution, which was the simple exponentiated version of any arbitrary baseline model. Accordingly, the first credit goes to Gupta et al.¹⁶ because they applied L – I on exponential distribution. The associated cumulative distribution function (CDF) is given by

F (x; α; ξ) = G^{α} (x; ξ)

Cordeiro et al.⁴ deserves to be acknowledged as they use dual transformation that yielded the Lehmann Type – II (L–II) G class of distributions. The associated CDF is given by

F (x; α; ξ) = 1 - {(1 - G (x; ξ))}^{α}

The closed-form feature of the L–II distribution assists one to derive and study its numerous properties and in literature, both the approaches (L–I and L–II) have been extensively utilized to study the unexplored characteristics of the classical and modified models.

We develop a new G class, known as a modified Lehmann Type–II (ML–II) G class of distributions. The corresponding CDF is given by

(1)

F_{ML - II - G} (x; α; β; ξ) = 1 - {(\frac{1 - G (x; ξ)}{1 - αG (x; ξ)})}^{β}, x ϵ ℝ

where $G (x; ξ) ϵ (0, 1)$ is a CDF of any arbitrary baseline model based on the parametric vector $ξ$ , dependent on (r x 1) with $- \infty < α < 1,$ and $α, β > 0$ are the scale and shape parameters, respectively. Let $g (x; ξ) = dG (x; ξ) / dx$ be the density function of any baseline model. The associated probability density function (PDF), hazard rate function (HRF), and quantile function of ML–II–G class of distributions are given by, respectively

(2)

f_{ML - II - G} (x; α; β; ξ) = β (1 - α) g (x; ξ) {(1 - G (x; ξ))}^{β - 1} {(1 - αG (x; ξ))}^{- (1 + β)}

(3)

h_{ML - II - G} (x; α; β; ξ) = \frac{β (1 - α) g (x; ξ) {(1 - G (x; ξ))}^{β - 1} {(1 - αG (x; ξ))}^{- (1 + 2 β)}}{{(1 - αG (x; ξ))}^{β} - {(1 - G (x; ξ))}^{β}}

(4)

Q_{ML - II - G} (p; α; β; ξ) = G^{- 1} (\frac{{(1 - p)}^{\frac{1}{β}} - 1}{α {(1 - p)}^{\frac{1}{β}} - 1}), p \in (0, 1)

Now and onward, the modified Lehmann Type–II–G random variable X corresponding to $f_{ML - II - G} (x; α; β; ξ)$ will be denoted by X~ML– II– $G (x; α; β; ξ) .$

This study aimed to propose a new G class of distributions that generates more flexible alternative continuous models relative to its parent distribution. From a computational point of view, new models are very simple to interpret. New models offer greater distributional flexibility and can provide a better fit over the complex random phenomena that exclusively arise in engineering sciences. However, to the best of our knowledge, no study has been conducted previously that discusses our new G class, deliberates the unexplained complex random phenomena so well and advances the fit to a diverse range of sophisticated lifetime data.

Linear combination provides a much more informal approach to discuss the CDF and PDF than the conventional integral computation when determining the mathematical properties. For this, binomial expansion is given as follows:

{(1 - z)}^{β} = \sum_{i = 0}^{\infty} {(- 1)}^{i} (\begin{matrix} β \\ i \end{matrix}) z^{i}, |z| < 1

Infinite linear combinations of CDF (1) and PDF (2) of ML–II–G class are given by, respectively

F (x; α; β; ξ) = 1 - \sum_{i, j = 0}^{\infty} {(- 1)}^{i + j} α^{j} (\begin{matrix} β \\ i \end{matrix}) (\begin{matrix} - β \\ j \end{matrix}) G^{i + j} (x; α; β; ξ)

(5)

F (x; α; β; ξ) = 1 - \sum_{i, j = 0}^{\infty} ω_{ij}^{*} G^{i + j} (x; α; β; ξ)

f (x; α; β; ξ) = β (1 - α) \sum_{i, j = 0}^{\infty} {(- 1)}^{i + j} (\begin{matrix} β - 1 \\ i \end{matrix}) (\begin{matrix} - β - 1 \\ j \end{matrix}) G^{i + j} (x; α; β; ξ) g (x; α; β; ξ)

(6)

f (x; α; β; ξ) = β (1 - α) \sum_{i, j = 0}^{\infty} ω_{ij} G^{i + j} (x; α; β; ξ) g (x; α; β; ξ)

The r-th ordinary moment (say $μ_{r}^{/}$ ) of X is given by

(7)

μ_{r, I}^{/} = β (1 - α) \sum_{i, j = 0}^{\infty} ω_{ij} I_{r, i + j} (x; ξ)

where coefficient $ω_{ij}^{*} = {(- 1)}^{i + j} α^{j} (\begin{matrix} β \\ i \end{matrix}) (\begin{matrix} - β \\ j \end{matrix})$ , $ω_{ij} = \sum_{i, j = 0}^{\infty} {(- 1)}^{i + j} (\begin{matrix} β - 1 \\ i \end{matrix}) (\begin{matrix} - β - 1 \\ j \end{matrix})$ , and $I_{r, i + j} (x; ξ) = \int_{- \infty}^{\infty} x^{r} G^{i + j} (x; ξ) g (x; ξ) dx .$

Further, by (1), Lehmann Type–II–G class and baseline $G (x)$ model can be traced back at $α = 0$ , and $α = 0, β = 1$ , respectively. The expansions of CDF (5) and PDF (6) provide us with the Exp-G class, which is quite useful for the generalization of models.

Order statistics

In reliability analysis and life testing of a component in quality control, order statistics (OS) and its moments are considered as a noteworthy measure. Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample of size n that follows the Lehmann Type–II–G class and $X_{(1 : n)} < X_{(2 : n)} < \dots < X_{(n : n)}$ be the corresponding OS. The random variables $X_{(i)}, X_{(1)},$ and $X_{(n)}$ be the i-th, minimum, and maximum OS of $X$ .

The PDF of $X_{(i)}$ is given by

f_{(i : n)} (x; ξ) = \frac{1}{B (i, n - i + 1)!} {(F (x; ξ))}^{i - 1} {(1 - F (x; ξ))}^{n - i} f (x; ξ)

where $= 1, 2, 3, \dots, n, F (x; ξ)$ , and $f (x; ξ)$ are the associated CDF (5) with corresponding PDF (6) of the Lehmann Type–II–G family. Using the fact that

{(1 - F (x; ξ))}^{n - i} = \sum_{m = 0}^{n - i} {(- 1)}^{m} (\begin{matrix} n - i \\ m \end{matrix}) F {(x; ξ)}^{m}

By placing the last expression in $f_{(i : n)} (x; ξ)$ , we get the most refined form of OS PDF and one may determine it by integrating (5–6) and the expression may be given as follows

f_{(i : n)} (x; ξ) = \frac{f (x; ξ)}{B (i, n - i + 1)!} \sum_{m = 0}^{n - i} {(- 1)}^{m} (\begin{matrix} n - i \\ m \end{matrix}) F {(x; ξ)}^{i + m - 1}

A modified Lehmann Type – II Exponential (Ml–II–Exp) distribution (sub-model)

In this section, we introduce a sub-model of ML–II–G class of distributions, known as a ML–II–Exp distribution. For this, we have the CDF and PDF of the exponential distribution $G_{Exp} (x; θ) = 1 - e^{- θx},$ and $g_{Exp} (x; θ) = {θe}^{- θx}, for θ > 0, 0 < x < \infty,$ respectively. The associated CDF and corresponding PDF of the ML–II–Exp distribution are obtained by following (1–2) and its analytical expressions are given by respectively:

(8)

F_{ML - II - Exp} (x; α; β; θ) = 1 - {(\frac{e^{- θx}}{1 - α + {αe}^{- θx}})}^{β}, - \infty < α < 1

(9)

f_{ML - II - Exp} (x; α; β; θ) = \frac{(1 - α) βθ e^{- θβx}}{{(1 - α + {αe}^{- θx})}^{β + 1}}

where $α$ is a scale and $β, θ > 0$ are the shape parameters, respectively. By following (1–2), linear representation of CDF and PDF are given as follows, respectively

(10)

F (x; α; β; θ) = 1 - \sum_{i, j = 0}^{\infty} {(- 1)}^{j} θ^{j} {(β + i)}^{j} (\begin{matrix} - β \\ i \end{matrix}) \frac{α^{j} {(1 - α)}^{- β - i}}{j!} x^{j}

F (x; α; β; θ) = 1 - \sum_{i, j = 0}^{\infty} ζ_{ij} x^{j}

(11)

f (x; α; β; θ) = βθ \sum_{i = 0}^{\infty} α^{i} {(1 - α)}^{- β} (\begin{matrix} - β - 1 \\ i \end{matrix}) e^{- (i + β) θx}

(12)

f (x; α; β; θ) = β \sum_{i, j = 0}^{\infty} {(- 1)}^{j} {(1 - α)}^{- β - i} α^{i} θ^{j + 1} (\begin{matrix} - β - 1 \\ i \end{matrix}) \frac{{(β + i)}^{j}}{j!} x^{j}

f (x; α; β; θ) = β \sum_{i, j = 0}^{\infty} ζ_{ij}^{*} x^{j} .

where $ζ_{ij} = {(- 1)}^{j} θ^{j} {(β + i)}^{j} (\begin{matrix} - β \\ i \end{matrix}) \frac{α^{j} {(1 - α)}^{- β - i}}{j!}$ , $ζ_{ij}^{*} = {(- 1)}^{j} {(1 - α)}^{- β - i} α^{i} θ^{j + 1} (\begin{matrix} - β - 1 \\ i \end{matrix}) \frac{{(i + β)}^{j}}{j!}$

Expression in (11) is expected to be quite supportive in the forthcoming computations of various mathematical properties of the ML–II–Exp distribution.

Mathematical properties

Reliability characteristics

One of the imperative roles of probability distribution in reliability engineering is to analyze and predict the life of a component. One may define the reliability function as the probability that a component survives until the time x and analytically it can be written as $R (x) = 1 - F (x)$ .

The reliability function of X is given by

(13)

R_{M L - I I - E x p} (x; α; β; θ) = {(\frac{e^{- θx}}{1 - α + {αe}^{- θx}})}^{β}

In reliability theory, the significant contribution of a function that measures the failure rate of a component in a particular time t is sometimes referred to as the HRF, failure rate function, or the force of mortality, and mathematically it can be written as $h (x) = f (x) / R (x) .$

The HRF of X is given by

(14)

h_{ML - II - Exp} (x; α; β; θ) = \frac{(1 - α) βθ}{(1 - α + {αe}^{- θx})}

Numerous notable reliability measures for the ML–II–Exp distribution can be discussed and derived, such as reverse HRF by $h_{r} (x; α; β; θ) = f (x; α; β; θ) / R (x; α; β; θ),$ Mills ratio by $M (x; α; β; θ) = R (x; α; β; θ) / f (x; α; β; θ),$ and Odd function by

O (x; α; β; θ) = F (x; α; β; θ) / R (x; α; β; θ) .

Shapes

Different plots of PDF and HRFs of the ML–II–Exp distribution are sketched over the selected and fixed combinations of the model parameters, respectively. Figure 1 (a, b, c) presents the reversed-J, constant, unimodal, and right-skewed shapes of the PDF and Figure 2 (a, b, c) illustrates the decreasing and increasing HRF. However, an increasing HRF with some interesting facts are identified when suddenly spikes arise at the tail end of HRF is unexpectedly detected. Such kinds of trends are often observed in non-stationary time series lifetime phenomena.

Table 1. Some special models and corresponding G(x, ξ) and S(x, ξ).

Model	G(x)	S(x)	ξ
Rayleigh ( $x > 0$ )	$1 - e^{- η x^{2}}$	${(\frac{e^{- η x^{2}}}{1 - α (1 - e^{- η x^{2}})})}^{β}$	$α, β, η$
Gompertz ( $x > 0$ )	$1 - e^{- η (e^{γx} - 1)}$	${(\frac{e^{- η (e^{γx} - 1)}}{1 + α (1 - e^{- η (e^{γx} - 1)})})}^{β}$	$α, β, η, γ$
Pareto ( $x > m$ )	$1 - {(\frac{m}{x})}^{η}$	${(\frac{{(\frac{m}{x})}^{η}}{1 - α (1 - {(\frac{m}{x})}^{η})})}^{β}$	$α, β, η$
Fréchet ( $x > 0$ )	$e^{- η x^{- γ}}$	${(\frac{1 - e^{- η x^{- γ}}}{1 - α e^{- η x^{- γ}}})}^{β}$	$α, β, η, γ$
Burr X ( $x > 0$ )	${(1 - e^{- {(ηx)}^{2}})}^{γ}$	${(\frac{1 - {(1 - e^{- {(ηx)}^{2}})}^{γ}}{1 - α {(1 - e^{- {(ηx)}^{2}})}^{γ}})}^{β}$	$α, β, η, γ$
Weibull ( $x > 0$ )	$1 - e^{- η x^{γ}}$	${(\frac{e^{- η x^{γ}}}{1 - α (1 - e^{- η x^{γ}})})}^{β}$	$α, β, η, γ$
Lomax ( $x > 0$ )	$1 - {(1 + \frac{x}{η})}^{- γ}$	${(\frac{{(1 + \frac{x}{η})}^{- γ}}{1 - α (1 - {(1 + \frac{x}{η})}^{- γ})})}^{β}$	$α, β, η, γ$

Figure 1. Density function plots of ML–II–Exponential distribution.

Figure 2. Hazard rate function plots of ML–II–Exponential distribution.

Limiting behavior

Here we study the limiting behavior of CDF, PDF, reliability, and HRFs of the ML–II–Exp distribution present in (8), (9), (13), and (14) at x $\to 0$ and x $\to \infty$ .

Proposition-1: Limiting behavior of CDF, PDF, reliability, and HRFs of the ML–II–Exp distribution at x $\to 0$ is followed by

\begin{array}{l} F (0; α; β; θ) \sim 0 \\ f (0; α; β; θ) \sim (1 - α) β θ \\ R (0; α; β; θ) \sim 1 \\ h (0; α; β; θ) \sim (1 - α) β θ \end{array}

Proposition-2: Limiting behavior of CDF, PDF, reliability, and HRFs of the ML–II–Exp distribution at x $\to \infty$ is followed by

\begin{array}{l} F (\infty α β θ) \sim 1 \\ f (\infty α β θ) \sim 0 \\ R (\infty α β θ) \sim 0 \\ h (\infty α β θ) \sim 0 \end{array}

Limiting behaviors developed in the above expressions may illustrate the effect of parameters on the tail of the ML–II–Exp distribution.

Moments and associated measures

Moments have a remarkable role in the discussion of the distribution theory, to study the significant characteristics of a probability distribution such as mean; variance; skewness, and kurtosis.

Theorem 1: If X $\sim$ ML–II–Exp ( $x; α, β, θ$ ), for $x; β, θ > 0$ , with $- \infty < α < 1$ , then the r-th ordinary moment (say $μ_{r}^{/}$ ) of X is given by

μ_{r}^{/} = \frac{β}{θ^{r}} \sum_{i = 0}^{\infty} \frac{η_{i}}{{(β + i)}^{r + 1}} Γ (r + 1)

where $η_{i} = α^{i} {(1 - α)}^{- β - i} (\begin{matrix} - β - 1 \\ i \end{matrix})$

Proof: $μ_{r}^{/}$ can be written by following (11), as

(15)

μ_{r}^{/} = βθ \sum_{i = 0}^{\infty} α^{i} {(1 - α)}^{- β - i} (\begin{matrix} - β - 1 \\ i \end{matrix}) \int_{0}^{\infty} x^{r} e^{- θx (β + i)} dx

Let’s suppose $θx (β + i) = y; x = \frac{y}{θ (β + i)} ⟹ dx = \frac{dy}{θ (β + i)}$

limits: as $x ⟶ 0 \Rightarrow x ⟶ \infty; y ⟶ 0 \Rightarrow y ⟶ \infty .$

By placing the above information in (15), we get

μ_{r}^{/} = βθ \sum_{i = 0}^{\infty} α^{i} {(1 - α)}^{- β - i} (\begin{matrix} - β - 1 \\ i \end{matrix}) \int_{0}^{\infty} {(\frac{y}{θ (β + i)})}^{r} \frac{e^{- y}}{θ (β + i)} d y

by making simple computation on the last expression leads us to the r-th ordinary moment, in terms of the gamma function and it is given by

(16)

μ_{r}^{/} = \frac{β}{θ^{r}} \sum_{i = 0}^{\infty} \frac{η_{i}}{{(β + i)}^{r + 1}} Γ (r + 1), r = 1, 2, \dots n

where $Γ (.)$ is a gamma function, $Γ (x) = \int_{0}^{\infty} t^{x - 1} e^{- t} d t, η_{i} = α^{i} {(1 - α)}^{- β - i} (\begin{matrix} - β - 1 \\ i \end{matrix}),$ and $- \infty < α < 1 .$

The derived expression in (16) may serve a supportive and useful role in the development of several statistical measures. For instance: to deduce the mean of X, placer = 1 in (16). For the higher-order ordinary moments of X approximating to 2^nd, 3^rd, and 4^th, and higher, these moments can be formulated by setting r = 2, 3, and 4 in (16), respectively. Additionally, to discuss the variability in X, the Fisher index (F. I = ( $Var (X) / E (X)$ )) may play a significant role in this scenario. To derive the negative moments of X, simply substitute r by –w in (16). One may perhaps further determine the well-established statistics such as skewness ( $τ_{1} = μ_{3}^{2} / μ_{2}^{3}$ ), and kurtosis ( $τ_{2} = μ_{4} / μ_{2}^{2}$ ), of X by integrating (16). A well-established relationship between the central moments $(μ_{s})$ and cumulants ( $K_{s}$ ) of X may easily be defined by ordinary moments $μ_{s} = \sum_{k = 0}^{s} (\begin{matrix} s \\ k \end{matrix}) {(- 1)}^{k} {(μ_{1}^{/})}^{s} μ_{s - k}^{/}$ . Hence, the first four cumulants can be calculated by $K_{1} = μ_{1}^{/}$ , $K_{2} = μ_{2}^{/} - μ_{1}^{/ 2}$ , $K_{3} = μ_{3}^{/} - 3 μ_{2}^{/} μ_{1}^{/} + 2 μ_{1}^{/ 3}$ , and $K_{4} = μ_{4}^{/} - 4 μ_{3}^{/} μ_{1}^{/} - 3 μ_{2}^{/ 2} + 12 μ_{2}^{/} μ_{1}^{/ 2} - 6 μ_{1}^{/ 4}$ , etc., respectively.

In Table 2, some numerical results of the first eight ordinary moments, $ε^{2}$ = variance, $τ_{1} =$ skewness, and $τ_{2} =$ kurtosis for some chosen parameters are presented in S-I ( $α = 0.9, θ = 0.1, β = 1.1$ ), S-II ( $α = 0.9, θ = 0.5, β = 1.5$ ), S-III ( $α = 0.8, θ = 0.09, β = 3.1$ ), and S-IV ( $α = 0.5, θ = 0.009, β = 5.1$ ).

Table 2. Some numerical results of moments, variance, skewness, and kurtosis.

${μ^{/}}_{r}$	S-I	S-II	S-III	S-IV
${μ^{/}}_{1}$	1.0669	1.0953	0.4328	0.3951
${μ^{/}}_{2}$	2.2184	2.0624	0.3682	0.3119
${μ^{/}}_{3}$	6.8258	5.3306	0.4638	0.3688
${μ^{/}}_{4}$	27.8026	17.3849	0.7714	0.5811
${μ^{/}}_{5}$	141.0179	68.4275	1.5920	1.1437
${μ^{/}}_{6}$	856.5819	316.0155	3.9204	2.6994
${μ^{/}}_{7}$	6063.8300	1678.4290	11.2159	7.4292
${μ^{/}}_{8}$	49031.2300	10095.8100	36.5526	23.3579
$ε^{2}$	1.9427	1.7261	0.2578	0.2129
$τ_{1}$	2.9596	1.8225	1.9987	2.0879
$τ_{2}$	5.0391	3.3552	4.5027	4.7385

Residual and reversed residual life functions

The residual life function/conditional survivor function of random variable $R (t) = X + t / X > t$ X is the probability that a component whose life says x, survives in an additional interval at $t \geq 0$ . Analytically it can be written as

S_{R (t)} (x) = \frac{S (x + t)}{S (t)}

The residual life function of X is given by

(17)

S_{R (t) - ML - II - Exp} (x; α; β; θ) = \frac{e^{- θβ (x + t)} {(1 - α + {αe}^{- θt})}^{β}}{e^{- θβt} {(1 - α + {αe}^{- θ (x + t)})}^{β}}, x > 0

with associated CDF

(18)

F_{R (t) - ML - II - Exp} (x; α; β; θ) = \frac{e^{- θβt} {(1 - α + {αe}^{- θ (x + t)})}^{β} - e^{- θβ (x + t)} {(1 - α + {αe}^{- θt})}^{β}}{e^{- θβt} {(1 - α + {αe}^{- θ (x + t)})}^{β}}

PDF and HRF corresponding to (18) are given as follows, respectively

(19)

f_{R (t) - ML - II - Exp} (x; α; β; θ) = \frac{βθ (1 - α) {(1 - α + α e^{- tθ})}^{β} e^{- βθ (x + t) + θ (x + t) + βtθ}}{{(1 - α + α e^{- θ (x + t)})}^{β} (α + (1 - α) e^{θ (x + t)})}

(20)

h_{R (t) - ML - II - Exp} (x; α; β; θ) = \frac{βθ (1 - α) e^{- βθ (x + t)}}{1 - α + {αe}^{- θ (x + t)}}

Mean residual life function is given by

E (S_{R} (x)) = \frac{1}{S (t)} (μ_{1}^{/} - \int_{0}^{t} x f (x) d x) - t, t \geq 0

Moreover, the reverse residual life can be defined as: $\bar{R} (t) = t - X / X \leq t$ .

S_{\bar{R} (t)} (x) = \frac{F (t - x)}{F (t)}, t \geq 0

The reverse residual life function of X is given by

(21)

S_{\bar{R} (t) - ML - II - Exp} (x; α; β; θ) = \frac{({(1 - α + {αe}^{- θ (t - x)})}^{β} - e^{- βθ (t - x)}) {(1 - α + {αe}^{- θt})}^{β}}{({(1 - α + {αe}^{- θt})}^{β} - e^{- βθt}) {(1 - α + {αe}^{- θ (t - x)})}^{β}}

with associated CDF

(22)

F_{\bar{R} (t) - ML - II - Exp} (x; α; β; θ) = \frac{(\begin{matrix} ({(1 - α + {αe}^{- θt})}^{β} - e^{- βθt}) {(1 - α + {αe}^{- θ (t - x)})}^{β} - \\ ({(1 - α + {αe}^{- θ (t - x)})}^{β} - e^{- βθ (t - x)}) {(1 - α + {αe}^{- θt})}^{β} \end{matrix})}{({(1 - α + {αe}^{- θt})}^{β} - e^{- βθt}) {(1 - α + {αe}^{- θ (t - x)})}^{β}}

PDF and HRF corresponding to (22) are given as follows, respectively

(23)

f_{\bar{R} (t) - ML - II - Exp} (x; α; β; θ) = \frac{- βθ (1 - α) {(1 - α + α e^{- tθ})}^{β} e^{- βθ (t - x) + θ (t - x) + βtθ}}{(\begin{matrix} (1 - {(1 - α + α e^{- tθ})}^{β} e^{βtθ}) {(1 - α + α e^{- θ (t - x)})}^{β} \\ (α + (1 - α) e^{θ (t - x)}) \end{matrix})}

(24)

h_{\bar{R} (t) - ML - II - Exp} (x; α; β; θ) = \frac{- βθ (1 - α) e^{- βθ (t - x)}}{1 - α + {αe}^{- θ (t - x)}}

Mean reversed residual life function/mean waiting time is given by

E (S_{\bar{R}} (x)) = t - \frac{1}{F (t)} \int_{0}^{t} x f (x) d x, t \geq 0

One may derive the strong mean inactivity time of X by following

M (t) = t^{2} - \frac{1}{f (t)} \int_{0}^{t} x^{2} f (x) dx for t \geq 0

where $Φ_{r} (t) = \int_{0}^{t} x^{r} f (x) dx,$ is the r–th incomplete moment, $μ_{1 t}^{/} = \int_{0}^{t} xf (x) dx = \frac{β}{θ} \sum_{i = 0}^{\infty} \frac{η_{i}}{{(β + i)}^{2}} Γ_{t} (2),$ and $μ_{2 t}^{/} = \int_{0}^{t} x^{2} f (x) dx = \frac{β}{θ^{2}} \sum_{i = 0}^{\infty} \frac{η_{i}}{{(β + i)}^{3}} Γ_{t} (3) .$ By following (16), we may derive directly the $μ_{1 t}^{/}$ and $μ_{2 t}^{/}$ by holding “ $t = lθ (β + i)$ ” as the upper bound. Furthermore, $μ_{1 t}^{/}$ and $μ_{2 t}^{/}$ are termed as the first and second lower incomplete moments of X, respectively, with $η_{i} = α^{i} {(1 - α)}^{- β - i} (\begin{matrix} - β - 1 \\ i \end{matrix}),$ and $- \infty < α < 1 .$

Entropy

When a system is quantified by disorderedness, randomness, diversity, or uncertainty, in general, it is known as entropy.

Rényi¹⁷ entropy of X is given by

(25)

H_{ζ} (X) = \frac{1}{1 - ζ} lo g \int_{0}^{\infty} f^{ζ} (x) dx, ζ > 0 and ζ \neq 1

By following (9), we simplify $f (x)$ in terms of $f^{ζ} (x)$ , we get

H_{ζ - M L - I I - E x p} (X) = {((1 - α) β θ)}^{ζ} {(1 - α)}^{- ζ (β + 1)} e^{- βθζx} {(1 + \frac{α}{1 - α} e^{- θ x})}^{- ζ (β + 1)}

by placing the above expression in (25), we get

H_{ζ - M L - I I - E x p} (X) = \frac{1}{1 - ζ} lo g ({((1 - α) βθ)}^{ζ} {(1 - α)}^{- ζ (β + 1)} \int_{0}^{\infty} e^{- βθζx} {(1 + \frac{α}{1 - α} e^{- θx})}^{- ζ (β + 1)} dx)

H_{ζ - M L - I I - E x p} (X) = \frac{1}{1 - ζ} lo g ({(βθ)}^{ζ} {(1 - α)}^{- ζβ} \sum_{j = 0}^{\infty} (\begin{matrix} - ζ (β + 1) \\ j \end{matrix}) {(\frac{α}{1 - α})}^{j} \int_{0}^{\infty} e^{- θ ζ (β + 1) x} dx)

hence, solving simple mathematics on the previous equation leads us to the most simplified form of the Rényi entropy of X and it is given by

H_{ζ - M L - I I - E x p} (X) = \frac{1}{1 - ζ} lo g (τ_{ζ} \sum_{j = 0}^{\infty} v_{j})

where $v_{j} = {(\frac{α}{1 - α})}^{j} (\begin{matrix} - ζ (β + 1) \\ j \end{matrix}), τ_{ζ} = \frac{1}{θ ζ (β + 1)}, - \infty < α < 1 .$

Quantile function, and mode

The q^th quantile function of ML–II–Exp distribution is obtained by inverting the CDF. Quantile function is defined as $q = F (x_{q}) = P (X \leq x_{q}), q \in (0, 1) .$

The quantile function of X is given by

(26)

x_{q - M L - I I - E x p} = \frac{1}{θ} lo g (\frac{1 - α {(1 - q)}^{\frac{1}{β}}}{(1 - α) {(1 - q)}^{\frac{1}{β}}})

To obtain the 1^st quartile, median and 3^rd quartile of X, place q = 0.25, 0.5, and 0.75 respectively in (26). Henceforth, to generate random numbers, one may assume that the CDF in (8) follows the uniform distribution u = U (0, 1).

The modal value of X is calculated by following the constraint $f^{/} (x; α; β; θ) = \frac{df (x; α; β; θ)}{dx} = 0 .$ For convenience, $f (x; α; β; θ)$ can be rewritten as

f_{M L - I I - E x p} (x; α; β; θ) = \frac{(1 - α) β θ e^{- β θ x}}{{(1 - α + {α e}^{- θ x})}^{β + 1}}

The simplified form of $f^{/} (x; α; β; θ)$ is given by

{f^{/}}_{M L - I I - E x p} (x; α; β; θ) = \frac{α e^{- θ x} - (1 - α) β}{{(1 - α + α e^{- θ x})}^{β + 1}}

Hence, solving simple algebra on the previous equation may provide us with the most suitable form of the mode of X in support of $f^{/} (x; α; β; θ) = 0$ and it is given by

{\hat{x}}_{M L - I I - E x p} = \frac{1}{θ} lo g (\frac{α}{β (1 - α)}), - \infty < α < 1 .

Stress – strength reliability

Let X₁ and X₂ be defined to discuss the strength and stress of a component, respectively, followed by the same uni-variate family of distributions, which will work in order if $X_{2} < X_{1}$ . To discuss the reliability (say R) of X, it is given by $R = P (X_{2} < X_{1}) .$

Theorem 2: Let $X_{1} \sim$ ML–II–Exp ( $x; α, β_{1}, θ$ ) and $X_{2} \sim$ ML–II–Exp ( $x; α, β_{2}, θ$ ) be independent random variables following the ML–II–Exp distribution; then the reliability is given by

$R = \frac{β_{1}}{β_{1} + β_{2}}$

Proof: Reliability (R) is defined as

R = P (X_{2} < X_{1}) = \int_{0}^{\infty} f_{1} (x) F_{2} (x) d x .

R of X can be written by following (9), as

(27)

R_{M L - I I - E x p} = \int_{0}^{\infty} (\frac{(1 - α) β_{1} θ e^{- θ β_{1} x}}{{(1 - α + {αe}^{- θx})}^{β_{1} + 1}}) (1 - {(\frac{e^{- θx}}{1 - α + {αe}^{- θx}})}^{β_{2}}) dx,

Let’s suppose $t = {(\frac{e^{- θx}}{1 - α + {αe}^{- θx}})}^{β_{1}} \Rightarrow - dt = \frac{(1 - α) β_{1} θ e^{- θ β_{1} x}}{{(1 - α + {αe}^{- θx})}^{β_{1} + 1}} dx; and t^{\frac{β_{2}}{β_{1}}} = {(\frac{e^{- θx}}{1 - α + {αe}^{- θx}})}^{β_{2}}$

limits: as $x ⟶ 0 \Rightarrow t ⟶ 1; x ⟶ \infty \Rightarrow t ⟶ 0 .$

By placing the above information in (27), we have

R_{M L - I I - E x p} = \int_{0}^{1} (1 - t^{\frac{β_{2}}{β_{1}}}) d t,

Hence, the simple computation of the above expression provides us with the reduced form of R in terms of $β_{1}$ and $β_{2}$ , as we presume that the R is a function of $β_{1}$ with increasing behavior and it is given by

R_{M L - I I - E x p} = \frac{β_{2}}{β_{1} + β_{2}}

Order statistics

In reliability analysis and life testing of a component in quality control, order statistics OS and its moments are considered as a noteworthy measure. Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample of size n following the ML–II–Exp distribution and $X_{(1 : n)} < X_{(2 : n)} < \dots < X_{(n : n)}$ be the corresponding OS. The random variables $X_{(i)}, X_{(1)},$ and $X_{(n)}$ be the i-th, minimum, and maximum OS of $X$ .

The PDF of $X_{(i)}$ is given by

f_{(i : n)} (x) = \frac{1}{B (i, n - i + 1)!} {(F (x))}^{i - 1} {(1 - F (x))}^{n - i} f (x), i = 1, 2, 3, \dots, n

By following (8) and (9), the PDF of $X_{(i)}$ takes the form

(28)

f_{(i : n) - M L - I I - E x p} (x; α; β; θ) = (\begin{matrix} \frac{1}{B (i, n - i + 1)!} {(1 - {(\frac{e^{- θx}}{1 - α + {αe}^{- θx}})}^{β})}^{i - 1} \\ {({(\frac{e^{- θx}}{1 - α + {αe}^{- θx}})}^{β})}^{n - i} \\ \frac{(1 - α) βθ e^{- θβx}}{{(1 - α + {αe}^{- θx})}^{β + 1}} \end{matrix})

by utilizing some the techniques of binomial expansion (mentioned in the new modified Lehmann Type – II–G Class of distributions) to simplify (28), we get the reduced form of $f_{(i : n)} (x; α; β; θ)$ and it is given as follows

(29)

f_{(i : n) - M L - I I - E x p} (x) = \frac{βθ}{B (i, n - i + 1)!} \sum_{j = 0}^{\infty} \sum_{k = 0}^{\infty} η_{jk} e^{- θxv}

and we determine the linear representation of (29) and it can be written as

(30)

f_{(i : n) M L - I I - E x p} (x) = \frac{β}{B (i, n - i + 1)!} \sum_{j = 0}^{\infty} \sum_{k = 0}^{\infty} \sum_{l = 0}^{\infty} \nabla_{jkl} x^{l}

Indeed, (29) has a supportive role in the calculation of r-th moment OS and hereafter, straightforward computation of (29) leads us to the r-th moment OS of X and it is given as follows

(31)

μ_{OS - M L - I I - E x p}^{r} = \frac{β}{B (i, n - i + 1)!} \sum_{j = 0}^{\infty} \sum_{k = 0}^{\infty} \frac{η_{jk}}{{(θv)}^{r + 1}} Γ (r + 1)

where $η_{jk} = {(- 1)}^{j} (\begin{matrix} i - 1 \\ j \end{matrix}) (\begin{matrix} λ \\ k \end{matrix}) α^{k} {(1 - α)}^{λ - k}, λ = - β (n - i + j + 1) + 1, v = βθ (n - i + j + 1)$

\nabla_{jkl} = {(- 1)}^{j + k} (\begin{matrix} i - 1 \\ j \end{matrix}) (\begin{matrix} λ \\ k \end{matrix}) v^{l} / l!

The CDF of $X_{(i)}$ is given by

F_{(i : n)} (x) = \sum_{r = i}^{n} (\begin{matrix} n \\ r \end{matrix}) {(F (x))}^{r} {(1 - F (x))}^{n - r}, i = 1, 2, 3, \dots, n

(32)

F_{(i : n) - M L - I I - E x p} (x) = \sum_{r = i}^{n} (\begin{matrix} n \\ r \end{matrix}) {(1 - {(\frac{e^{- θx}}{1 - α + {αe}^{- θx}})}^{β})}^{r} {({(\frac{e^{- θx}}{1 - α + {αe}^{- θx}})}^{β})}^{n - r}

Furthermore, the minimum and maximum OS of X follows directly from (28) with i = 1 and i = n, respectively.

Inference

In this section, we estimate the parameters of the ML–II–Exp distribution by following the method of MLE, as this method provides the maximum information about the unknown model parameter. Let $X_{1}, X_{2}, X_{3}, \dots, X_{n}$ be a random sample of size n from the ML–II–Exp distribution, then the likelihood function $L_{M L - I I - E x p} (ψ = α, β, θ) = \prod_{i = 1}^{n} f_{ML - II - Exp} (x_{i}; ψ)$ of X is given by

L_{M L - I I - E x p} (ψ) = \prod_{i = 1}^{n} \frac{(1 - α) β θ e^{- β θ x_{i}}}{{(1 - α + {α e}^{- θ x_{i}})}^{β + 1}}

The log-likelihood function, $l (ψ)$ is given by

(33)

l_{M L - I I - e x p} (ψ) = n (lo g (1 - α) + lo g β + lo g θ) - βθ \sum_{i = 1}^{n} x_{i} - (β + 1) \sum_{i = 1}^{n} lo g (1 - α + {αe}^{- θ x_{i}})

Partial derivatives of (33) w.r.t. $α, θ,$ and $β$ yield, respectively

\frac{\partial l_{M L - I I - E x p} (ψ)}{∂α} = - \frac{n}{1 - α} - (β + 1) \sum_{i = 1}^{n} (\frac{- 1 + e^{- θ x_{i}}}{(1 - α + {αe}^{- θ x_{i}})}) = 0

\frac{\partial l_{M L - I I - E x p} (ψ)}{\partial β} = \frac{n}{β} - θ \sum_{i = 1}^{n} x_{i} - \sum_{i =}^{n} lo g (1 - α + {αe}^{- θ x_{i}}) = 0

\frac{\partial l_{M L - I I - E x p} (ψ)}{∂θ} = \frac{n}{θ} - β \sum_{i = 1}^{n} x_{i} - (β + 1) \sum_{i = 1}^{n} (\frac{- α θ e^{- θ x_{i}}}{(1 - α + {αe}^{- θ x_{i}})}) = 0

The maximum likelihood estimates ( ${\hat{ψ}}_{i} = \hat{α}, \hat{β}, \hat{θ}$ ) of the ML–II–Exp distribution can be obtained by maximizing (33) or by solving the above non-linear equations simultaneously. These non-linear equations, however, do not provide an analytical solution for the MLEs and the optimum value of $α$ , $β$ and $θ$ . Consequently, iterative techniques such as the Newton-Raphson type algorithm are an appropriate choice in the support of MLEs.

Simulation study

In this sub-section, we discuss the performance of MLEs using the following algorithm.

Step 1: A random sample x₁, x₂, x₃ , ..., x_n of sizes n = 25, 50, 100, 200, 300, 400, and 500 are generated from (26).

Step 2: The required results are obtained based on the different combinations of the model parameters placed in S-V ( $α = 0.9, θ = 1.1, β = 0.5$ ), S-VI ( $α = 0.5, θ = 1.2, β = 0.6$ ), and S-VII ( $α = 0.9, θ = 0.2, β = 0.06$ ), S-VIII ( $α = 0.5, θ = 0.09, β = 0.2$ ), S-IX ( $α = 0.1, θ = 0.1, β = 0.2$ ) and S-X ( $α = 0.9, θ = 0.1, β = 0.9$ ).

Step 3: Average MLEs and their corresponding standard errors (SEs) (in parenthesis) are presented in Table 4.

Step 4: Estimated bias, root mean square error (RMSE), variance, and mean values are presented in Table 5.

Step 5: Each sample is replicated N = 500 times.

Step 6: A gradual decrease in SEs, biases, RMSE, variances, means, and MLEs pretty close to the true parameters are observed with increase in the sample sizes.

Step 7: Finally, the estimates present in Tables 4 and 5 help us to specify that the method of maximum likelihood works consistently for the ML–II–Exp distribution.

The in-practice measures for the development of average estimate (AE), SE, bias, and RMSE are given as follows:

$AE (\hat{ψ}) = \frac{1}{N} \sum_{i = 1}^{N} \hat{ψ},$	$SE (\hat{ψ}) = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {({\hat{ψ}}_{i} - \bar{ψ})}^{2}},$	$RMSE (\hat{ψ}) = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {({\hat{ψ}}_{i} - ψ)}^{2}},$
$Bias (\hat{φ}) = \frac{1}{N} \sum_{i = 1}^{N} ({\hat{ψ}}_{i} - ψ),$	$and CI (\hat{ψ}) = \hat{ψ} \pm Z_{\frac{α}{2}} \sqrt{var (\hat{ψ})}$

Four real data applications

In this section, we explore four suitable lifetime datasets to model the ML–II–Exp distribution. These datasets are associated with the engineering sector. The first dataset relates to the study of failure times of 84 windshields for a particular model of aircraft (the unit for measurement is 1000 hours) that was first discussed by Ramos et al.¹⁸ The second dataset relates to the study of service times of 63 aircraft windshields (the unit for measurement is 1000 hours) that was discussed by Tahir et al.¹⁹ The third dataset follows the discussion of the breaking stress of carbon fibers (in Gba) that was initially developed by Nicholas and Padgett²⁰ and finally the fourth one relates to the study of fatigue life of 6061 - T6 aluminum coupons cut parallel to the direction of rolling and oscillated at 18 cycles per second. The data set consists of 101 observations with maximum stress per cycle of 31,000 psi. This dataset was pioneered by Birnbaum and Saunders.²¹ Subsequently, this data was discussed by Shanker et al.,²² after subtracting 65 from each observation. The datasets are given in the underlying data.

The ML–II–Exp distribution is compared with the well-known models (CDF list is mentioned in Table 6). We follow some recognized selection criteria including -Log-likelihood (-LL), Anderson-Darling (A*), Cramer-Von Mises (W*), and Kolmogorov-Smirnov (K-S) test statistics. Some common results of descriptive statistics such as minimum value, 1^st quartile, means, 3^rd quartile, 95% confidence interval, and the maximum value, are tabulated in Table 3. The parameter estimates, standard errors (in parenthesis), and the goodness-of-fit are confirmed in Tables 7-10, respectively. The minimum value of the goodness-of-fit is the criteria of the better fit model that the ML–II–Exp distribution perfectly satisfies. Hence, we affirm that the ML–II–Exp distribution is a better fit than its competitors.

Table 3. Descriptive statistics.

Dataset	Minimum	1^st Quartile	Mean	3^rd Quartile	95% Confidence Interval	Maximum
1	0.040	1.866	2.563	3.376	(2.32,2.80)	4.663
2	0.046	1.122	2.085	2.820	(1.77,2.40)	5.140
3	0.390	1.840	2.621	3.220	(2.42,2.82)	5.560
4	5.000	55.00	68.33	80.25	(63.89,72.77)	147.0

Table 4. Average MLEs and standard errors (in parenthesis).

$n$	S-V Estimates (Standard Errors)			S-VI Estimates (Standard Errors)			S-VII Estimates (Standard Errors)
$n$	$\hat{α}$	$\hat{β}$	$\hat{θ}$	$\hat{α}$	$\hat{β}$	$\hat{θ}$	$\hat{α}$	$\hat{β}$	$\hat{θ}$
25	0.8895 (0.0979)	0.4935 (0.4787)	1.3017 (0.9226)	0.4036 (0.4076)	0.6612 (1.9725)	0.9195 (2.4506)	0.8031 (0.3244)	0.0764 (0.0725)	0.1573 (0.1428)
50	0.9303 (0.0397)	0.6702 (0.3945)	1.1816 (0.4331)	0.5499 (0.2228)	0.5939 (0.5831)	1.4548 (1.2127)	0.9785 (0.0399)	0.0854 (0.6502)	0.2136 (0.1452)
100	0.8997 (0.0408)	0.5616 (0.2859)	1.1073 (0.3819)	0.4599 (0.1847)	0.8215 (0.8319)	1.0985 (0.9456)	0.9696 (0.0535)	0.0597 (0.0481)	0.1929 (0.1445)
200	0.9261 (0.0236)	0.4642 (0.1344)	1.2698 (0.2561)	0.3174 (0.1657)	0.4879 (0.5244)	1.2621 (1.2587)	0.8903 (0.0803)	0.0573 (0.0239)	0.1927 (0.0761)
300	0.9199 (0.0235)	0.3574 (0.0871)	1.3011 (0.2391)	0.5149 (0.1087)	0.3991 (0.2764)	1.6572 (1.0288)	0.9198 (0.0476)	0.0700 (0.0221)	0.1755 (0.0516)
400	0.9098 (0.0224)	0.3528 (0.0772)	1.3579 (0.2252)	0.4738 (0.0886)	0.6453 (0.3588)	1.1241 (0.5378)	0.9415 (0.0391)	0.0552 (0.0183)	0.2169 (0.0678)
500	0.9281 (0.0145)	0.4973 (0.0996)	1.1945 (0.1629)	0.5620 (0.0659)	0.7147 (0.3690)	1.0924 (0.4628)	0.9606 (0.0240)	0.0547 (0.0139)	0.2175 (0.0521)

Table 5. Bias, Root Mean Square Error (RMSE), variance, and mean.

$n$	Est.	S-VIII			S-IX			S-X
$n$	Est.	$α$	$β$	$θ$	$α$	$β$	$θ$	$α$	$β$	$θ$
25	Bias	0.1963	1.5078	1.1519	0.4068	1.2949	1.0298	-0.0001	2.0924	1.4949
	RMSE	0.1879	29.3727	5.4729	0.3575	25.1695	5.7784	0.0282	42.1068	12.3474
	Var.	0.1494	27.0993	4.1459	0.1921	23.4925	4.7180	0.0282	37.7286	10.1124
	Mean	0.6926	1.5978	1.3519	0.5068	1.3949	1.2298	0.8998	2.1924	2.3949
50	Bias	0.1717	0.7480	1.0657	0.3391	0.8762	0.8289	0.0125	0.4615	1.2245
	RMSE	0.1431	11.0845	6.1472	0.2802	14.4779	5.1008	0.0118	6.3089	9.8381
	Var.	0.1135	10.5251	5.0115	0.1652	13.7103	4.4136	0.0116	6.0959	8.3388
	Mean	0.6717	0.8380	1.2657	0.4391	0.9761	1.0289	0.9125	0.5615	2.1245
100	Bias	0.1453	0.6293	0.9190	0.2736	0.6040	0.8120	0.0077	0.1359	0.5918
	RMSE	0.1038	8.1899	6.7048	0.2046	8.8975	7.0779	0.0070	1.5761	3.3296
	Var.	0.0827	7.7939	5.8601	0.1297	8.5325	6.4185	0.0069	1.5575	2.9795
	Mean	0.6453	0.7193	1.1191	0.3736	0.7041	1.0120	0.9077	0.2359	1.4918
200	Bias	0.0929	0.2690	0.4411	0.1972	0.3435	0.5571	0.0047	0.0075	0.2014
	RMSE	0.0673	2.1801	3.3810	0.1351	3.7753	6.3946	0.0045	0.0046	0.4451
	Var.	0.0587	2.1078	3.1864	0.0962	3.6573	6.0841	0.0044	0.0046	0.4046
	Mean	0.5929	0.3590	0.6411	0.2972	0.4435	0.7571	0.9047	0.1075	1.1014
300	Bias	0.0793	0.1599	0.2405	0.1558	0.2281	0.4264	0.0049	0.0027	0.1191
	RMSE	0.0490	0.9116	0.8901	0.0993	1.5787	5.4937	0.0028	0.0018	0.1716
	Var.	0.0427	0.8861	0.8322	0.0751	1.5267	5.3118	0.0028	0.0018	0.1574
	Mean	0.5794	0.2499	0.4405	0.2558	0.3281	0.6264	0.9049	0.1028	1.0191
400	Bias	0.0606	0.1091	0.1665	0.1354	0.2417	0.3892	0.0027	0.0021	0.0736
	RMSE	0.0363	0.4973	0.4553	0.0843	1.3273	4.3197	0.0021	0.0012	0.0872
	Var.	0.0327	0.4854	0.4276	0.0660	1.2688	4.1683	0.0021	0.0012	0.0818
	Mean	0.5606	0.1991	0.3665	0.2354	0.3417	0.5892	0.9027	0.1021	0.9736
500	Bias	0.0506	0.0529	0.1483	0.1131	0.1231	0.3047	0.0029	0.0006	0.0611
	RMSE	0.0295	0.0968	0.4917	0.0642	0.3419	3.8841	0.0016	0.0007	0.0624
	Var.	0.0269	0.0968	0.4697	0.0514	0.3268	3.7912	0.0015	0.0007	0.0586
	Mean	0.5506	0.1429	0.3482	0.2131	0.2231	0.5047	0.9029	0.1007	0.9612

Table 6. List of CDFs for some competitive models.

Abbr.	Model	Parameter/ variable range $0 < x < \infty$	Reference
HL-Exp	$G_{I} (x) = \frac{1 - e^{- αx}}{1 + e^{- αx}}$	$α > 0$	Balakrishnan²³
E-Exp	$G_{II} (x) = {(1 - e^{- αx})}^{β}$	$α, β > 0$	Ahuja and Nash²⁴
Exp	$G_{III} (x) = 1 - e^{- αx}$	$α > 0$	Nadarajah and Kotz²⁵
ETr-Exp	$G_{IV} (x) = 1 - e^{- α (1 - e^{- β}) x}$	$α, β > 0$	El-Alosey²⁶
Logis-Exp	$G_{V} (x) = 1 - \frac{1}{1 + {(e^{αx} - 1)}^{β}}$	$α, β > 0$	Lan and Leemis²⁷
MO-Exp	$G_{VI} (x) = 1 - \frac{α e^{- x}}{1 - (1 - α) e^{- x}}$	$α > 0$	Salah et al.²⁸
NH-Exp	$G_{VII} (x) = 1 - e^{1 - {(1 + αx)}^{β}}$	$α, β > 0$	Nadarajah and Haghighi²⁹
Ts-Exp	$G_{VIII} (x) = (1 + λ) (1 - e^{- αx}) - λ {(1 - e^{- αx})}^{2}$	$α > 0, \|λ\| \leq 1$	Merovci and Puka³⁰
DUS-Exp-Exp	$G_{IX} (x) = \frac{e^{(1 - e^{- αx})} - 1}{e - 1}$	$α > 0$	Kumar et al.¹¹
Alp-Exp	$G_{X} (x) = \frac{α^{(1 - e^{- βx})} - 1}{α - 1}$	$α, β > 0$	Mahdavi and Kundu¹²

Table 7. Parameter estimates and standard errors (in parenthesis) along with goodness-of-fit for the failure time of 84 windshield dataset.

Model	Parameters (Standard Errors)				Goodness-of-fit
Model	$\hat{α}$	$\hat{β}$	$\hat{θ}$	$\hat{λ}$	-LL	W*	A*	K-S
ML-II-Exp	0.8344 (0.1142)	0.9941 (0.0025)	15.3305 (9.570)	-	128.7357	0.1353	0.8537	0.0888
Logis-Exp	0.3946 (0.0193)	3.9081 (0.3917)	-	-	131.3591	0.1064	0.7073	0.0862
MO-Exp	11.2200 (1.9813)	-	-	-	134.2496	0.0479	0.4734	0.0988
Alp-Exp	91.0358 (45.7226)	0.8149 (0.0607)	-	-	135.6613	0.0693	0.6914	0.1191
E-Exp	0.7593 (0.0764)	3.5949 (0.6139)	-	-	141.3958	0.2187	1.7391	0.1215
NH-Exp	0.0062 (0.0073)	44.9430 (52.8504)	-	-	145.4100	0.0613	0.6058	0.2588
Ts-Exp	0.5717 (0.0470)	-	-	−0.9970 (0.0503)	146.8122	0.1818	1.5000	0.1855
HL-Exp	0.5776 (0.0497)	-	-	-	153.6200	0.1000	0.9368	0.2632
Exp-Exp	0.5137 (0.0469)	-	-	-	156.1826	0.1169	1.0577	0.2699
Exp	2.5609 (0.2776)	-	-	-	164.9877	0.1664	1.3972	0.3033
ETr-Exp	0.5431 (-)	1.2678 (-)	-	-	164.9877	0.1664	1.3972	0.3033

Table 8. Parameter estimates and standard errors (in parenthesis) along with goodness-of-fit for the service times of 63 aircraft windshield dataset.

Model	Parameters (Standard Errors)				Goodness-of-fit
Model	$\hat{α}$	$\hat{β}$	$\hat{θ}$	$\hat{λ}$	-LL	W*	A*	K-S
ML-II-Exp	0.7013 (0.4333)	0.9260 (0.0884)	3.4208 (6.852)	-	98.1497	0.0350	0.2350	0.0657
MO-Exp	5.9950 (1.2886)	-	-	-	99.1004	0.0553	0.3428	0.0784
Alp-Exp	28.4623 (22.5320)	0.8771 (0.1036)	-	-	100.3555	0.0975	0.5932	0.1060
NH-Exp	0.0118 (0.0098)	26.4046 (21.4902)	-	-	100.1846	0.0638	0.3902	0.1436
Ts-Exp	0.6790 (0.0734)	-	-	−0.8693 (0.1465)	102.9673	0.1679	1.0249	0.1426
E-Exp	0.6921 (0.0942)	1.8979 (0.3402)	-	-	103.5466	0.2034	1.2315	0.1438
Logis-Exp	0.4922 (0.0379)	2.8590 (0.3664)	-	-	103.5850	0.0488	0.4020	0.0804
HL-Exp	0.6871 (0.0701)	-	-	-	103.8485	0.1201	0.7302	0.1643
Exp-Exp	0.6178 (0.0665)	-	-	-	105.0672	0.1393	0.8460	0.1699
ETr-Exp	0.6989 (7.1557)	1.1588 (22.3832)	-	-	109.2985	0.1861	1.1264	0.2077
Exp	2.0848 (0.2626)	-	-	-	109.2985	0.1861	1.1264	0.2078

Table 9. Parameter estimates and standard errors (in parenthesis) along with goodness-of-fit for the breaking stress of carbon fibers dataset.

Model	Parameters (Standard Errors)				Goodness-of-fit
Model	$\hat{α}$	$\hat{β}$	$\hat{θ}$	$\hat{λ}$	-LL	W*	A*	K-S
ML-II-Exp	1.6982 (0.3168)	0.9869 (0.0056)	0.9675 (0.4008)	-	142.1201	0.0651	0.4142	0.0602
Logis-Exp	0.3850 (0.0148)	4.5316 (0.4141)	-	-	143.3108	0.0668	0.4723	0.0575
E-Exp	1.0133 (0.0875)	7.7926 (1.4973)	-	-	146.1823	0.2267	1.1861	0.1077
Ts-Exp	0.7651 (0.0704)	-	-	−2.0285 (0.1786)	150.0987	-	-	0.1841
MO-Exp	12.0519 (1.9083)	-	-	-	153.6640	0.0647	0.3791	0.1450
Alp-Exp	106.8843 (41.7742)	0.8280 (0.0528)	-	-	153.8885	0.1299	0.6566	0.1431
NH-Exp	0.0058 (0.0034)	48.1147 (28.6443)	-	-	171.4835	0.0691	0.4253	0.2896
H-Log-Exp	0.5709 (0.0451)	-	-	-	181.5941	0.1147	0.5874	0.2904
Exp-Exp	0.5061 (0.0425)	-	-	-	184.8960	0.1271	0.6482	0.2949
Exp	2.6210 (0.2621)	-	-	-	196.3709	0.1493	0.7643	0.3206
ETr-Exp	0.5467 (-)	1.1969 (-)	-	-	196.3709	0.1493	0.7643	0.3206

Table 10. Parameter estimates and standard errors (in parenthesis) along with goodness-of-fit for the fatigue life of 6061 - T6 aluminum coupons dataset.

Model	Parameters (Standard Errors)				Goodness-of-fit
Model	$\hat{α}$	$\hat{β}$	$\hat{θ}$	$\hat{λ}$	-LL	W*	A*	K-S
ML-II-Exp	0.0786 (0.0129)	0.9954 (0.0015)	1.0426 (0.4383)	-	450.4221	0.0313	0.2384	0.0446
Logis-Exp	0.0147 (0.0004)	5.5022 (0.4889)	-	-	450.7880	0.0323	0.2447	0.0442
E-Exp	0.0413 (0.0033)	9.6723 (1.8416)	-	-	463.8919	0.2790	1.6812	0.1162
Ts-Exp	0.0232 (0.0015)			−1.2479 (0.0444)	488.3510	-	-	0.2308
MO-Exp	24668.86 (-)	-	-	-	5831.908	4.8685	22.758	0.9899
Alp-Exp	119.9019 (44.8694)	0.0316 (0.0019)	-	-	473.8240	0.1063	0.6380	0.1863
NH-Exp	0.0025 (0.0003)	4.7152 (0.6756)	-	-	500.5346	0.0910	0.6214	0.3567
H-Log-Exp	0.0221 (0.0017)	-	-	-	506.3137	0.1165	0.7363	0.3365
Exp-Exp	0.0195 (0.0016)	-	-	-	509.9945	0.1308	0.8179	0.3407
Exp	68.31 (6.8289)	-	-	-	522.4349	0.1646	1.0197	0.3667
ETr-Exp	0.02015 (0.0209)	1.2924 (2.7465)			522.4349	0.1646	1.0199	0.3662

Furthermore, for a visual comparison the fitted density and distribution functions, Kaplan-Meier survival and probability-probability (PP) plots, total time on test transform (TTT), and box plots, are presented in Figures 3–6 (a, b, c, d, e, f, g, h), respectively. These plots provide sufficient information about the closest fit to the data. All the numerical results in the subsequent tables are calculated with the assistance of statistical software RStudio-1.2.5033. with its package AdequacyModel. The explored datasets are given in the underlying data.

Figure 3. Failure time of 84 windshields dataset.

Figure 4. Service time of 63 aircraft windshield dataset.

Figure 5. Breaking stress of carbon fibers dataset.

Figure 6. Fatigue life of 6061 - T6 aluminum coupons dataset.

Conclusion

In this article, we introduced and studied a more flexible G class, called the modified Lehmann Type–II (ML–II) G class of distributions along with explicit expressions for the moments, quantile function, and OS. The exponential distribution was used as the baseline distribution for ML–II–G class, known as ML–II–Exp distribution. It was discussed comprehensively, which demonstrated the reversed-J, constant, unimodal, and right-skewed shapes of a density function. The method of MLE along with the simulation was carried out to investigate the performance of the proposed method. The efficiency of the ML–II–G class was evaluated when the most efficient and consistent results of ML–II–Exp distribution competed the well-known models and explored the dominance along with a better fit in four real-life datasets.

We hope that in the future, the proposed class and its sub-models will explore the wider range of applications in diverse areas of applied research and will be considered as a choice against the baseline models.

Data availability

Figshare. four_dataset.csv. DOI: https://doi.org/10.6084/m9.figshare.14518383.v1³¹

This project contains the following extended data:

• Failure Time of 84 Windshield, Service Times of 63 Aircraft Windshield, Breaking Stress of Carbon Fibers and Fatigue Life of 6061 - T6 Aluminum Coupons used for the article titled “A New Modified Lehmann Type – II G Class of Distributions: Exponential Distribution with Theory, Simulation, and Applications to Engineering Sector”
• Ramos et al. dataset
• Tahir et al. dataset
• Nicholas and Padgett dataset
• Birnbaum and Saunders dataset

Data are available under the terms of the Creative Commons Zero “No rights reserved” data waiver (CC BY 4.0 Public domain dedication).

Acknowledgments

The authors are grateful to the Editor-in-Chief and anonymous referrers for their constructive comments and valuable suggestions which certainly improved the quality of the paper. We dedicate this work to our mentor Professor Dr. Munir Ahmad (Late) who was the Founding President of the Islamic Society of Statistical Sciences (ISOSS), Editor-in-Chief of Pakistan Journal Statistics, and the paramedic staff that are scarifying their lives against the COVID-19 pandemic war.

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4. Cordeiro GM, De Castro M: A new family of generalized distributions. J. Stat. Comput. Simul. 2011; 81: 883–893. Publisher Full Text
5. Ristic MM, Balakrishnan N: The gamma-exponentiated exponential distribution. J. Stat. Comput. Simul. 2012; 82: 1191–1206. Publisher Full Text
6. Cordeiro GM, Ortega EMM, da-Cunha DCC: The exponentiated generalized class of distributions. Data Sci. J. 2013; 11: 777–803. Publisher Full Text
7. Alzaatreh A, Lee C, Famoye F: A new method for generating families of continuous distributions. Metron. 2013; 71: 63–79. Publisher Full Text
8. Bourguignon M, Silva RB, Cordeiro GM: The Weibull-G family of probability distributions. Data Sci. J. 2014; 12: 53–68. Publisher Full Text
9. Alizadeh M, Cordeiro GM, de-Brito E, et al.: The beta Marshall-Olkin family of distributions. IJSDA. 2015; 2: Art 4. Publisher Full Text
10. Tahir MH, Cordeiro GM, Alzaatreh A, et al.: The Logistic-X family of distributions and its applications. Commun. Stat. Theory Methods. 2016; 45: 732–7349. Publisher Full Text
11. Kumar D, Singh U, Singh SK: A Method of Proposing New Distribution and its Application to Bladder Cancer Patients Data. JSAPL. 2015; 2: 235–245.
12. Mahdavi A, Kundu D: A new method for generating distributions with an application to exponential distribution. Commun. Stat. Theory Methods. 2017; 46: 6543–6557. Publisher Full Text
13. Elbatal I, Ahmad Z, Elgarhy M, et al.: A new alpha power transformed family of distributions: properties and applications to the Weibull model. J. Nonlinear Sci. Appl. 2019; 12: 1–20. Publisher Full Text
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¹ School of Computing, University of Eastern Finland, Kuopio, Northern Europe, 70211, Finland
² Department of Mathematics and Statistics, University of Agriculture, Faisalabad, Faisalabad, Pakistan, Pakistan
³ Department of Mathematics and Statistics, University of Agriculture, Faisalabad, Pakistan, University of Agriculture, Faisalabad, Faisalabad, Faisalabad, Pakistan
⁴ Department of Mathematics and Statistics, University of Agriculture, Faisalabad, Faisalabad, Faisalabad, Pakistan

Oluwafemi Samson Balogun
Roles: Funding Acquisition, Supervision, Writing – Original Draft Preparation, Writing – Review & Editing

Muhammad Zeshan Arshad
Roles: Conceptualization, Data Curation, Methodology, Software, Writing – Original Draft Preparation, Writing – Review & Editing

Muhammad Zafar Iqbal
Roles: Conceptualization, Supervision, Writing – Original Draft Preparation, Writing – Review & Editing

Madiha Ghamkhar
Roles: Methodology, Supervision, Writing – Original Draft Preparation, Writing – Review & Editing

Competing interests

No competing interests were disclosed.

Grant information

The author(s) declared that no grants were involved in supporting this work.

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Balogun OS, Arshad MZ, Iqbal MZ and Ghamkhar M. A new modified Lehmann type – II G class of distributions: exponential distribution with theory, simulation, and applications to engineering sector [version 1; peer review: 1 approved with reservations, 1 not approved]. F1000Research 2021, 10:483 (https://doi.org/10.12688/f1000research.52494.1)

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Reviewer Report 22 Feb 2022

Nofiu Idowu Badmus, Department of Mathematics, University of Lagos, Akoka, Nigeria

Approved with Reservations

https://doi.org/10.5256/f1000research.55779.r120000

My comments on the manuscript are as follows:

Generally, the paper is on non-normal data (i.e skewed data) as one of the challenges in Statistics in which several authors in literature have developed many suitable distributions/models to ... Continue reading

My comments on the manuscript are as follows:

Generally, the paper is on non-normal data (i.e skewed data) as one of the challenges in Statistics in which several authors in literature have developed many suitable distributions/models to capture the excessive skewness and kurtosis of the data. Although, the authors of this paper based their work on a part/segment in parametric distribution using G-model as the basic. They discussed and explored numerous G-models in literature to support their research. Properties of the ML-II G model are presented. The model is tested using both simulation and real-life phenomena. The results show that their model supersedes other models considered.

The title should be “A new modified Lehmann type – II G class of distributions: Theory, Simulation, and Applications”.
The abstract contains the problem, methods used, findings, and result which is good and ok.
In the introduction, the CDFs of cited papers are not necessary to appear in the context except the CDF and PDF of the work they followed in their work. Therefore, the introduction should dwell more on the paper(s) they used. This implies that there is a need for them to upgrade/add more to the introduction.
A new G class is developed which is known as a new modified Lehmann Type-II-(ML-II) G class of distributions. Why a new G class? Since we can see many G classes in the literature. What makes your G class better than others?
Under Order Statistics, after the second expression where you wrote “one may determine it by integrating (5–6)” – with respect to? This is missing.
Under limiting behavior, it should be: “Here, we obtain the limiting behavior of PDF, CDF…”.
In Table 2, it should be that some chosen parameters with several values are presented.
In the paper, it was discussed that lessing/setting one or two parameter(s) in the model led to some special cases emanating from the proposed model, One of those special cases is Lehmann Type II. Due to the fact that the author's work targeted Lehmann Type II they need to include/cite some Lehmann Type II research works from the literature review. The following articles can serve as a guide for authors:
- N. I. Badmus, T. A. Bamiduro and S. G. Ogunobi (2014). Lehmann Type II Weighted Weibull Distribution¹.
- Badmus, N. Idowu and Bamiduro, T. Adebayo. (2015). Log-Lehmann Type II Weighted Weibull (LLWW) Regression Model: Theory and Method².
- Badmus, N. I., Bamiduro, T. A., Akingbade, A. A and Rauf-Animasaun, M. A (2016). Estimation of Parameters in Generalized Modified Weighted Weibull Distribution³.
- Badmus, N. I., Alakija, T. O.and Olanrewaju, G. O. (2017). The Beta-Modified Weighted Rayleigh Distribution: Application to Virulent Tubercle Disease⁴.
Authors may source for other ones online.

The manuscript can be accepted subject to the corrections listed above. Thanks.

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Yes
Are all the source data underlying the results available to ensure full reproducibility?

Partly
Are the conclusions drawn adequately supported by the results?

Yes

References

1. N. I, T. A, S. G: Lehmann Type II weighted Weibull distribution. International Journal of Physical Sciences. 2014; 9 (4): 71-78 Publisher Full Text
2. Badmus NI, Bamiduro TA: Log-Lehmann Type II Weighted Weibull (LLWW) Regression Model: Theory and Method. International Journal of Advanced Research in Statistics, Management and Finance. 2015; 3 (1): 59-70 Reference Source
3. Badmus NI, Bamiduro TA, Akingbade AA, Rauf-Animasaun MA: Estimation of Parameters in Generalized Modified Weighted Weibull Distribution. International Journal of Science.
4. Badmus NI, Alakija TO, Olanrewaju GO: The Beta-Modified Weighted Rayleigh Distribution: Application to Virulent Tubercle Disease. International Journal of Mathematical Sciences and Optimization: Theory and Applications. 2017. 154-167 Reference Source

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Inferential statistics, Computational Statistics, Distribution Theory

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

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Reviewer Report 22 Dec 2021

Frank Gomes-Silva, Postgraduate Program of Biometrics and Applied Statistics, Federal Rural University of Pernambuco, Recife, Brazil

Not Approved

https://doi.org/10.5256/f1000research.55779.r102485

The authors propose a new probabilistic class. Then, starting from this class, they propose a new distribution called ML-II-Exp. Present mathematical properties, simulation, and application for this new distribution. Here are some notes:

The introduction can

The authors propose a new probabilistic class. Then, starting from this class, they propose a new distribution called ML-II-Exp. Present mathematical properties, simulation, and application for this new distribution. Here are some notes:

The introduction can be improved. More motivation is needed to introduce a new generator. More references should be added, for example:
- Parameter induction in continuous univariate distributions: Well established G families, by Tahir and Nadarajah¹; Compounding of distributions: a survey and new generalized classes, by Tahir and Cordeiro², among others. These two references are important in giving readers an overview of the area.
- The authors should have explored Lehmann's distributions - see, for example, My Musings on a Pioneering Work by Erich Lehmann and Its Rediscoveries on Some Families of Distributions, by Narayanaswamy Balakrishnan³. Professor Balakrishnan gives an explanation of Lehmann's contribution to the theory of distributions.
- Another interesting article: Method for generating distributions and classes of probability distributions: the univariate case⁴, where authors generalize (and introduce new generators) to the vast majority of classes; class T-X is a particular case of this new method. It is worth noting that for the first time new generators are proposed from multiple baselines.
What do you mean by DUS (in the introduction)?
On page 3 (just after Eq. 1), appears \alpha <1 and then \alpha > 0. What does that mean?
Identifiable problems are inconvenient when estimating the parameters of a model. A section to present proof of the model's (or even class') identifiability would be of interest. The following article presents an interesting section for identifiable issues: Normal-G Class of Probability Distributions: Properties and Applications⁵.
Equation (22) appears in an unconventional form. The same goes for writing Equations (23) and (28). Writing in a conventional way!
At the end of the equations, put a period. Equations are part of the text.
In the simulation section, some discussions appear as steps. This can be confused with steps in an algorithm (which is usually placed in this type of section). The authors have presented an algorithm explaining a simulation. For example, what was done with the simulations that didn't converge?
The simulation can be improved. Note that the mean (for the \theta parameter) is not close to the true value of the parameter.
Standard errors play an important role in terms of accuracy. Standard errors are high in the first application for Alp-Exp and NH-Exp models. Where are the standard errors for the Etr-Exp model (in the first application)?
The W* and A* statistics are important measures of goodness-of-fit in distribution theory. But the proposed model (in the first application) loses to several others based on these indicators.
In Table 8, the standard error of the Etr-Exp model is too high.
In Table 9, the estimate for the parameter \lambda is inconsistent since by definition \lambda must be between -1 and 1.
In Table 9, where are the standard errors for the Etr-Exp model?
Table 10 gives an estimate (for \lambda) of -1.2479. This estimate is inconsistent since \lambda must be between -1 and 1. For this same model (in this same line), the W* and A* statistics do not appear. Again, a very high standard error appears for the Etr-Exp model.

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Partly
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Partly

References

1. Tahir MH, Nadarajah S: Parameter induction in continuous univariate distributions: Well-established G families.An Acad Bras Cienc. 87 (2): 539-68 PubMed Abstract | Publisher Full Text
2. Tahir M, Cordeiro G: Compounding of distributions: a survey and new generalized classes. Journal of Statistical Distributions and Applications. 2016; 3 (1). Publisher Full Text
3. Balakrishnan N: My musings on a pioneering work of Erich Lehmann and its rediscoveries on some families of distributions. Communications in Statistics - Theory and Methods. 2021. 1-8 Publisher Full Text
4. Chaves Rego L, Rosa de Oliveira W, Gomes-Silva F, Ramos de Brito C: Method for generating distributions and classes of probability distributions: the univariate case. Hacettepe Journal of Mathematics and Statistics. 2018; 48 (2). Publisher Full Text
5. Silveira F, Gomes-Silva F, Brito C, Cunha-Filho M, et al.: Normal-G Class of Probability Distributions: Properties and Applications. Symmetry. 2019; 11 (11). Publisher Full Text

Competing Interests: No competing interests were disclosed.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

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VERSION 1 PUBLISHED 17 Jun 2021

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Version 1 17 Jun 21	read	read

Frank Gomes-Silva, Federal Rural University of Pernambuco, Recife, Brazil
Nofiu Idowu Badmus, University of Lagos, Akoka, Nigeria

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Reviewer Report

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22 Feb 2022 | for Version 1

Nofiu Idowu Badmus, Department of Mathematics, University of Lagos, Akoka, Nigeria

18 Views Cite this report Responses(0)

Approved With Reservations

My comments on the manuscript are as follows:

Generally, the paper is on non-normal data (i.e skewed data) as one of the challenges in Statistics in which several authors in literature have developed many suitable distributions/models to capture the excessive skewness and kurtosis of the data. Although, the authors of this paper based their work on a part/segment in parametric distribution using G-model as the basic. They discussed and explored numerous G-models in literature to support their research. Properties of the ML-II G model are presented. The model is tested using both simulation and real-life phenomena. The results show that their model supersedes other models considered.

The title should be “A new modified Lehmann type – II G class of distributions: Theory, Simulation, and Applications”.
The abstract contains the problem, methods used, findings, and result which is good and ok.
In the introduction, the CDFs of cited papers are not necessary to appear in the context except the CDF and PDF of the work they followed in their work. Therefore, the introduction should dwell more on the paper(s) they used. This implies that there is a need for them to upgrade/add more to the introduction.
A new G class is developed which is known as a new modified Lehmann Type-II-(ML-II) G class of distributions. Why a new G class? Since we can see many G classes in the literature. What makes your G class better than others?
Under Order Statistics, after the second expression where you wrote “one may determine it by integrating (5–6)” – with respect to? This is missing.
Under limiting behavior, it should be: “Here, we obtain the limiting behavior of PDF, CDF…”.
In Table 2, it should be that some chosen parameters with several values are presented.
In the paper, it was discussed that lessing/setting one or two parameter(s) in the model led to some special cases emanating from the proposed model, One of those special cases is Lehmann Type II. Due to the fact that the author's work targeted Lehmann Type II they need to include/cite some Lehmann Type II research works from the literature review. The following articles can serve as a guide for authors:
- N. I. Badmus, T. A. Bamiduro and S. G. Ogunobi (2014). Lehmann Type II Weighted Weibull Distribution¹.
- Badmus, N. Idowu and Bamiduro, T. Adebayo. (2015). Log-Lehmann Type II Weighted Weibull (LLWW) Regression Model: Theory and Method².
- Badmus, N. I., Bamiduro, T. A., Akingbade, A. A and Rauf-Animasaun, M. A (2016). Estimation of Parameters in Generalized Modified Weighted Weibull Distribution³.
- Badmus, N. I., Alakija, T. O.and Olanrewaju, G. O. (2017). The Beta-Modified Weighted Rayleigh Distribution: Application to Virulent Tubercle Disease⁴.
Authors may source for other ones online.

The manuscript can be accepted subject to the corrections listed above. Thanks.

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Yes
Are all the source data underlying the results available to ensure full reproducibility?

Partly
Are the conclusions drawn adequately supported by the results?

Yes

References

1. N. I, T. A, S. G: Lehmann Type II weighted Weibull distribution. International Journal of Physical Sciences. 2014; 9 (4): 71-78 Publisher Full Text
2. Badmus NI, Bamiduro TA: Log-Lehmann Type II Weighted Weibull (LLWW) Regression Model: Theory and Method. International Journal of Advanced Research in Statistics, Management and Finance. 2015; 3 (1): 59-70 Reference Source
3. Badmus NI, Bamiduro TA, Akingbade AA, Rauf-Animasaun MA: Estimation of Parameters in Generalized Modified Weighted Weibull Distribution. International Journal of Science.
4. Badmus NI, Alakija TO, Olanrewaju GO: The Beta-Modified Weighted Rayleigh Distribution: Application to Virulent Tubercle Disease. International Journal of Mathematical Sciences and Optimization: Theory and Applications. 2017. 154-167 Reference Source

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Inferential statistics, Computational Statistics, Distribution Theory

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

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Reviewer Report

22 Views

22 Dec 2021 | for Version 1

Frank Gomes-Silva, Postgraduate Program of Biometrics and Applied Statistics, Federal Rural University of Pernambuco, Recife, Brazil

22 Views Cite this report Responses(0)

Not Approved

The authors propose a new probabilistic class. Then, starting from this class, they propose a new distribution called ML-II-Exp. Present mathematical properties, simulation, and application for this new distribution. Here are some notes:

The introduction can be improved. More motivation is needed to introduce a new generator. More references should be added, for example:
- Parameter induction in continuous univariate distributions: Well established G families, by Tahir and Nadarajah¹; Compounding of distributions: a survey and new generalized classes, by Tahir and Cordeiro², among others. These two references are important in giving readers an overview of the area.
- The authors should have explored Lehmann's distributions - see, for example, My Musings on a Pioneering Work by Erich Lehmann and Its Rediscoveries on Some Families of Distributions, by Narayanaswamy Balakrishnan³. Professor Balakrishnan gives an explanation of Lehmann's contribution to the theory of distributions.
- Another interesting article: Method for generating distributions and classes of probability distributions: the univariate case⁴, where authors generalize (and introduce new generators) to the vast majority of classes; class T-X is a particular case of this new method. It is worth noting that for the first time new generators are proposed from multiple baselines.
What do you mean by DUS (in the introduction)?
On page 3 (just after Eq. 1), appears \alpha <1 and then \alpha > 0. What does that mean?
Identifiable problems are inconvenient when estimating the parameters of a model. A section to present proof of the model's (or even class') identifiability would be of interest. The following article presents an interesting section for identifiable issues: Normal-G Class of Probability Distributions: Properties and Applications⁵.
Equation (22) appears in an unconventional form. The same goes for writing Equations (23) and (28). Writing in a conventional way!
At the end of the equations, put a period. Equations are part of the text.
In the simulation section, some discussions appear as steps. This can be confused with steps in an algorithm (which is usually placed in this type of section). The authors have presented an algorithm explaining a simulation. For example, what was done with the simulations that didn't converge?
The simulation can be improved. Note that the mean (for the \theta parameter) is not close to the true value of the parameter.
Standard errors play an important role in terms of accuracy. Standard errors are high in the first application for Alp-Exp and NH-Exp models. Where are the standard errors for the Etr-Exp model (in the first application)?
The W* and A* statistics are important measures of goodness-of-fit in distribution theory. But the proposed model (in the first application) loses to several others based on these indicators.
In Table 8, the standard error of the Etr-Exp model is too high.
In Table 9, the estimate for the parameter \lambda is inconsistent since by definition \lambda must be between -1 and 1.
In Table 9, where are the standard errors for the Etr-Exp model?
Table 10 gives an estimate (for \lambda) of -1.2479. This estimate is inconsistent since \lambda must be between -1 and 1. For this same model (in this same line), the W* and A* statistics do not appear. Again, a very high standard error appears for the Etr-Exp model.

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Partly
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Partly

References

1. Tahir MH, Nadarajah S: Parameter induction in continuous univariate distributions: Well-established G families.An Acad Bras Cienc. 87 (2): 539-68 PubMed Abstract | Publisher Full Text
2. Tahir M, Cordeiro G: Compounding of distributions: a survey and new generalized classes. Journal of Statistical Distributions and Applications. 2016; 3 (1). Publisher Full Text
3. Balakrishnan N: My musings on a pioneering work of Erich Lehmann and its rediscoveries on some families of distributions. Communications in Statistics - Theory and Methods. 2021. 1-8 Publisher Full Text
4. Chaves Rego L, Rosa de Oliveira W, Gomes-Silva F, Ramos de Brito C: Method for generating distributions and classes of probability distributions: the univariate case. Hacettepe Journal of Mathematics and Statistics. 2018; 48 (2). Publisher Full Text
5. Silveira F, Gomes-Silva F, Brito C, Cunha-Filho M, et al.: Normal-G Class of Probability Distributions: Properties and Applications. Symmetry. 2019; 11 (11). Publisher Full Text

Competing Interests

No competing interests were disclosed.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

Respond to this report

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