Mathematical and Statistical Properties for a New IDNAI Distribution[a]

Abbas Najim Salman; Iden Hassan

doi:10.12688/f1000research.172616.1

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Research Article

Mathematical and Statistical Properties for a New IDNAI Distribution[a]

[version 1; peer review: awaiting peer review]

Abbas Najim Salman ¹, Iden Hassan²

PUBLISHED 03 Feb 2026

Author details Author details

¹ Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, Baghdad Governorate, 10053, Iraq
² Mathematics, University of Baghdad Al-Jaderyia Campus College of Science, Baghdad, Baghdad Governorate, 10070, Iraq

Abbas Najim Salman
Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing

Iden Hassan
Roles: Conceptualization, Formal Analysis, Methodology, Project Administration, Supervision

OPEN PEER REVIEW

REVIEWER STATUS AWAITING PEER REVIEW

This article is included in the Fallujah Multidisciplinary Science and Innovation gateway.

Abstract

Over the past decade, the statistical and reliability literature has seen an enormous rise in the number of various probability distributions. Many of them are designed to integrate, enhance, or add to traditional models. This rise shows that people are becoming more aware of empirical integrity, flexibility, and adaptability, especially when it comes to modeling income distributions, dependability difficulties, and longevity statistics. In this paper, more general and flexible distributions are provided by employing a technique that generates the T−R {Y} family distributions as well as a brief overview for this kind of family. This technique may be applied to agree with certain data distribution types, including very left, right, thin or heavy tailed distribution, or to develop new distributions that are very applicable and flexible. In this context, produces a distinct member of a new sub-family called the IDNAI distribution, which is referred to the Lomax–Rayleigh {Exponential} distribution. The most important statistical and mathematical characteristics of the new family distribution are discussed. In order to estimate the parameters of the new class distribution, two estimation techniques will be additionally presented: the maximum likelihood method (MLE) and the least square method (LS). In order to analyze the effectiveness of suggested estimation techniques, simulation tools have been constructed. The results of the comparison indicated that the maximum likelihood method performed better than the least squares approach in terms of bias and mean squared error criteria. However, the application of a real dataset showed that, depending on several kinds of information criteria, the new class distribution fit the data more effectively than the specific classical and moderate distributions.

Keywords

Transformed-transformer family, Quantile function, Statistical properties, Entropies, Estimation Methods, Monte Carlo simulation and Fitting real data.

Corresponding author: Abbas Najim Salman

Competing interests: No competing interests were disclosed.

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

Copyright: © 2026 Salman AN and Hassan I. 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: Salman AN and Hassan I. Mathematical and Statistical Properties for a New IDNAI Distribution[a] [version 1; peer review: awaiting peer review]. F1000Research 2026, 15:177 (https://doi.org/10.12688/f1000research.172616.1) First published: 03 Feb 2026, 15:177 (https://doi.org/10.12688/f1000research.172616.1) Latest published: 03 Feb 2026, 15:177 (https://doi.org/10.12688/f1000research.172616.1)

1-Introduction

Statistical modeling is significant compared to other aspects of life, especially in probability theory, which is commonly used to identify variation and make inferences from observable data. Several new generalized distributions have been created and have developed in significance during the past few decades. Developing these distributions was based mostly on the notion that they had more parameters. The data in every part of life is being modeled by a number of extensions of conventional statistical models. Flexible statistical models are still needed in many practical domains to better fit the data, and they usually have significant levels of skewness and kurtosis. Probability theory has developed more quickly because it may be applied to many different aspects of life (such as engineering sciences, survival, and actuarial sciences). Every new breakthrough broadens the scope of the probability theory’s effect.

Most new distributions are created using modern techniques, even if researchers are currently looking into the techniques used for creating new distributions since 1980; see for example Refs. 1–10.

The transformed-transformer technique, created by Alzaatreh et al.,¹¹ will be employed in this study among the more current approaches. In order to obtain the distribution function of a new class of distributions, they transformed additional variable through (-∞, ∞) using the density and distribution functions of a r.v. X with $r (t)$ and $R (t)$ , respectively.

(1)

G (x) = \int_{c}^{Z (F (x))} r (t) dt = R {Z (F (X))}

Where Z(F(X)) belong to [c,d] is monotone increasing, $Z ́ (F (X))$ less than infinity, $lim_{x \to - \infty} Z (F (X)) = c$ and $lim_{x \to \infty} Z (F (X)) = d$ , with the following matching pdf

(2)

g (x) = \overset{´}{Z} (F (X)) r {Z (F (X))}

The transformed-transformer technique has been used in a number of studies, see for instance, Refs. 12, 13, and 14.

By using quantile function of additional r. v. Y, in relations of $F (x)$ instead of the function, $Z (F (x)),$ a novel family of distributions was created by Aljarrah et al.¹⁵; which gave the $T - X {Y}$ distribution class.

(3)

G (x) = \int_{c}^{Q_{Y} (F (x))} r (t) dt = R {Q_{Y} (F (X))}

When F(x) equal to λ and p(y) greater than 0 for all y in neighborhood of Q_y (λ), such that 0 < λ < 1, this implies that $\overset{´}{Q}$ _Y (λ) less than infinity and it is equivalent to P [Q_y (λ)]⁻¹.

Therefore, the related pdf of G(x) is

(4)

g (x) = \frac{f (x)}{p {Q_{Y} (F (x))}} . r [Q_{Y} (F (x))]

Assume T, R, and Y denote random variables with quantile functions QT(p), QR(p), and QY(p), respectively; suppose that their cdf^s are F_T(x), F_R(x) and F_Y(x), and their density functions, which is represented as f_T(x), f_R(x) and f_Y(x) respectively, with T, Y∈[c, d], and R∈[e, f] for -∞ < c < d < ∞ and -∞ < e < f < ∞.

Afterward, the distribution function of the novel class is provided as

(5)

F_{X} (x) = \int_{c}^{Q_{Y} (F_{R} (x))} f_{T} (t) dt = F_{T} {Q_{Y} (F_{R} (x))}

With the associated pdf as

(6)

f_{X} (x) = f_{R} (x) . {\overset{´}{Q}}_{Y} (F_{R} (x)) . f_{T} {Q_{Y} (F_{R} (x))} = f_{R} (x) f_{T} (Q_{Y} (F_{R} (x))) / f_{Y} (Q_{Y} (F_{R} (x))) .

In 2021, Ibeh et al.¹⁶; introduce new Weibull -Exponential {Rayleigh} distribution based the idea of T-X {Y} family distribution.

A contemporary statistical technique that provides an adaptable framework for producing novel probability distributions beyond the constraints of traditional models is the Transformed-Transformer (T-X) Family method, which is based on the quantile function. By fusing the quantile function of a generator distribution with the cumulative distribution function (CDF) of a base distribution, this method allows for the creation of personalized statistical models with exact control over shape properties like sleekness, kurtosis, and tail behavior. Its importance stems from its capacity to precisely depict non-standard data, offer direct control over the form of the resulting distribution, and support risk analysis and simulation. This approach is a potent tool for academics and applied statisticians since it has proven successful in a variety of domains, such as industrial engineering, medical statistics, financial modeling, and environmental sciences.

This work aims to present and examine the key mathematical and statistical features of a novel sub-family known as the IDNAI distribution, expressive of the Lomax–Rayleigh {Exponential} distribution.

The following is the structure of the other sections in this article. The Lomax-Rayleigh {Exponential} distribution, which we shall refer to as the IDNAI distribution, is created in Subsection 2-1. Most of the distribution’s characteristics are analyzed and shown in Subsection 2-2. The suggested estimation techniques for the parameters of the IDNAI distribution are discussed in Subsection 2-3. Subsection 2-4 presents the comparison of the estimators with the Monty Carlo simulation study. In Subsection 2-5, the data set’s applicability and fitting real-world dataset are demonstrated. The results are presented in Section 3. Section 4 provides the discussion of the results. Section 5 presents the summary and conclusions of the paper.

2- Methods

2-1 Creating a novel IDNAI distribution

Assume a random variable T with the Lomax distribution, including parameter (λ), in addition to cdf, pdf and quantile function which are provided as

(7)

F_{T} (x; λ) = 1 - {(1 + X)}^{- λ}

(8)

f_{T} (x; λ) = λ {(1 + X)}^{- (λ + 1)}

(9)

Q_{T} (p) = [{(1 - P)}^{- 1 / λ} - 1]

Where λ > 0, x > 0, 0 < P < 1.

In the same way, let a random variable R follow the Rayleigh distribution with parameter α as well as the distribution function, probability density function and quantile function of R, are respectively given as

(10)

F_{R} (x; α) = 1 - exp (- {αx}^{2})

(11)

f_{R} (x; α) = 2α \times exp (- {αx}^{2})

(12)

Q_{R} (p) = {(ln (1 - p)) / α}^{1 / 2}

Where α > 0, x > 0, 0 < P < 1.

Moreover, suppose that a random variable Y follows the Exponential distribution with the parameter β, then the cdf, pdf and quantile function are provided as

(13)

F_{Y} (x) = 1 - exp (- x / β)

(14)

f_{Y} (x) = (1 / β) exp (- x / β)

(15)

Q_{Y} (p) = - β log (1 - p)

Where, β > 0, x > 0, 0 < P < 1.

Equation (10), together with Equation (15), gives us

Q_{Y} (F_{R} (x)) = - β log (1 - (1 - exp (- {αx}^{2}))) = {αβX}^{2}

Consequently, based on Equation (7) we obtain

F_{T} (Q_{Y} (F_{R} (x))) = 1 - {(1 + {αβX}^{2})}^{- λ}

Hence, the cdf of the Lomax-Rayleigh {exponential} distribution or IDNAI distribution is s provided by

(16)

F_{IDNAI} (x; α, β, λ) = 1 - {(1 + {αβX}^{2})}^{- λ} for x > 0 and α, β, λ > 0

Henceforward, from (8) we get

(17)

f_{T} (Q_{Y} (F_{R} (x))) = λ {(1 + {αβX}^{2})}^{- (λ + 1)}

Subsequently, from Equation (14), we obtained

(18)

\begin{array}{c} F_{Y} (Q_{Y} (F_{R} (x))) = (1 / β) exp (- αβ X^{2} / β) \\ F_{Y} (Q_{Y} (F_{R} (x))) = (1 / β) exp (- α X^{2}) \end{array}

When Equations (11), (17) and (18) are substituted into Equation (6), the matching IDNAI distribution pdf is obtained as

f_{X} (x) = f_{IDNAI} (x) = [2α \times exp (- {αx}^{2})] [λ {(1 + {αβx}^{2})}^{- (λ + 1)}] / [(1 / β) exp (- {αx}^{2})],

Accordingly,

(19)

f_{IDNAI} (x; α, β, λ) = (2αβλ) \times {(1 + {αβX}^{2})}^{- (λ + 1)} for x > 0 and α, β, λ > 0

We deduce that the new class IDNAI distribution’s cdf and pdf with three parameters (α,β,λ) are provided by Equations (16) and (19), respectively.

Additionally, based on Equation (16), the IDNAI distribution’s reliability function will be

(20)

R_{IDNAI} (x; α, β, λ) = {(1 + {αβx}^{2})}^{- λ} for x > 0 and α, β, λ > 0

2-2 Properties of the IDNAI distribution

Furthermost, some fundamental statistical and mathematical characteristics of the IDNAI distribution are presented in this section, as follows:

2-2-1 Additional probability functions of IDNAI distribution

In addition to the three previously discussed functions in Equations (16), (19) and (20), we introduce the majority of probability functions in this section.

a- Hazard function of IDNAI distribution h_IDNAI(x;α,β,λ)
(21)
$\begin{matrix} h_{IDNAI} (x; α, β, λ) = f (x; α, β, λ) / R (x; α, β, λ) for x > 0 and α, β, λ > 0 \\ h_{IDNAI} (x; α, β, λ) = (2 αβλ) (x) / (1 + αβ x^{2}) for x > 0 and α, β, λ > 0 \end{matrix}$
b- Cumulative hazard function of IDNAI distribution H_IDNAI (x; α,β,λ)
(22)
$H_{IDNAI} (x; α, β, λ) = λ Ln (1 + {αβx}^{2}) for x > 0 and α, β, λ > 0$
c- Reverse Hazard Function of IDNAI distribution Ф_IDNAI (x; α,β,λ)
(23)
$\begin{matrix} Ф_{IDNAI} (x; α, β, λ) = f (x; α, β, λ) / F (x; α, β, λ) \\ Ф_{IDNAI} (x; α, β, λ) = [(2 αβλ) \times {(1 + αβ x^{2})}^{- (λ + 1)}] / [1 - {(1 + αβ x^{2})}^{- λ}], for x > 0 and α, β, λ > 0 \end{matrix}$
d- Odd Function of IDNAI distribution ʘ _IDNAI (x; α,β,λ)
(24)
$\begin{matrix} Θ_{IDNAI} (x; α, β, λ) = F (x; α, β, λ) / R (x; α, β, λ) \\ Θ_{IDNAI} (x; α, β, λ) = {(1 + αβ x^{2})}^{λ} - 1 for x > 0 and α, β, λ > 0 \end{matrix}$

2-2-2 The quantile function of IDNAI distribution

The q^th quantile function of the IDNAI distribution may be found as

(25)

\begin{matrix} F (x) = 1 - {(1 + αβ X^{2})}^{- λ} = q; (0 < q < 1) \\ x_{q} = {[{(1 - q)}^{- 1 / λ} - 1] / αβ}^{1 / 2} \end{matrix}

Consequently, the median (x_med) of the IDNAI distribution may be found by setting q = 0.5 as

(26)

x_{med} = {[2^{1 / λ} - 1] / αβ}^{1 / 2}

2-2-3 The mode of IDNAI distribution

The mode of the IDNAI distribution can be calculated as follows

(27)

\begin{matrix} f (x; α, β, λ) = (2 αβλ) \times {(1 + αβ X^{2})}^{- (λ + 1)} \\ Ln f (x; α, β, λ) = Ln (2 αβλ) + LnX - (λ + 1) Ln (1 + αβ X^{2}) \\ \frac{\partial Lnf (x; α, β, λ)}{\partial x} = 0 \to \frac{1}{x} - (λ + 1) \frac{(2 αβx)}{(1 + αβ x^{2})} = 0 \\ \frac{(λ + 1) (2 αβ x^{2})}{(1 + αβ x^{2})} = 1 \to (λ + 1) \frac{2 αβ}{\frac{1}{x^{2}} + αβ} = 1 \to \frac{1}{x^{2}} = (2 αβ) (λ + 1) - αβ \\ x^{2} = \frac{1}{(2 αβ) (λ + 1) - αβ} \\ x_{mode} = {[αβ (2 λ + 1]}^{- 1 / 2} \end{matrix}

2-2-4 Limit of the pdf and cdf of IDNAI distribution

a- The Limit of pdf $f (x; α; β; λ)$

1-

(28)

\begin{matrix} {lim}_{x \to 0} f (x; α; β; λ) = lim_{x \to 0} (2 αβλ) \times {(1 + αβ X^{2})}^{- (λ + 1)} = (2 αβλ) lim_{x \to 0} x lim_{x \to 0} {(1 + αβ X^{2})}^{- (λ + 1)} \\ {lim}_{x \to 0} f (x; α; β; λ) = (2 αβλ) * 0^{*} 1 \\ \to {lim}_{x \to 0} f (x; α; β; λ) = 0 . \end{matrix}

2-

(29)

\begin{matrix} {lim}_{x \to \infty} f (x; α; β; λ) = lim_{x \to \infty} (2 αβλ) \times {(1 + αβ X^{2})}^{- (λ + 1)} = (2 αβλ) lim_{x \to \infty} \frac{x}{{(1 + αβ x^{2})}^{(λ + 1)}} \\ = (2 αβλ) lim_{x \to \infty} \frac{\frac{x}{{(x^{2})}^{(λ + 1)}}}{\frac{{(1 + αβ x^{2})}^{(λ + 1)}}{{(x^{2})}^{(λ + 1)}}} = (2 αβλ) lim_{x \to \infty} \frac{\frac{1}{{(x^{2})}^{(λ + \frac{1}{2})}}}{{(\frac{1}{x^{2}} + αβ)}^{(λ + 1)}} \\ = (2 αβλ) \frac{\frac{1}{\infty}}{{(\frac{1}{\infty} + αβ)}^{(λ + 1)}} = (2 αβλ) \frac{0}{{(0 + αβ)}^{(λ + 1)}} = 0 \\ \to {lim}_{x \to \infty} f (x; α; β; λ) = 0 . \end{matrix}

b- The Limit of cdf $F (x; α; β; λ)$

1-

(30)

\begin{matrix} {lim}_{x \to 0} F (x; α; β; λ) = {lim}_{x \to 0} [1 - {(1 + αβ x^{2})}^{- λ}] = 1 - {(1 + 0)}^{- λ} = 0 \\ \to {lim}_{x \to 0} F (x; α; β; λ) = 0 . \end{matrix}

2-

(31)

\begin{matrix} {lim}_{x \to \infty} F (x; α; β; λ) = lim_{x \to \infty} [1 - {(1 + αβ x^{2})}^{- λ}] \\ = lim_{x \to \infty} [1 - \frac{1}{{(1 + αβ x^{2})}^{λ}}] = 1 - \frac{1}{\infty} = 1 - 0 = 1 \\ \to {lim}_{x \to \infty} F (x; α; β; λ) = 1 . \end{matrix}

2-2-5 Moments

In this section, we present the r^th moments about origin and the r^th moments about mean for IDNAI distribution

a- The r^th moments about origin
$\begin{matrix} E (x^{r}) = \int_{0}^{\infty} x^{r} (2 αβλ) \times {(1 + αβ X^{2})}^{- (λ + 1)} dx \\ = (2 αβλ) \int_{0}^{\infty} x^{r + 1} {(1 + αβ X^{2})}^{- (λ + 1)} dx \end{matrix}$

$Put y = αβ X^{2} \to^{0} < y < \infty \to x = {(\frac{y}{αβ})}^{\frac{1}{2}} \to dx = (\frac{1}{2}) {(αβy)}^{\frac{- 1}{2}}$

(32)
$\begin{matrix} E (x^{r}) = λ {(α β)}^{\frac{- r}{2}} \int_{0}^{\infty} y^{\frac{r}{2}} {(1 + y)}^{- (λ + 1)} dy \\ = {(α β)}^{\frac{- r}{2}} \int_{0}^{\infty} y^{(\frac{r}{2} + 1) - 1} {(1 + y)}^{- (\frac{r}{2} + 1) - [- (\frac{r}{2} + 1) + (λ + 1)]} dy \\ = λ {(α β)}^{\frac{- r}{2}} \int_{0}^{\infty} y^{(\frac{r}{2} + 1) - 1} {(1 + y)}^{- (\frac{r}{2} + 1) - (λ - \frac{r}{2})} dy \\ = λ {(αβ)}^{\frac{- r}{2}} . B (\frac{r}{2} + 1, λ - \frac{r}{2}), B (., .) refer to Beta type II distribution \\ = λ {(αβ)}^{\frac{- r}{2}} \frac{Γ (\frac{r}{2} + 1) Γ (λ - \frac{r}{2})}{Γ (λ + 1)} \\ E (X^{r}) = \frac{{(\frac{1}{α β})}^{\frac{r}{2}} (\frac{r}{2}) Γ (\frac{r}{2}) Γ (λ - \frac{r}{2})}{Γ (λ)} \end{matrix}$
b- The r^th moments about mean μ
$\begin{matrix} E {(X - μ)}^{r} = \int_{0}^{\infty} {(x - μ)}^{r} f (x; \underline{ϑ}) dx, \underline{ϑ} = (α, β, λ) \\ = \int_{0}^{\infty} \sum_{i = 0}^{r} (\begin{array}{c} r \\ i \end{array}) {(- 1)}^{i} μ^{i} (X^{r - i}) f (x; \underline{ϑ}) dx \\ = \sum_{i = 0}^{r} (\begin{array}{c} r \\ i \end{array}) {(- 1)}^{i} μ^{i} \int_{0}^{\infty} (X^{r - i}) f (x; \underline{ϑ}) dx \\ = \sum_{i = 0}^{r} (\begin{array}{c} r \\ i \end{array}) {(- 1)}^{i} μ^{i} E (X^{r - i}), from (a) above we get \end{matrix}$

(33)
$E {(X - μ)}^{r} = \frac{\sum_{i = 0}^{r} (\frac{r}{i}) {(- 1)}^{i} μ^{i} {(\frac{1}{α β})}^{\frac{r - i}{2}} (\frac{r - i}{2}) Γ (\frac{r - i}{2}) Γ (λ - \frac{r - i}{2})}{Γ (λ)}$

2-2-6 Moment generating function M_X(t)

It is simple to describe the moment generating function (MGF) of the IDNAI distribution as

(34)

\begin{matrix} M_{X} (t) = E (e^{tx}) \\ = \int_{0}^{\infty} e^{tx} f (x; \underline{ϑ}) dx \\ = \int_{0}^{\infty} \sum_{i = 0}^{\infty} \frac{{(tx)}^{i}}{i!} f (x; \underline{ϑ}) dx \\ = \sum_{i = 0}^{\infty} \frac{t^{i}}{i!} \int_{0}^{\infty} x^{i} f (x; \underline{ϑ}) dx, \\ = \sum_{i = 0}^{\infty} \frac{t^{i}}{i!} E (x^{i}), from (a) above we get \\ = \sum_{i = 0}^{\infty} \frac{(\frac{t^{i}}{i!}) {(\frac{1}{α β})}^{\frac{i}{2}} (\frac{i}{2}) Γ (\frac{i}{2}) Γ (λ - \frac{i}{2})}{Γ (λ)} \end{matrix}

2-2-7 Characteristic function Ф_X(t)

The characteristic function of the IDNAI distribution can be found as

(35)

\begin{matrix} Ф_{X} (t) = E (e^{itx}), i = \sqrt{- 1} \\ = \int_{0}^{\infty} e^{itx} f (x; \underline{ϑ}) dx \\ = \int_{0}^{\infty} \sum_{r = 0}^{\infty} \frac{{(itx)}^{r}}{r!} f (x; \underline{ϑ}) dx \\ = \sum_{r = 0}^{\infty} \frac{{(it)}^{r}}{r!} \int_{0}^{\infty} x^{r} f (x; \underline{ϑ}) dx \\ = \sum_{r = 0}^{\infty} \frac{{(it)}^{r}}{r!} E (x^{r}), from (a) above we get \\ = \sum_{r = 0}^{\infty} \frac{{(it)}^{r}}{r!} \frac{{(\frac{1}{α β})}^{\frac{r}{2}} (\frac{r}{2}) Γ (\frac{r}{2}) Γ (λ - \frac{r}{2})}{Γ (λ)} \end{matrix}

2-2-8 Factorial generating function З_X(t)

One may determine the factorial generating function of the IDNAI distribution as follows

(36)

\begin{matrix} З_{X} (t) = E (t^{x}) \\ = \int_{0}^{\infty} t^{x} f (x; \underline{ϑ}) dx \\ = \int_{0}^{\infty} e^{x ln t} f (x; \underline{ϑ}) dx \\ = \int_{0}^{\infty} \sum_{r = 0}^{\infty} \frac{{(x ln t)}^{r}}{r!} f (x; \underline{ϑ}) dx \\ = \sum_{r = 0}^{\infty} \frac{{(ln t)}^{r}}{r!} \int_{0}^{\infty} x^{r} f (x; \underline{ϑ}) dx \\ = \sum_{r = 0}^{\infty} \frac{{(ln t)}^{r}}{r!} E (x^{r}), from (a) above we get \\ = \sum_{r = 0}^{\infty} \frac{{(ln t)}^{r}}{r!} \frac{{(\frac{1}{α β})}^{\frac{r}{2}} (\frac{r}{2}) Γ (\frac{r}{2}) Γ (λ - \frac{r}{2})}{Γ (λ)} \end{matrix}

2-2-9 Coefficients

From sub-sections 2-2-5, we infer the following conclusions. This section lists the IDNAI distribution’s Skewness, Kurtosis, and Variation coefficients.

a- Coefficient of Skewness (C.S.)
The definition of the skewness coefficient (C.S.) is as follows
$C . S . = {E {(X - μ)}^{3} / {[E {(X - μ)}^{2}]}^{3 / 2}}$

Thus, skewness coefficient for the IDNAI distribution will be

(37)

C . S . = {\begin{cases} [\sum_{i = 0}^{3} (\begin{array}{c} 3 \\ i \end{array}) {(- 1)}^{i} μ^{i} {(\frac{1}{αβ})}^{\frac{3 - i}{2}} (\frac{3 - i}{2}) Γ (\frac{3 - i}{2}) Γ (λ - \frac{3 - i}{2}) / Γ (λ)] / \\ {[\sum_{i = 0}^{2} (\begin{array}{c} 2 \\ i \end{array}) {(- 1)}^{i} μ^{i} {(\frac{1}{αβ})}^{\frac{2 - i}{2}} (\frac{2 - i}{2}) Γ (\frac{2 - i}{2}) Γ (λ - \frac{2 - i}{2}) / Γ (λ)]}^{3 / 2} \end{cases}}

b- Coefficient of Kurtosis (C.K.)

The Kurtosis coefficient (C.K.) is defined as follows.

C . K . = {E {(X - μ)}^{4} / {[E {(X - μ)}^{2}]}^{2}}

Thus, Kurtosis coefficient for the IDNAI distribution will be

(38)

C . K . = {\begin{cases} [\sum_{i = 0}^{4} (\begin{array}{c} 4 \\ i \end{array}) {(- 1)}^{i} μ^{i} {(\frac{1}{αβ})}^{\frac{4 - i}{2}} (\frac{4 - i}{2}) Γ (\frac{4 - i}{2}) Γ (λ - \frac{4 - i}{2}) / Γ (λ)] / \\ {[\sum_{i = 0}^{2} (\begin{array}{c} 2 \\ i \end{array}) {(- 1)}^{i} μ^{i} {(\frac{1}{αβ})}^{\frac{2 - i}{2}} (\frac{2 - i}{2}) Γ (\frac{2 - i}{2}) Γ (λ - \frac{2 - i}{2}) / Γ (λ)]}^{2} \end{cases}} .

c- Coefficient of Variation (C.V.)

The Variation coefficient (C.V.) is defined as follows.

C.V. = {[E(X-μ)²]^1/2/E(X)}

Consequently, the IDNAI distribution’s variation coefficient will be

(39)

C . V . = {\begin{cases} {[\sum_{i = 0}^{2} (\begin{array}{c} 2 \\ i \end{array}) {(- 1)}^{i} μ^{i} {(\frac{1}{αβ})}^{\frac{2 - i}{2}} (\frac{2 - i}{2}) Γ (\frac{2 - i}{2}) Γ (λ - \frac{2 - i}{2}) / Γ (λ)]}^{1 / 2} / \\ [{(\frac{1}{αβ})}^{\frac{1}{2}} (\frac{1}{2}) Γ (\frac{1}{2}) Γ (λ - \frac{1}{2}) / Γ (λ)] \end{cases}} .

2-2-10 Probability density functions of order statistic (O.S.)

Let x₁,x₂, …, x_n be a random sample of size n from IDANI distribution with cdf F(x; $\underline{ϑ}$ ) and pdf f(x; $\underline{ϑ}$ ), where $\underline{ϑ}$ = (α,β,λ). Let y₁< y₂ < … < y_n denote to order statistic for this sample. Then the probability density function of y_r (r =1, 2, …, n) is given by f (y_r) = ( $\frac{1}{B (r; n - r + 1)}$ )= $f (x; \underline{ϑ}) {[F (x)]}^{r - 1} {[1 - F (x)]}^{n - r}$ , for (1 < r < n), hence we get

a-

(40)

\begin{matrix} f (y_{r}) = (\frac{1}{B (r; n - r + 1)}) (2 αβλ) (y_{r}) {(1 + αβ y_{r}^{2})}^{- (λ + 1)} {[1 - {(1 + αβ y_{r}^{2})}^{- λ}]}^{r - 1} {[{(1 + αβ y_{r}^{2})}^{- λ}]}^{n - r} \\ = (\frac{1}{B (r; n - r + 1)}) (2 αβλ) (y_{r}) {(1 + αβ y_{r}^{2})}^{- λ - 1 - nλ + rλ} \sum_{i = 0}^{r - 1} (\begin{array}{c} r - 1 \\ i \end{array}) {(- 1)}^{i} {(1 + αβ y_{i}^{2})}^{- iλ} \\ = (\frac{1}{B (r; n - r + 1)}) (2 αβλ) (y_{r}) \sum_{i = 0}^{r - 1} (\begin{array}{c} r - 1 \\ i \end{array}) {(- 1)}^{i} {(1 + αβ y_{r}^{2})}^{- λ - 1 - nλ + rλ - iλ} \\ f (y_{r}) = (\frac{1}{B (r; n - r + 1)}) (2 αβλ) (y_{r}) \sum_{i = 0}^{r - 1} (\begin{array}{c} r - 1 \\ i \end{array}) {(- 1)}^{i} {(1 + αβ y_{r}^{2})}^{- [(1 + n + i - r) λ + 1}, for (1 < r < n) . \end{matrix}

b-

(41)

\begin{matrix} f (y_{1}) = n f (x; \underline{ϑ}) {[1 - F (x)]}^{n - 1} \\ f (y_{1}) = 2 nαβλ y_{1} {(1 + αβ y_{1}^{2})}^{- (nλ + 1)} \end{matrix}

According to this, y₁~IDNAI (α,β,nλ)

c-

(42)

\begin{matrix} f (y_{n}) = n f (x; \underline{ϑ}) {[1 - F (x)]}^{n - 1} \\ = n (2 αβλ) (y_{n}) {(1 + αβ y_{n}^{2})}^{- (λ + 1)} {[1 - {(1 + αβ y_{n}^{2})}^{- λ}]}^{n - 1} \\ = n (2 αβλ) (y_{n}) {(1 + αβ y_{n}^{2})}^{- (λ + 1)} \sum_{i = 0}^{n - 1} (\begin{array}{c} n - 1 \\ i \end{array}) {(- 1)}^{i} {(1 + αβ y_{n}^{2})}^{- [(1 + i) λ + 1]} \\ = 2 nαβλ y_{n} \sum_{i = 0}^{n - 1} (\begin{array}{c} n - 1 \\ i \end{array}) {(- 1)}^{i} {(1 + αβ y_{n}^{2})}^{- [(1 + i) λ + 1]} \end{matrix}

2-2-11 Cumulative distribution function of order statistic (O.S.)

This section presents the order statistic’s cumulative distribution.

The common formula for the cdf of order statistic is

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F_{i} (t) = \sum_{k = i}^{n} (\frac{n}{k}) {[F (t)]}^{k} {[1 - F (t)]}^{n - k}

Consequently,

\begin{matrix} F (y_{r}) = \sum_{j = r}^{n} (\begin{array}{c} n \\ j \end{array}) {[1 - {(1 + αβ y_{r}^{2})}^{- λ}]}^{j} {[{(1 + αβ y_{r}^{2})}^{- λ}]}^{n - j} \\ = \sum_{j = r}^{n} (\begin{array}{c} n \\ j \end{array}) {(1 + αβ y_{r}^{2})}^{- λ (n - j)} \sum_{i = 0}^{j} (\begin{array}{c} j \\ i \end{array}) {(- 1)}^{i} {[{(1 + αβ y_{r}^{2})}^{- λ}]}^{i} \\ = \sum_{j = r}^{n} \sum_{i = 0}^{j} (\begin{array}{c} n \\ j \end{array}) (\begin{array}{c} j \\ i \end{array}) {(- 1)}^{i} {(1 + αβ y_{r}^{2})}^{- λ (n - j + i)} \\ = \sum_{j = r}^{n} \sum_{i = 0}^{j} (\begin{array}{c} n \\ j \end{array}) (\begin{array}{c} j \\ i \end{array}) {(- 1)}^{i} \sum_{k = 0}^{\infty} {(- 1)}^{k} (\begin{array}{c} λ (n - j + i) + k - 1 \\ k \end{array}) {(αβ y_{r}^{2})}^{k} \end{matrix}

Hence, the cumulative distribution function of r^th order statistic will be

(44)

F (y_{r}) = \sum_{k = 0}^{\infty} \sum_{J = r}^{n} \sum_{i = 0}^{j} (\frac{n}{j}) (\frac{j}{i}) (\frac{λ (n - j + i) + k - 1}{k}) {(- 1)}^{i + k} {({αβy}_{r}^{2})}^{k}, {αβy}_{r}^{2} < 1

2-2-12 Mean time to failure (death); MTTF

In this section, we introduce the mean time to failure (death)

\begin{array}{l} MTTF = \int_{0}^{\infty} R (x; \underline{ϑ}) dx; R (x; \underline{ϑ}) = {(1 + αβ x^{2})}^{- λ} refer to reliability function of IDNAI distribution \\ = \int_{0}^{\infty} {(1 + αβx 2)}^{- λ} dx \end{array}

Put y = αβ X^{2} \to^{0} < y < \infty \to x = {(\frac{y}{α β})}^{\frac{1}{2}} \to dx = (\frac{1}{2}) {(α β y)}^{\frac{- 1}{2}} dy

(45)

\begin{matrix} MTTF = (\frac{1}{2}) {(α β)}^{\frac{- 1}{2}} \int_{0}^{\infty} {(1 + y)}^{- λ} {(y)}^{\frac{- 1}{2}} dy \\ = (\frac{1}{2}) {(α β)}^{\frac{- 1}{2}} \int_{0}^{\infty} {(y)}^{(\frac{- 1}{2} + 1) - 1} {(1 + y)}^{- λ} dy \\ = (\frac{1}{2}) {(α β)}^{\frac{- 1}{2}} \int_{0}^{\infty} {(y)}^{(\frac{- 1}{2} + 1) - 1} {(1 + y)}^{- λ} dy \\ = (\frac{1}{2}) {(α β)}^{\frac{- 1}{2}} \int_{0}^{\infty} {(y)}^{(\frac{- 1}{2} + 1) - 1} {(1 + y)}^{\frac{- 1}{2} - (λ - \frac{1}{2})} dy \\ = (\frac{1}{2}) {(αβ)}^{\frac{- 1}{2}} \frac{Γ (\frac{1}{2}) Γ (λ - \frac{1}{2})}{Γ (λ)} \\ = \frac{{(\frac{1}{α β})}^{\frac{1}{2}} (\frac{1}{2}) Γ (\frac{1}{2}) Γ (λ - \frac{1}{2})}{Γ (λ)} \end{matrix}

It was observed that; Γ(1/2) = $\sqrt{π}$

2-2-13 The Average failure rate (AFR)

a- The Average failure rate over time (0, t)
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$AFR (0, t) = L (t) = \frac{H (t)}{t} = \frac{λ Ln (1 + α β t^{2})}{t}$
b- The Average failure rate over time ( $t_{1}$ , $t_{2}$ )
(47)
$AFR (t_{1}, t_{2}) = \frac{1}{t_{2} - t_{1}} {H (u)]}_{t_{1}}^{t_{2}} = \frac{λ [Ln (1 + αβ t_{2}^{2}) - Ln (1 + αβ t_{1}^{2})]}{t_{2} - t_{1}}$

2-2-14 Self-reproducing property

let y₁< y₂ < … < y_n denote to order statistic for the sample x₁,x₂, …, x_n of size n from IDNAI distribution with pdf and cdf defined in Equations (16) and (19).

According to Equation (41) in property 10, the pdf of y₁ is; f(y₁) =2nαβλ y₁ (1+ αβ y₁²)^{- (n λ+1)}

It is clear that $y_{1}$ follows $IDNAI distribution with parameters (α, β, nλ)$ . As a result, the self-reproducing property is satisfied by the IDNAI distribution.

2-2-15 Entropies

We present several uncertainty measurements, such as Shannon and Rényi entropies, that may be employed to measure the total amount of data in the system.

a- Shannon Entropy SH_IDNAI(x)

Shannon’s entropy has been used in many technical fields, such as physics and economics.

According to Shannon’s 1948 formulation, the entropy of a random variable X with density function f(x) is¹⁶

η_{x} = E {- ln [f (X)]}; .

The Shannon entropy for T-R {Y) family distribution was defined as below.¹⁷

{SH}_{IDNA} (x) = {SH}_{lomax} (x) + E (ln (f_{Y} (T))) - E ((ln (f_{R} (X))) .

Additionally, as was previously indicated; T~ Lomax (λ); R~ Rayleigh(α) and Y~ Exponential (β).

Where, the Shannon entropy of T is known as¹⁸

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{SH}_{lomax} (x) = 1 / λ + ln (1 / λ) + 1

\begin{array}{l} ln (f_{Y} (T)) = ln (1 / β) - (T / β) \to E (ln (f_{Y} (T))) = ln (1 / β) - (1 / β) μ_{T} \\ ln (f_{R} (X)) = ln (2 α) + ln (x) - α x^{2} \to E ((ln (f_{R} (X)))) = ln (2 α) + E (ln (x)) - α E (X^{2}), E (X^{2}) = \frac{1}{α} \\ E (ln (x)) = \int_{0}^{\infty} ln (x) (2 α x e^{- α x^{2}}) dx = 2 α \int_{0}^{\infty} x e^{- α x^{2}} ln (x) dx \end{array}

Recall; \int_{0}^{\infty} x^{m} e^{- {sx}^{b}} ln (x) dx = \frac{Γ (a)}{b^{2} s^{a}} [ψ (a) - ln (s)] .

\begin{matrix} E (ln (x)) = 2 α [\frac{Γ (1)}{4 α} (ψ (1) - ln (α)] \\ = \frac{1}{2} [ψ (1) - ln (α)] \end{matrix}

Hence, Shannon entropy of IDNAI distribution will be as follow

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{SH}_{IDNA} (x) = [1 / λ + ln (1 / λ) + 1] + ln (1 / β) - (1 / β) μ_{T} - ln (2α) - (1 / 2) [ψ (1) - ln (α)] - 1

Where, μ_T is the mean of the random variable T.

b- Rényi Entropy RE_IDNAI(x)

The measure of uncertainty of a random variable X follows IDNAI distribution is determined by its entropy. The Rényi entropy is defined as follows¹⁹:

{RE}_{IDNAI} (θ) = (\frac{1}{1 - θ}) Ln {\int_{0}^{\infty} {[f (x; α; β; λ)]}^{θ}] dx .

f_{IDNAI} (x; α, β, λ) = (2 αβλ) \times {(1 + αβ X^{2})}^{- (λ + 1)}

\int_{0}^{\infty} {[f (x; α, β, λ)]}^{θ} dx = {(2 α βλ)}^{θ} \int_{0}^{\infty} x^{θ} {(1 + α β X 2)}^{- θ (λ + 1)} dx

Put y = αβ X^{2} \to 0 < y < \infty \to x = {(\frac{y}{α β})}^{\frac{1}{2}} \to dx = (\frac{1}{2}) {(α β y)}^{\frac{- 1}{2}} dy

\begin{matrix} \int_{0}^{\infty} {[f (x; α, β, λ)]}^{θ} dx = {(2 α βλ)}^{θ} \int_{0}^{\infty} {(\frac{y}{α β})}^{\frac{θ}{2}} {(1 + y)}^{- θ (λ + 1)} (\frac{1}{2}) {(α βλ)}^{\frac{- 1}{2}} dy \\ = {(2 α βλ)}^{θ} (\frac{1}{2}) {(α β)}^{\frac{- 1}{2}} {(α β)}^{\frac{θ}{2}} \int_{0}^{\infty} y^{\frac{θ - 1}{2}} {(1 + y)}^{- θ (λ + 1)} dy \\ = 2^{θ - 1} {(α β)}^{\frac{θ}{2} - 1} λ^{θ} \int_{0}^{\infty} y^{\frac{θ + 1}{2} - 1} {(1 + y)}^{- \frac{θ + 1}{2} - [- \frac{θ + 1}{2} + θ (λ + 1)]} dy \\ = 2^{θ - 1} {(αβ)}^{\frac{θ - 2}{2}} λ^{θ} B (\frac{θ + 1}{2}, - \frac{θ + 1}{2} + θ (λ + 1)), \\ = 2^{θ - 1} {(αβ)}^{\frac{θ - 1}{2}} λ^{θ} B (\frac{θ + 1}{2}, θ (λ + \frac{1}{2}) - \frac{1}{2}) \end{matrix}

Consequently, The Rényi entropy IDNAI distribution will be as follows

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{RE}_{IDNAI} (θ) = (\frac{1}{1 - θ}) {Ln [2^{θ - 1} {(α β)}^{\frac{θ - 1}{2}} λ^{θ}] + Ln [B (\frac{θ + 1}{2}, θ (λ + \frac{1}{2}) - \frac{1}{2})]}

2-3 Estimation methods for the parameters of IDNAI distribution

In this section, we present two estimation methods for estimate the parameters of $IDNAI (α, β, λ) distribution$

2-3-1 Maximum Likelihood Estimator (MLE)

Because of its invariance property, the Maximum Likelihood is one of the most important estimation techniques. We look at estimating the unknown parameters of the IDNAI $(α, β, λ)$ distribution using a maximum likelihood function.

Let x₁,x₂, …. x_n be observed values of a random sample of size n from the $IDNAI (α, β, λ) distribution$ .

The pdf of $IDNAI (α, β, λ)$ distribution is as below;

f (x; α, β, λ) = (2 αβλ) (x) {(1 + αβ x^{2})}^{- (λ + 1)} : x, α, β, λ > 0 .

The likelihood function with the vector $ϑ$ = (α, β, λ) of parameters can be written as:

L (x_{i}; α, β, λ) = \prod_{i = 1}^{n} f (x; α, β, λ) = {(2 αβλ)}^{n} \prod_{i = 1}^{n} x_{i} \prod_{i = 1}^{n} {(1 + αβ {x_{i}}^{2})}^{- (λ + 1)}

The log likelihood function will be

LnL (x_{i}; α, β, λ) = nLn + nLnα + nLnβ + nLnλ + \sum_{i = 1}^{n} Ln x_{i} - (λ + 1) \sum_{i = 1}^{n} Ln (1 + αβ {x_{i}}^{2})

The log likelihood function with respect to α, β, and λ is derived as follows:

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\frac{∂LnL (x_{i}; α, β, λ)}{∂α} = \frac{n}{α} - (λ + 1) \sum_{i = 1}^{n} \frac{β {x_{i}}^{2}}{(1 + αβ {x_{i}}^{2})}

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\frac{∂LnL (x_{i}; α, β, λ)}{∂β} = \frac{n}{β} - (λ + 1) \sum_{i = 1}^{n} \frac{α {x_{i}}^{2}}{(1 + αβ {x_{i}}^{2})}

(53)

\frac{∂LnL (x_{i}; α, β, λ)}{∂λ} = \frac{n}{λ} - \sum_{i = 1}^{n} Ln (1 + αβ {x_{i}}^{2})

Setting the above nonlinear equations to zero and solve them using Newton-Raphson method, the maximum likelihood estimates of the three parameters $\hat{α}$ , $\hat{β}$ and $\hat{λ}$ were obtained.

2-3-2 Least Squared Estimator (LS)

In this subsection, we discuss the estimator of Least Squares method. This method used for many mathematical and engineering application.²⁰

The main idea for this method is to minimize the sum of square error between the values and the expected value.

To find the estimation of the three parameter of $IDNAI (α, β, λ) distribution$ via mentioned least squares method, let: x₁, x₂, …. x_n be observed values of a random sample of size n from the $IDNAI (α, β, λ) distribution$ , and

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S = \sum_{i = 1}^{n} [F (x_{i}) - E (F (x_{i}))]^{2}

Where, $F (x_{i})$ refers to the CDF of $IDNAI (α, β, λ) distribution$ which is defined as follows:

F (x; α, β, λ) = 1 - {(1 + {αβx}^{2})}^{- λ} for x > 0 and α, β, λ > 0

And, $E (F (x_{i}))$ = $P_{i}$ , $P_{i}$ = $\frac{i}{n + 1}$ , $i$ =1,2,..,n.

Derive S w.r.t. $α, β and λ$ , we obtained the following three nonlinear equations respectively

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\frac{∂S}{∂α} = 2βλ \sum_{i = 1}^{n} {(1 - p_{i}) x_{i}^{2} {(1 + α β x_{i}^{2})}^{- (λ + 1)} - x_{i}^{2} {(1 + α β x_{i}^{2})}^{- (2 λ + 1)}}

(56)

\frac{∂S}{∂β} = 2αλ \sum_{i = 1}^{n} {(1 - p_{i}) x_{i}^{2} {(1 + α β x_{i}^{2})}^{- (λ + 1)} - x_{i}^{2} {(1 + α β x_{i}^{2})}^{- (2 λ + 1)}}

(57)

\frac{∂S}{∂λ} = 2 \sum_{i = 1}^{n} {(1 - p_{i}) {(1 + α β x_{i}^{2})}^{- λ} log (1 + α β x_{i}^{2}) - {(1 + α β x_{i}^{2})}^{- 2 λ} log (1 + α β x_{i}^{2})}

Setting the above nonlinear equations to zero and solve them using Newton-Raphson method, the Least Squares estimates of the three parameters $\hat{α}$ , $\hat{β}$ and $\hat{λ}$ will be obtained.

2-4 Comparison of estimators

The simulation study will be advanced to show estimators’ behaviour of parameters by suggested two mentioned estimation methods and then compare results using the two statistical criteria bias and mean squared error (MSE). The simulation study is repeated (1000) times to obtain independent samples of different sizes.

2-4-1 Generating random variables

Let U be a random variable with a Uniform distribution on the interval (0,1). The data for the IDNAI distribution can be generated using the inverse transformation method for the cumulative distribution function (CDF), where if²¹:

F (x_{i}; α, β, λ) = 1 - {(1 + αβ {x_{i}}^{2})}^{- λ} \to U_{i} = 1 - {(1 + αβ {x_{i}}^{2})}^{- λ}

Then $x_{i}$ = $F {(x_{i}; α, β, λ)}^{- 1} =$ {[(1- $U_{i}$ )^-1/λ -1] /αβ}^1/2

2-4-2 Simulation study

The Monte Carlo simulation software, created in MATLAB 2022, enables the comparison of reliable estimators over the steps that come next:

1. Random samples x₁, x₂, …, x_n, of sizes n, where n = 10, 20, 50, 100, are generated from the IDNAI distribution.
2. The real parameter values are considered for two experiments (α, β, λ) in Table 1:
3. The Bias for all proposed estimators about the parameters was evaluated as follows, utilizing L = 1000 duplicates, which illustrated in Tables 2 and 3:
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$Bias = \frac{1}{L} \sum_{i = 1}^{L} | {\hat{ϑ}}_{i} - ϑ |, ϑ = (α, β, λ)$
4. The MSE for all suggested estimators w.r.t. parameters was calculated as below using L = 1000 duplicates: as illustrated in Tables 2 and 3:
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$MSE = \frac{1}{L} \sum_{i = 1}^{L} {({\hat{ϑ}}_{i} - ϑ)}^{2} .$

Table 1. Shown two cases of real parameters.

Case	α	β	λ
1	0.7	2.0	2.0
2	1.5	0.5	1.8

Table 2. Bias and MSE of case 1.

∝=0.7 β=2.0 λ= 2.0
n	Mse/Bias	MLE	LS	Best I
10	Mse $\hat{\propto}$	0.0020	0.1941	MLE
	Bias $\hat{\propto}$	6.4500e-04	0.0125	MLE
	Mse $\hat{β}$	0.0160	0.8239	MLE
	Bias $\hat{β}$	0.0055	0.0128	MLE
	Mse $\hat{λ}$	0.0280	8.2454	MLE
	Bias $\hat{λ}$	0.0962	0.2553	MLE
20	Mse $\hat{\propto}$	0.0020	0.3194	MLE
	Bias $\hat{\propto}$	2.1100e-04	0.0069	MLE
	Mse $\hat{β}$	0.0099	2.0694	MLE
	Bias $\hat{β}$	0.0038	0.0132	MLE
	Mse $\hat{λ}$	0.6482	7.9692	MLE
	Bias $\hat{λ}$	0.0509	0.2762	MLE
50	Mse $\hat{\propto}$	0.0007	0.1792	MLE
	Bias $\hat{\propto}$	3.1300e-04	0.0181	MLE
	Mse $\hat{β}$	0.0050	0.7214	MLE
	Bias $\hat{β}$	0.0042	0.0235	MLE
	Mse $\hat{λ}$	0.2159	7.7296	MLE
	Bias $\hat{λ}$	0.0146	0.2486	MLE
100	Mse $\hat{\propto}$	0.0003	0.4536	MLE
	Bias $\hat{\propto}$	6.8000e-04	0.0114	MLE
	Mse $\hat{β}$	0.0018	3.0239	MLE
	Bias $\hat{β}$	0.0025	0.0217	MLE
	Mse $\hat{λ}$	0.0084	7.3317	MLE
	Bias $\hat{λ}$	0.0040	0.2556	MLE

Table 3. Bias and MSE of case 2.

∝=1.5 β=0.5 λ= 1.8
n	Mse/Bias	MLE	LS	Best I
10	Mse $\hat{\propto}$	0.0102	0.0440	MLE
	Bias $\hat{\propto}$	0.0045	0.0109	MLE
	Mse $\hat{β}$	0.0016	0.0303	MLE
	Bias $\hat{β}$	3.3800e-04	0.0118	MLE
	Mse $\hat{λ}$	0.4731	8.1057	MLE
	Bias $\hat{λ}$	0.0590	0.2184	MLE
20	Mse $\hat{\propto}$	0.0045	0.0952	MLE
	Bias $\hat{\propto}$	0.0012	0.0090	MLE
	Mse $\hat{β}$	0.0007	0.0986	MLE
	Bias $\hat{β}$	8.3700e-04	0.0112	MLE
	Mse $\hat{λ}$	0.3975	7.0912	MLE
	Bias $\hat{λ}$	0.0155	0.2311	MLE
50	Mse $\hat{\propto}$	0.0019	0.5751	MLE
	Bias $\hat{\propto}$	0.0032	0.0122	MLE
	Mse $\hat{β}$	0.0002	0.2269	MLE
	Bias $\hat{β}$	1.2900e-04	0.0155	MLE
	Mse $\hat{λ}$	0.0373	6.6451	MLE
	Bias $\hat{λ}$	0.0115	0.2003	MLE
100	Mse $\hat{\propto}$	0.0013	0.5473	MLE
	Bias $\hat{\propto}$	0.0017	0.0090	MLE
	Mse $\hat{β}$	0.0002	0.2770	MLE
	Bias $\hat{β}$	8.2100e-04	0.0122	MLE
	Mse $\hat{λ}$	0.0157	5.6964	MLE
	Bias $\hat{λ}$	0.0035	0.2492	MLE

2-5 Application for comparison and fitting real-world dataset

This section relates to Tables 4 and 5, which present survival times (in days) and descriptive data, respectively, for 72 guinea pigs infected with virulent tubercle bacilli, as documented by Olubiyi et al.,²² to illustrate the flexibility and applicability of the IDNAI distribution. To determine how well the IDNAI distribution fits the data, seven models that comprise the baseline distribution would be compared: the Weibull Gumbel type II distribution (WGuTII), the Exponentiated Gumbel type II (EGuTII) distribution, the Gumbel type II (GuTII) distribution, the Weibull Rayleigh distribution (WRD), the Exponential Lomax distribution (ELD), the Exponential Rayleigh distribution (ERD), and the Logistic distribution. Akaike information criteria (AIC), Bayesian information criteria (BIC), corrected Akaike information criterion (AICC), Hannan-Quinn information criterion (HQIC), consistent Akaike information criterion (CAIC), Kolmogorov-Smirnov K-S distance with P-values, and a minus two times of the negative log-likelihood value are used to confirm that the IDNAI distribution is a suitable model for the data set. The basis for comparing the models would be the reduced log-likelihood estimate and the mentioned information criteria. Having the lowest minimized log-likelihood and information statistics value is the ideal model. Accordingly, based on the findings in Table 6, the IDNAI might be chosen as the dataset’s best fit model.

Table 4. Dataset of the survival times (in days).

0.1	0.33	0.44	0.56	0.59	0.72	0.74	0.77	0.92	0.93
0.96	1	1	1.02	1.05	1.07	1.07	1.08	1.08	1.08
1.09	1.12	1.13	1.15	1.16	1.2	1.21	1.22	1.22	1.24
1.3	1.34	1.36	1.39	1.44	1.46	1.53	1.59	1.6	1.63
1.63	1.68	1.71	1.72	1.76	1.83	1.95	1.96	1.97	2.02
2.13	2.15	2.16	2.22	2.3	2.31	2.4	2.45	2.51	2.53
2.54	2.54	2.78	2.93	3.27	3.42	3.47	3.61	4.02	4.32
4.58	5.55

Table 5. Descriptive statistics of dataset.

Stats names	Stats values
Mean	1.7682
Median	1.4950
Mode	1.0800
Std. Dev.	1.0345
Skewness	1.3419
Kurtosis	4.9911
Sample size	72

Table 6. Parameter estimates with different information criteria.

Distribution	Parameters estimate	Neg2LogL	AIC	BIC	HQIC	CAIC	AICC	K-S Stat	P-Value
IDNAI	alpha=0.135, beta=0.500, lambda=4.537	187.9999	193.9999	200.8298	196.7194	194.6461	194.3528	0.1116	0.3081
Logistic	mu=1.637, sigma=0.541	202.1306	206.1306	210.6840	207.9433	206.3046	206.3046	0.0989	0.4525
WRD	alpha=1.825, beta=1.996	191.5796	197.5796	204.4096	200.2987	197.9326	197.9326	0.0978	0.4824
ELD	alpha=272457603.679, lambda=481771451.918	476.1900	482.1900	489.0200	484.9091	492.020023	482.54297	0.2945	0.070
ERD	alpha=1172907863446629791710576640, beta=1.861	193.6750	197.6750	202.2283	199.4877	197.8489	197.8489	0.1166	0.2606
WGuTII	Beta=0.599,theta=15.230,alfa=2566.429,lambda=0.231	189.202	197.202	206.305	200.823	210.309	197.799	0.3146	0.0944
EGuTII	Theta=9.234,alfa=2582.288, lambda=0.230	193.604	199.604	206.434	202.320	209.434	199.957	0.3491	0.0821
GuTII	Theta=1.069, lambda=1.173	236.332	240.332	244.885	242.142	246.885	240.506	0.3770	0.0791

3- Results

3-1 Special cases of the new IDNAI distribution

The following may be concluded from the proposed IDNAI distribution’s PDF in Equation (19).

(i) The IDNAI (α,β,λ) distribution follows the Beta prime distribution with parameters 1 and λ for the random variable y = αβX².
(ii) When the random variable Z = βX², then the IDNAI (α,β,λ) distribution is interpreted to Lomax distribution with two parameters (α,λ).
(iii) When the random variable W = α X², then the IDNAI (α,β,λ) distribution reduce to Lomax distribution with two parameters (β,λ).
(iv) When the random variable V = X^2,then the IDNAI (α,β,λ) distribution reduce to Lomax distribution with three parameters (α,β,λ).
(v) When the random variable T = αβx², then the IDNAI (α,β,λ) distribution reduce to Power function distribution with one parameter λ.

3-2 Figures of probability functions of IDNAI distribution

Figure 1 shows the graph of the probability density function for the IDNAI distribution with certain parameter values. The PDF graph in Figure 1 above demonstrates that the density of the IDNAI distribution might be symmetric, adjacent symmetric, right-skewed, or bimodal.

Figure 1. IDNAI distribution's pdf according to different values of α, β and λ.

Figure 2 shows the cumulative distribution function graph for the IDNAI distribution with certain parameter values. The CDF graph in Figure 2 shows that the cumulative distribution function of the IDNAI distribution is a function that does not decrease.

Figure 2. The CDF of IDNAI distribution for different values of α, β and λ.

The graph in Figure 3 shows the Reliability function for the IDNAI distribution with certain parameter values. The reliability function graph in Figure 3 shows that the reliability function of IDNAI distribution is a non-increasing function, which is what most people already know.

Figure 3. The Reliability function of IDNAI distribution for different values of α, β and λ.

Figure 4’s graph of the Hazard function illustrates that the IDNAI distribution’s hazard rate has a convex form for different values of the parameters.

Figure 4. The Hazard function of IDNAI distribution for different values of α, β and λ.

Figure 5 shows the cumulative hazard function of the IDNAI distribution. This shows how it may be used in survival analysis and reliability theory by showing the overall failure risk up to time x. It demonstrates that when x increases, the cumulative danger escalates, conforming to standard survival model expectations. When λ is larger, the hazard accumulation is higher, which means the failure rates are higher. When α and β are removed, the curves are flatter, which means the risk increase is lower. The graphic shows how the properties of the IDNAI distribution change based on different dependability scenarios.

Figure 5. The IDNAI distribution’s cumulative hazard function for various values of α, β and λ.

The graph in Figure 6 shows how the reverse hazard function of the IDNAI distribution changes when different parameters are used, especially λ, which affects failure rates. This function is very important for survival modeling and reliability analysis since it shows how strong the distribution is in situations when reliability is important.

Figure 6. The Reverse hazard function of IDNAI distribution for different values of α, β and λ.

Figure 7 shows that the odd function grows quickly as x gets bigger, especially when the parameters α, β, and λ are big. Solid curves with λ = 1 expand a lot faster than those with λ = 0.5, which shows that the tail behavior is stronger. This function is a good way to show how flexible a model is and how well it controls its tail because it is very sensitive to changes in its parameters.

Figure 7. The Odd function of IDNAI distribution for different values of α, β and λ.

The graph in Figure 8 shows how well a model fits by comparing the empirical cumulative distribution function (CDF) of a dataset with many fitted CDFs from theoretical models. The empirical CDF and IDNAI and LOGISTIC are very near, while WGUTII, EGUTII, and GUTII are only moderately close. WRD and ELD, on the other hand, are very different from each other.

Figure 8. Empirical, fitted distributions.

The graph in Figure 9 shows the p-values and Kolmogorov-Smirnov (K-S) statistics for eight fitted distributions side by side. The IDNAI and logistic models provide a strong fit to the data, as evidenced by their minimal K-S value and maximal p-value. On the other hand, the models WRD, ELD, ERD, WGUTII, EGUTII, and GUTII have low p-values, which mean they don’t fit the real data as well. The way the models are set up makes it easy to compare them and choose the best one.

Figure 9. Histogram for K-S and P-value of fitted distributions.

3-3 Tables of Bias, MSE, dataset, and parameter estimations utilizing different information criteria

The proposed real parameters are shown in Table 1, and the MSE and Bias calculations for the proposed estimators are provided in Tables 2 and 3. Tables 4 and 5 contain the dataset and its descriptive statistics, respectively. Table 6 displays the information criterion and estimation parameters for each fitted distribution.

4- Discussions

4-1 According to the tables of Bias and mean squared errors (MSEs), we identify

1. Bias of the two estimators for the parameters:
The bias Tables 2, 3 obviously demonstrated the outlines across the two cases:
- – The MLE method reliably displayed the least bias, regardless of all the sample size.
- – LS displayed varying degrees of bias performance, particularly in the estimation of λ.
2. Mean Squared Error (MSE) of the two estimators for the parameters:
The MSE Tables 2, 3 of the estimators of parameters for each method across two cases indicate:
- – MLE had the lowest mean squared error across all parameter settings.
- – LS have minimal efficacy, particularly when sample sizes were limited or when settings were chosen to reduce their value.

Consequently, the MLE method was very stable and efficient in a wide range of situations. It is a strong nominee for use in the real world especially when the data is complex or not normal.

4-2 The basis for comparing the models would be the reduced log-likelihood estimate and the mentioned information criteria

Having the lowest minimized log-likelihood and information statistics value is the ideal model. Accordingly, the IDNAI may be selected as the dataset’s optimal fit model in the sense of minimum information criteria based on the Table 6.

5- Conclusion

Over the past ten years, numerous novel distributions introduced in the literature appear to emphasize more general and adaptable types. By utilizing the method that creates the T-X family, it is possible to establish creative distributions that can either be exceptionally general and flexible or specifically tailored to fit certain kinds of data distributions, such as those that are highly left-tailed (right-tailed, thin-tailed, or heavy-tailed) as well as those that are bimodal. This study presents a novel family of probability distributions known as the T–R {Y} distribution family. The resulting family distribution has a number of statistical and mathematical characteristics. Numerous statistical and mathematical characteristics of these distributions are derived, including the related probability functions, the moments, the failure rate, the cumulative hazard functions, the entropies, the moments, order statistics … etc. To encourage the new family members’ suitability for fitting real-world data sets, we developed a specific member of the new family called the Lomax–Rayleigh {Exponential} and named it by IDNA distribution. the literature on the family of probability distributions for complex applications is expected to use the new family. Two approaches were used to estimate parameters for IDNAI, and their effectiveness was then assessed using Monte Carlo simulation. According to the results, the Maximum likelihood technique performed better than the least square method when evaluated in terms of mean squared error and bias. After applying the suggested distribution to empirical data, a comparison showed that it performed better than some models based on goodness-of-fit standards like the Bayesian Information Criteria (BIC), Akaike Information Criteria (AIC), and others.

Data and software availability

This study does not generate or analyze any new data. The data utilized were obtained from the published article by “Olubiyi AO, Olayemi MS, Olajide OO, Fadugba SE. Modification of a new probability distribution with applications to real life datasets. In: 2024 Int Conf Sci Eng Bus Driving Sustainable Dev Goals (SEB4SDG). IEEE; 2024. pp. 1–6”. DOI: 10.1109/SEB4SDG60871.2024.10629851.

Accessible at: https://ieeexplore.ieee.org/document/10629851²²

The datasets utilized in that study were employed here to assess the efficacy of the suggested IDNAI distribution. All data were acquired from publically available sources and utilized exclusively for comparative modeling purposes.

Figshare- MatLab codes of IDNAI distribution. https://doi.org/10.6084/m9.figshare.30408598.v2²³

Data are available under the terms of the Creative Commons Attribution 4.0 International license (CC-BY 4.0).

Extended data

Figshare: Supplementary Files.

Figshare. Figures and tables of IDNAI distribution. https://doi.org/10.6084/m9.figshare.30408343.v4²⁴

Data are available under the terms of the Creative Commons Attribution 4.0 International license (CC-BY 4.0).

AI Disclosure Statement: Generative AI tools were utilized to help with language refinement and structural clarity while writing this article. We used Microsoft Copilot (October 2025 edition) to assist us write technical descriptions, manage documentation, and make sure everything followed open science norms. The authors individually created all of the scientific information, derivations, and interpretations.

Acknowledgements

The authors appreciate the anonymous reviewers for their constructive feedback that enhanced the quality of this work. The authors would like to acknowledge the support provided by the University of Al-Fallujah Conference, which kindly covered the article processing charges for the publishing of this work.

References

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16. Shannon CE: A mathematical theory of communication. Bell Syst Tech J. 1948; 27(3): 379–423. Publisher Full Text
17. Ibeh GC, Ekpenyoung EJ, Anyiam K, et al.: The Weibull-exponential Rayleigh distribution: Theory and applications. Earthline J Math Sci. 2021; 6(1): 65–86.
18. Moll VH: The integrals in Gradshteyn and Ryzhik. Part 3: Combinations of logarithms and exponentials. Sci Ser A Math Sci (N.S.). 2007; 15(1): 31–36. Reference Source
19. Rényi A: On measures of entropy and information. Proc 4th Berkeley Symp Math Stat Probab. University of California Press; 1961; 1. .
20. Swain JJ, Venkatraman S, Wilson JR: Least-squares estimation of distribution function in Johnson’s translation system. J Stat Comput Simul. 1988; 29(4): 271–297.
21. Rubinstein RY: Simulation and the Monte Carlo Method. Book Review. Technometrics. 1982; 24(2): 167–168.
22. Olubiyi AO, Olayemi MS, Olajide OO, et al.: Modification of a new probability distribution with applications to real life datasets. 2024 Int Conf Sci Eng Bus Driving Sustainable Dev Goals (SEB4SDG). IEEE; 2024; pp. 1–6.
23. Najim A: MatLab codes. figshare. Software. 2025. Publisher Full Text
24. Najim A: Figures of IDNAI distribution functions. figshare. Figure. 2025. Publisher Full Text

Footnotes

[1] Tables of the suggested real parameters, MSE and Bias, Dataset and information criteria

Comments on this article Comments (0)

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Author details Author details

¹ Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, Baghdad Governorate, 10053, Iraq
² Mathematics, University of Baghdad Al-Jaderyia Campus College of Science, Baghdad, Baghdad Governorate, 10070, Iraq

Abbas Najim Salman
Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing

Iden Hassan
Roles: Conceptualization, Formal Analysis, Methodology, Project Administration, Supervision

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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Salman AN and Hassan I. Mathematical and Statistical Properties for a New IDNAI Distribution[a] [version 1; peer review: awaiting peer review]. F1000Research 2026, 15:177 (https://doi.org/10.12688/f1000research.172616.1)

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[1] 1. Al-Hussaini EK, Abdel-Hamid AH: Generation of distribution functions: A survey. J Stat Appl Probab. 2018; 7: 91–103.

[2] 2. Mohammed MJ, Hussein IH: Some estimation methods for new mixture distribution with simulation and application. IOP Conf Ser Mater Sci Eng. 2019; 571: 012014.

[3] 3. Mohammed MJ, Hussein IH: Study of new mixture distribution. ARPN J Eng Appl Sci. 2019; 14(20): 7566–7573.

[4] 4. Shallan RN, Alkanani IH: Statistical properties of generalized exponential Rayleigh distribution. Palest J Math. 2023; 12(Special Issue I): 46–58.

[5] 5. Noaman AA, Suhad AA, Mohammed KH: Constructing a new mixed probability distribution with fuzzy reliability estimation. Period Eng Nat Sci. 2020; 8(2): 590–601.

[6] 6. Lamyaa KH, Rasheed HA, Iden HH: A class of exponential Rayleigh distribution and new modified weighted exponential Rayleigh distribution with statistical properties. Ibn Al-Haitham J Pure Appl Sci. 2023; 6(2): 390–406.

[7] 7. Ghasaq HS, Iden HH: Study of new exponential-Weibull distribution about children with leukemia. AIP Conf Proc. 2024; 3036(1): 040030.

[8] 8. Kalaf BA, Batah FS, AbdulNabi AI: Right truncated Shankar distribution and its properties. Ibn Al-Haitham J Pure Appl Sci. 2025; 38(1): 493–499.

[9] 9. Abdul-Ameer NJ, Kalaf BA: A new hybrid algorithm based on the firefly algorithm and cuckoo search to estimate survival analysis of truncated inverse Gompertz distribution. Iraqi J Sci. 2024; 65(2): 891–906. Publisher Full Text

[10] 10. Hameed BA, Salman AN, Kalaf BA: On estimation of in cased inverse Kumaraswamy distribution. Iraqi J Sci. 2020; 61(4): 845–853.

[11] 11. Alzaatreh A, Lee C, Famoye F: A new method for generating families of continuous distributions. Metron. 2013; 71(1): 63–79.

[12] 12. Alzaatreh A, Famoye F, Lee C: Weibull-Pareto distribution and its applications. Commun Stat Theory Methods. 2013; 42(9): 1673–1691.

[13] 13. Al-Aqtash R, Lee C, Famoye F: Gumbel-Weibull distribution: Properties and applications. J Mod Appl Stat Methods. 2014; 13(2): 11.

[14] 14. Osatohanmwen P, Oyegue FO, Ogbonmwan SM: A new member from the T-X family of distributions: the Gumbel-Burr XII distribution and its properties. Sankhya A. 2019; 81: 298–322.

[15] 15. Aljarrah MA, Lee C, Famoye F: On generating T-X family of distributions using quantile functions. J Stat Distrib Appl. 2014; 1: 1–17.

[16] 16. Shannon CE: A mathematical theory of communication. Bell Syst Tech J. 1948; 27(3): 379–423. Publisher Full Text

[17] 17. Ibeh GC, Ekpenyoung EJ, Anyiam K, et al.: The Weibull-exponential Rayleigh distribution: Theory and applications. Earthline J Math Sci. 2021; 6(1): 65–86.

[18] 18. Moll VH: The integrals in Gradshteyn and Ryzhik. Part 3: Combinations of logarithms and exponentials. Sci Ser A Math Sci (N.S.). 2007; 15(1): 31–36. Reference Source

[19] 19. Rényi A: On measures of entropy and information. Proc 4th Berkeley Symp Math Stat Probab. University of California Press; 1961; 1. .

[20] 20. Swain JJ, Venkatraman S, Wilson JR: Least-squares estimation of distribution function in Johnson’s translation system. J Stat Comput Simul. 1988; 29(4): 271–297.

[21] 21. Rubinstein RY: Simulation and the Monte Carlo Method. Book Review. Technometrics. 1982; 24(2): 167–168.

[22] 22. Olubiyi AO, Olayemi MS, Olajide OO, et al.: Modification of a new probability distribution with applications to real life datasets. 2024 Int Conf Sci Eng Bus Driving Sustainable Dev Goals (SEB4SDG). IEEE; 2024; pp. 1–6.

[23] 23. Najim A: MatLab codes. figshare. Software. 2025. Publisher Full Text

[24] 24. Najim A: Figures of IDNAI distribution functions. figshare. Figure. 2025. Publisher Full Text

Mathematical and Statistical Properties for a New IDNAI Distribution[a]

Abstract

Keywords

1-Introduction

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2- Methods

2-1 Creating a novel IDNAI distribution

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2-2 Properties of the IDNAI distribution

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2-3 Estimation methods for the parameters of IDNAI distribution

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2-4 Comparison of estimators

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Table 1. Shown two cases of real parameters.

Table 2. Bias and MSE of case 1.

Table 3. Bias and MSE of case 2.

2-5 Application for comparison and fitting real-world dataset

Table 4. Dataset of the survival times (in days).

Table 5. Descriptive statistics of dataset.

Table 6. Parameter estimates with different information criteria.

3- Results

3-1 Special cases of the new IDNAI distribution

3-2 Figures of probability functions of IDNAI distribution

Figure 1. IDNAI distribution's pdf according to different values of α, β and λ.

Figure 2. The CDF of IDNAI distribution for different values of α, β and λ.