Nonparametric Survival Analysis estimation and comparison with Algorithm

1. Introduction

Nonparametric estimation has become an important statistical approach for modeling complex data without imposing restrictive parametric assumptions. In survival and reliability analysis, observed data are frequently positively supported, skewed, and bounded below by zero, which makes flexible estimation methods particularly important. Kernel density estimation (KDE) is one of the most widely used smoothing techniques for estimating unknown probability density functions from observed samples. However, classical symmetric kernels may suffer from substantial boundary bias when applied to positive-support data, especially near zero. To overcome these limitations, asymmetric kernel estimation methods have been developed using positively supported distributions such as Gamma, Inverse-Gaussian, and related skewed families.^1–3 These kernels improve estimation accuracy in bounded domains and provide better adaptability for skewed survival and reliability data. Recent developments in asymmetric kernel estimation have demonstrated improved performance in survival applications, hazard estimation, and density reconstruction for nonnegative random variables.^4–6 In survival analysis, flexible nonparametric estimation of the survival function and hazard function is essential for accurately representing lifetime behavior without relying on restrictive parametric assumptions. Kernel-based survival estimation provides a useful alternative to classical approaches by combining smoothing flexibility with data-driven estimation.^7,8 In addition, several recent studies have emphasized the importance of transformed survival models and algorithm-based estimation methods in reliability and lifetime analysis.^9–11 This study adopts a kernel-family framework rather than relying on a single asymmetric kernel. Several positive-support kernel families derived from Log-Lindley, Birnbaum–Saunders, and Inverse-Weibull distributions are constructed and evaluated under unified comparison criteria. The proposed framework extends asymmetric kernel estimation from density estimation to survival and hazard estimation while integrating likelihood cross-validation and Silverman-type bandwidth selection methods. Recent related work and applications can be found in Refs. 12–20. The main contributions of this study can be summarized as follows:

2. Asymmetric Kernel Families on (0, ∞)

Kernel density estimation (KDE) is one of the most widely used nonparametric techniques for estimating unknown probability density functions. Given a random sample x₁, x₂, …, x_n from a positive-support distribution, the KDE provides a smooth estimate of the underlying density by averaging localized kernel functions., which leads to a smooth and flexible estimate of the probability density function by kernel functions.²¹ Given observations x₁, x₂, …, x_n with x _i > 0, the kernel density estimator is defined as:

where K (t; x _i, h) is a nonnegative asymmetric kernel centered around x_i and controlled by the bandwidth parameter h. The kernel integrates to one over (0, ∞), ensuring that the estimator remains a valid density function on positive support. In asymmetric KDE, K depends on x_i so that the kernel adapts locally to the positive support and reduces boundary bias.

The following asymmetric kernel families are considered in this study. Each kernel is defined on the positive semi-axis and parameterized locally through the observation x_i and the bandwidth parameter h.

Using the transformation Y = exp (−Z), where Z follows the Lindley distribution, the induced Log-Lindley density on (0,1) is obtained as:

Define $T_i=-x_{i}log\left(Y\right)\in (0,\infty ).$ Then a convenient log-Lindley-based kernel is:

A practical local bandwidth parameterization is adopted through θ _i = 1/h, allowing the kernel shape to adapt according to the smoothing level. Birnbaum-Sauders (fatigue-life) kernel.

Considering the standard normal pdf $\varnothing (.)$ and for the shape parameter $\alpha >0\mathrm{\text{and scale parameter}}\beta >0,$

One can use practically local ${\beta }_i=x_i,{\alpha }_i=h.$

The Birnbaum–Saunders kernel is suitable for lifetime and fatigue-type data due to its positive support and skewness flexibility.

For parameters $\beta >0,\gamma >0$ ,

These asymmetric kernels provide flexible local smoothing mechanisms while preserving the positive support of survival data. Compared with symmetric kernels, they reduce boundary distortion and improve estimation accuracy near zero.

3. Benchmark Kernel Families

To evaluate the performance of the proposed asymmetric kernel families, several benchmark kernels commonly used in positive-support density estimation are considered for comparison. These include Gamma, Inverse-Gaussian, Lindley-based, and symmetric Epanechnikov-type kernels. Consider the following kernels:

on the log-scale: let $u=(logt-log{x}_i)/h$ and

Then the positive support version is

The multiplicative factor 1/ t arises from the logarithmic transformation Jacobian and guarantees proper normalization on the positive semi-axis.

These benchmark kernels provide reference models for evaluating the flexibility and estimation performance of the proposed asymmetric kernel-family framework.

4. Bandwidth selection

Estimator performance is significantly impacted by bandwidth selection. We employ two complementary approaches: Silverman-type rule adapted to positive support (pilot scale estimate) and Likelihood cross-validation (LCV): choose h that maximizes the leave-one-out log-likelihood

5. Nonparametric survival and hazard estimation

Once the kernel density estimator $\widehat{f}\left(t\right)$ is obtained, the corresponding distribution and survival functions can be computed numerically.

The corresponding hazard function is estimated as

In the presence of censored observations, the Kaplan–Meier estimator ${\widehat{S}}_{\text{KM}}\left(t\right)$ is used as a benchmark nonparametric survival estimator and compared with the kernel-based survival estimate. Kernel-based survival estimation provides a smooth alternative to empirical survival estimation and allows flexible representation of lifetime behavior on positive support.

Algorithm of Asymmetric Kernel-Family Survival Estimation

Step 1: preprocess data and define minimum and maximum of data

Step 2: Preprocess data and define grid T on (min(t), max(t)).

Step 3: For each kernel family j:

    (a) Select bandwidth h _j via likelihood cross-validation (or Silverman-type rule).

    (b) Compute density ${\widehat{f}}_j\left(t\right)$ on T.

    (c) Numerically integrate to obtain ${\widehat{F}}_j\left(t\right)$ and ${\widehat{S}}_j\left(t\right)$  = 1− ${\widehat{F}}_j\left(t\right)$ .

    (d) Compute hazard ${\widehat{h}}_j\left(t\right)$ = ${\widehat{f}}_j\left(t\right)$ /max( ${\widehat{S}}_j\left(t\right)$ , ε).

Step 4: Compute Kaplan–Meier survival ${\widehat{S}}_{\text{KM}}\left(t\right)$ (benchmark, when censoring exists).

Step 5: Fit parametric models M _k by MLE under censoring and compute S _k(t), h _k(t).

Step 6: Evaluate kernels and models using multiple criteria (Section 6) and select the best performer.

Step 7: Report tables and figures for $\widehat{f}$ (t), $\widehat{S}$ (t), and $\widehat{h}$ (t).

6. Simulation study

We assess performance under two data-generating scenarios to represent different shapes and tail behaviors (e.g., Gamma-like and Lognormal-like). For each scenario, we consider several sample sizes (e.g., n = 25, 50, 100, 200) and repeat the experiment over R replications. For each replication, we compute the kernel estimates and evaluate them using integrated error measures and predictive (CV) scores.

Recommended criteria (replace/extend beyond ISE): Integrated Absolute Error (IAE) Integrated Mean Squared Error (IMSE), Hellinger distance, and likelihood cross-validation score (LCV).

This table reports the Integrated Mean Squared Error (IMSE) of the estimated density under Scenario A using two bandwidth selectors: a Silverman-type rule and likelihood cross-validation (LCV). For each kernel family, the corresponding selected bandwidth values are also reported. Lower IMSE indicates better estimation accuracy. As shown in Table 1, kernel-family performance under, Scenario A varies across bandwidth selection methods, highlighting the impact of the bandwidth choice on estimation accuracy.

This table reports the Integrated Absolute Error (IAE) under Scenario B using two bandwidth selectors (Silverman-type and LCV). The selected bandwidth values are included for each kernel family. Lower IAE indicates better estimation accuracy. Table 2 summarizes kernel-family performance under Scenario B, where accuracy is evaluated using IAE under both Silverman-type and LCV bandwidth selection.

7. Application to real survival data

7.2 Figures

Figures 1–4 summarize the real-data application of the proposed positive-support kernel-family framework. We present kernel-based density estimates under different bandwidth selection strategies and compare the resulting fitted curves with the empirical distribution of the data. In addition, we report normalized error/predictive measures to quantify performance across kernels and bandwidth selectors, and we compare survival curves to evaluate how well the nonparametric estimators reproduce the empirical survival pattern. Together, these figures illustrate the impact of bandwidth selection (Silverman vs LCV), the differences between kernel families on positive support, and the resulting consequences for density and survival estimation.

Figure 1. Displays kernel-family density estimates for the catheterization data using both Silverman’s rule and likelihood cross-validation (LCV) bandwidths; the histogram represents the empirical distribution, while the solid curves correspond to asymmetric kernel estimates.

Figure 2. Presents kernel-family density estimates for the real data using the LCV-selected bandwidth, highlighting differences among positive-support (asymmetric) kernel families.

Figure 3. Summarizes normalized error and predictive measures across kernel families and bandwidth selectors, including IMSE and IAE (computed against a lognormal reference fit on a dense grid) and the LCV log-likelihood (higher values indicate better fit).

Figure 4. Survival function comparison for the real data. The empirical survival (1 − ECDF; equivalent to Kaplan–Meier with no censoring) is contrasted with the best-performing kernel-family survival estimate and the best parametric survival model selected by information criteria.

8. Conclusions

This paper provided a kernel-family system for positive-support nonparametric estimation and applied it to survival evaluation by estimating survival and hazard functions. In contrast to single-kernel methods, the family-based design enables practitioners to choose kernels that correspond to the data’s tail characteristics and boundary behavior. An efficient, data-driven method for choosing bandwidth is likelihood cross-validation. Comparing kernel-based survival with Kaplan-Meier and non-Weibull parametric models in real survival analysis reveals the useful trade-off between interpretability/parsimonious structure (parametric) and flexibility (nonparametric).

All tables have been labeled sequentially ( Tables 1–8), cited in the text, and provided with complete.