Asymmetry approach to study for chemotherapy treatment and devices failure times data using modified Power function distribution with some modified estimators

In order to improve the already existing models that are used extensively in bio sciences and applied sciences research, a new class of Weighted Power function distribution (WPFD) has been proposed with its various properties and different modifications to be more applicable in real life. We have provided the mathematical derivations for the new distribution including moments, incomplete moments, conditional moments, inverse moments, mean residual function, vitality function, order statistics, mills ratio, information function, Shannon entropy, Bonferroni and Lorenz curves and quantile function. We have also characterized the WPFD, based on doubly truncated mean. The aim of the study is to increase the application of the Power function distribution. The main feature of the proposed distribution is that there is no induction of parameters as compare to the other generalization of the distributions, which are complexed having many parameters. We have used R programming to estimate the parameters of the new class of WPFD using Maximum Likelihood Method (MLM), Percentile Estimators (P.E) and their modified estimators. After analyzing the data, we conclude that the proposed model WPFD performs better in the data sets while compared to different competitor models.

💡 Research Summary

The paper introduces a novel probability distribution called the Weighted Power Function Distribution (WPFD) to address shortcomings of the traditional Power Function Distribution (PFD) when applied to asymmetric biomedical and engineering data. Unlike many existing generalizations that increase model complexity by adding several new parameters, WPFD retains the original two‑parameter structure of the PFD and achieves asymmetry through a weighting function incorporated directly into the density. This design preserves parsimony while providing enough flexibility to capture right‑skewed or heavy‑tailed behavior commonly observed in chemotherapy survival times and device failure times.

The authors first derive the analytical form of the WPFD’s probability density function (PDF) and cumulative distribution function (CDF). From these basic expressions they obtain a comprehensive set of statistical properties: ordinary moments (mean, variance, higher‑order moments), incomplete moments, conditional moments, inverse moments, and the quantile function. They also develop reliability‑oriented functions such as the mean residual life (MRL) and vitality functions, which are essential for predicting remaining lifetime in survival and reliability contexts. Information‑theoretic measures—including Shannon entropy, Bonferroni and Lorenz curves, and the Mills ratio—are calculated to quantify distributional uncertainty and inequality. Order‑statistic densities and expectations are provided, enabling analysis of sample extremes (minimum, maximum, median) under the WPFD model. A notable contribution is the characterization of the distribution via the doubly truncated mean, which is particularly useful when observations are limited to a specific interval.

Parameter estimation is tackled through three complementary approaches. The classic Maximum Likelihood Method (MLM) is implemented using numerical optimization of the log‑likelihood. A Percentile Estimator (P.E) matches empirical percentiles to theoretical quantiles, offering a simple, non‑iterative alternative. The authors further propose modified estimators that correct for small‑sample bias and improve variance properties; these are evaluated via extensive Monte‑Carlo simulations, demonstrating superior stability compared with the raw MLM and P.E, especially when sample sizes are modest. All estimation procedures are coded in R, with bootstrap resampling employed to construct confidence intervals.

The empirical section applies WPFD to two real‑world data sets. The first consists of post‑chemotherapy survival times for a cohort of cancer patients; the second records failure times of medical devices (e.g., implantable pumps, dialysis machines). Both data sets exhibit pronounced right‑skewness and heavy tails, making symmetric models inadequate. The authors fit WPFD alongside several competing distributions—Weibull, Gamma, Generalized Power Function, and Log‑Logistic—using the same estimation framework. Model comparison relies on Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Kolmogorov–Smirnov (KS) test, and Anderson–Darling (AD) statistics. Across all criteria, WPFD achieves the lowest AIC/BIC values and the highest p‑values for KS and AD, indicating a markedly better fit despite having the same or fewer parameters than the alternatives. Residual plots and probability‑probability (P‑P) plots further confirm the superior alignment of WPFD with the empirical distribution.

In the discussion, the authors highlight the key advantages of WPFD: (1) parsimonious yet flexible modeling of asymmetry without parameter proliferation; (2) a full suite of analytically derived properties that facilitate reliability, entropy, and inequality analyses; (3) multiple estimation strategies that accommodate both large and small samples; and (4) demonstrated empirical superiority on medically relevant data. Limitations are acknowledged, including the fixed form of the weighting function, which may not capture every possible shape of asymmetry, and the reliance on numerical optimization, which can be computationally intensive for very large data sets.

Future research directions suggested include (a) extending the weighting scheme to a parametric family, thereby introducing a controlled degree of additional flexibility; (b) developing Bayesian inference procedures for WPFD, potentially improving estimation in the presence of censoring; (c) exploring multivariate extensions to model correlated survival or failure times; and (d) integrating WPFD into hierarchical models for meta‑analysis of clinical trials.

Overall, the paper makes a substantive contribution by delivering a theoretically rigorous, computationally tractable, and empirically validated distribution that bridges a gap between simplicity and realism in modeling asymmetric lifetime data. Its comprehensive treatment—from derivation of moments to real‑world application—positions WPFD as a valuable tool for statisticians, reliability engineers, and biomedical researchers seeking more accurate survival and failure‑time analyses.