This article presents two novel goodness-of-fit tests for the Rayleigh distribution, specifically designed for analyzing Type-II censored data. The development of these tests is based on the Kullback–Leibler information criterion. Both tests exhibit consistency, and one of the test statistics possesses a nonnegative property akin to the Kullback–Leibler information. To assess the performance of the proposed tests, a comprehensive simulation study is conducted. The simulation results provide percentile points and power values, offering valuable insights into the effectiveness of the tests in detecting departures from the Rayleigh distribution. Furthermore, a real-life data analysis is included to demonstrate the practical application of the proposed tests. This application showcases how these tests can be utilized to assess the goodness-of-fit of the Rayleigh distribution when analyzing real-world data.

This paper proposes a novel goodness-of-fit test for the Weibull distribution, specifically tailored for Type-II progressively censored data. The critical points of the proposed test statistic are determined through Monte Carlo simulation, which is also used to calculate the Type-I error and power of the test. Finally, the effectiveness of the proposed test is demonstrated by applying it to a real dataset.

This paper introduces a novel divergence measure between two probability distributions, parameterized by a constant α∈ [0,1] . The proposed divergence generalizes well-known measures such as the Kullback–Leibler (KL) and Jensen–Shannon (JS) divergences, providing a flexible and unified framework for distribution comparison. We analyze its key mathematical properties, including convexity, differentiability, and symmetry, and explore its relationships with other divergence measures and invariance characteristics. Furthermore, we demonstrate its practical effectiveness through applications in clustering, anomaly detection, and machine learning, supported by experiments on the Iris and MNIST datasets. Detailed results and visualizations highlight the advantages of the proposed divergence, particularly in adaptive and dynamic scenarios.

We propose a novel entropy-based goodness-of-fit test for the Weibull distribution tailored to fuzzy lifetime data. By transforming fuzzy Weibull observations into exponential form, the method exploits the maximum entropy property of the exponential distribution and computes an integrated entropy-based statistic across fuzzy membership levels, effectively capturing data imprecision. The test’s critical values and statistical power are evaluated via extensive Monte Carlo simulations, demonstrating its reliability. Application to real-world fuzzy lifetime data illustrates the method’s practical effectiveness in reliability and risk assessment under uncertainty, highlighting its robustness and applicability.

Extropy, introduced as a complementary dual to Shannon entropy, provides an alternative measure of uncertainty with valuable interpretations in information theory, statistics, and signal processing. For a continuous random variable with density f, the extropy is defined by J(f) = -1/2∫ f(x)^2 dx , making its estimation closely connected to nonparametric density estimation. In this paper, we propose a new nonparametric estimator for extropy based on a locally tilted Kernel Density Estimator (KDE). The method applies an exponential tilting transformation to the classical KDE, reducing bias in squared-density functionals. We derive theoretical properties including consistency and asymptotic normality of the proposed estimator. Simulation studies illustrate its competitive performance relative to plug-in KDE and k-nearest-neighbor methods, and applications to real-world datasets demonstrate its practical utility. The paper contributes a novel bias-reduced approach to estimating extropy and other integral functionals of density functions.

This study investigates the performance of goodness-of-fit tests for the ARA(infinity)-PLP imperfect maintenance model, with a particular emphasis on entropy-and extropy-based test statistics. Test statistics are constructed using three different approaches: martingale residuals, probability integral transform, and information-based measures. Extensive simulation studies are conducted under several alternative hypotheses, including ARA1, ARA(infinity)-LLP, QR, EGP, and Brown-Proschan models, to evaluate the empirical power of the proposed tests. In addition to numerical power comparisons, graphical analyses are employed to illustrate the behavior of the test statistics and to provide further insight into their sensitivity under different repair scenarios. The simulation results demonstrate that entropy-and extropy-based statistics generally outperform classical goodness-of-fit tests, particularly in detecting deviations from the null model under moderate and severe imperfect repair effects. The consistency observed between graphical patterns and numerical findings further confirms the robustness and interpretability of the proposed procedures. An application to a real dataset related to automobile failure times illustrates the practical effectiveness of the methodology and supports the suitability of the ARA(infinity)-PLP model for real-world repairable systems.

center dot The Lindley distribution is one of the fundamental models applied for reliability models and in the present article, we propose some test statistics for testing the validity of Lindley model based on correcting moments of nonparametric probability density functions of entropy estimators. Critical points and type I error of the tests are obtained and power values of the tests are computed by Monte Carlo simulation. We show that the proposed tests are more powerful than competitor tests. Finally, the proposed tests are illustrated by a real data example.

Developing a powerful goodness-of-fit test for the Inverse Gaussian distribution is of significant importance due to its wide practical application. In this article, we introduce and investigate a novel goodness-of-fit test specifically designed for the Inverse Gaussian distribution. Our method employs a local linear regression approach to estimate Kullback–Leibler information, enhancing the accuracy of the test. The properties of the proposed test statistic are presented. To calculate the test statistic, maximum likelihood estimators, which are straightforward and explicit, are employed to estimate the parameters of the Inverse Gaussian distribution. Critical values and the actual sizes of the proposed test are determined by Monte Carlo simulation. A comprehensive simulation study is conducted to compare the power values of the proposed test with those of other well-known existing tests. Finally, two illustrative examples are presented and analyzed to showcase the practical application of the proposed test.

The Inverse Gaussian distribution finds application in various fields, such as finance, survival analysis, psychology, engineering, physics, and quality control. Its capability to model skewed distributions and non-constant hazard rates makes it a valuable tool for understanding a wide range of phenomena. In this paper, we present a goodness-of-fit test specifically designed for the Inverse Gaussian distribution. Our test uses an estimate of the Gini index, a statistical measure of inequality. We provide comprehensive details on the exact and asymptotic distributions of the newly developed test statistic. To facilitate the application of the test, we estimate the unknown parameters of the Inverse Gaussian distribution using maximum likelihood estimators. Monte Carlo methods are utilized to determine the critical points and assess the actual sizes of the test. A power comparison study is conducted to evaluate the performance of existing tests. Comparing its powers with those of other tests, we demonstrate that the Gini index-based test performs favorably. Finally, we present a real data analysis for illustrative purposes.

In this article, we introduce a new estimator of entropy of continuous random variable. Bias, variance and the mean squared error of the new estimator are obtained and compared with the other existing estimators. The results show that the proposed estimator has a lower mean squared error than its competitors. Then, we propose some goodness of fit tests for Weibull distribution based on the entropy estimators. To assess the effectiveness of the proposed tests, we utilize Monte Carlo simulation to evaluate their power against eighteen different alternatives with varying sample sizes. The results show that the tests are powerful and we can use them in practice. Finally, two real datasets are considered and modeled by the Weibull distribution.

This article presents a novel approach for conducting a goodness-of-fit (GOF) test on Type-II censored data. The method utilizes a new estimator of Kullback-Leibler (KL) information, which is derived from a local linear regression. The properties of the proposed test statistic are discussed. To test for exponentiality using Type-II censored data, the new test statistic is applied. Through a comprehensive simulation study, critical points of the test statistic are determined, and the power of the proposed test is compared with several well-known existing tests. Additionally, two real-life data analyses are presented to provide illustrative examples of the method's application.

In recent years, various goodness-of-fit tests have been developed to identify the underlying distribution of failure data. In this paper, we extend the application of such tests to evaluate the adequacy of imperfect maintenance models for engineering systems. Specifically, we investigate and compare three types of test statistics: those based on martingale residuals, the probability integral transform, and varentropy—a concept derived from information theory. The null hypothesis assumes that the failure times follow the $ARA_{\infty}$ model with a power law process (PLP) as the initial hazard rate. To evaluate the performance of the proposed tests, we conduct extensive simulation studies under different alternative maintenance models (e.g., $ARA_1$, $ARA_{\infty}$–Log Linear Process(LLP)) and varying parameter settings. Our findings show that the power of the tests varies depending on the nature of the alternatives, and varentropy-based statistics outperform others under certain conditions. Finally, we apply the proposed methods to a real dataset (Ambassador vehicle failure times) to assess their practical relevance. The results confirm the validity of the fitted model and demonstrate the usefulness of varentropy-based approaches for detecting subtle deviations in maintenance patterns.

In this paper we propose a new estimator of the entropy of a continuous random variable. The estimator is obtained by modifying the estimator proposed by Vasicek (1976). Consistency of the proposed estimator is proved, and comparisons are made with Vasicek’s estimator (1976), Ebrahimi et al.’s estimator (1994) and Correa’s estimator (1995). The results indicate that the proposed estimator has smaller mean squared error than considered alternative estimators. The proposed estimator is applied to a real data set for illustration.

In this paper, a novel test statistic is introduced to evaluate the goodness of fit of lifetime distributions when dealing with Type II censored data. The test statistic is derived from a local linear regression-based estimator of the Kullback-Leibler information. Extensive analysis is conducted to examine the properties of this test statistic, highlighting its nonnegative nature, as for KL information. The proposed test statistic is applied to various distributions, namely exponential, Weibull, log-normal, and Pareto. Critical values and Type I error rates for the tests are determined, demonstrating their exceptional accuracy and reliability. To further assess their performance, the proposed tests are subjected to Monte Carlo simulations, comparing their power values against alternative tests currently in use. Finally, the proposed method is applied to three real data sets from the engineering reliability aspect to prove their practical versatility.

The Rayleigh distribution is widely used to model events that occur in different fields such as medicine and natural sciences. In this article, we suggest some test statistics for examining the Rayleigh goodness of fit based on the empirical distribution function. Critical points and power of the tests are obtained by Monte Carlo simulation. We show that the proposed tests have a good performance against different alternatives and therefore these tests can be confidently used in practice. Finally, the proposed tests are illustrated by real data examples.

In many life-testing and reliability experiments, censoring of data is often employed to reduce costs and testing time. However, the conventional Type-I and Type-II censoring schemes may not offer the desired flexibility. To address this limitation, researchers have developed progressive censoring methods. In this article, we propose a general goodness-of-fit test for progressively Type-II censored data by utilizing a local linear regression estimate of the Kullback-Leibler information. We establish the consistency of the proposed test, demonstrating its reliability and accuracy. Furthermore, we employ the proposed test statistic to assess the exponentiality assumption based on progressively Type-II censored data. To evaluate the performance of the test, we compute power values through Monte Carlo simulations under various progressively Type-II censoring schemes. Our findings suggest that the proposed test demonstrates strong power and surpasses the performance of existing tests. To illustrate the practical application of our approach, we present two real datasets from the progressive censoring literature. These datasets serve as compelling examples to showcase the effectiveness and utility of our proposed test.

The logistic distribution has been used for various growth models, and is used in a certain type of regression, known appropriately as logistic regression. In this article, we propose a new goodness of fit test for the logistic distribution based on the negative cumulative residual extropy introduced by Tahmasebi and Toomaj (2022). The mean, variance and the other properties of the test statistic is presented. Percentage points of the test statistic are obtained and then power of the test against different alternatives are reported. The results of a simulation study show the test is competitive in terms of power. The proposed statistic is easy to compute and a real data set is used to illustrate the application of the proposed test.

This study focuses on developing goodness-of-fit test statistics for the Erlang-2 model. The proposed statistics utilize sample entropy as their foundation. We derive the properties and critical values of these tests for various sample sizes. Subsequently, we conduct an empirical power comparison through Monte Carlo simulations, exploring different alternatives and sample sizes. The new tests demonstrate superior power performance compared to existing tests. A discussion of the findings is included, highlighting the most effective tests for different categories of alternatives. Lastly, we analyze a real dataset to illustrate the application of our methods.