Different Classical and Bayesian Methods of Estimation of the Power Log-Logistic Distribution with Applications

In this article, we introduce an enhanced version of the log-logistic model, termed the Kumaraswamy alpha-power log-logistic (KAPLL) distribution. The KAPLL model expands upon the traditional log-logistic distribution and several well-established distributions. We investigate the mathematical properties of the KAPLL model, highlighting its ability to effectively model various aging and failure criteria. The KAPLL distribution exhibits remarkable flexibility in modeling various types of hazard rate behaviors. It is capable of accommodating a wide range of shapes, including increasing, decreasing, J-shaped, reversed J-shaped, bathtub-shaped, inverted bathtub-shaped, and even more complex forms such as decreasing–increasing–decreasing failure rates. The KAPLL distribution is characterized by its capacity to exhibit both symmetric and asymmetric shapes in its density function. The proposed KAPLL model overcomes key limitations of existing LL-based generalizations by offering enhanced flexibility in modeling diverse hazard rate shapes and tail behaviors. We estimate the KAPLL parameters using eight classical estimation methods. Comprehensive simulation results are presented and ranked to identify the most effective approach for estimating KAPLL parameters, which we believe will be of great interest to engineers and applied statisticians. To further demonstrate the versatility of the KAPLL distribution, we analyze five real-world datasets from reliability, engineering, biomedical, and environmental sciences, highlighting its flexibility relative to other extensions of the log-logistic model. Likelihood ratio tests conducted across five real datasets confirm that the KAPLL model provides a statistically significant improvement over the baseline log-logistic distribution.

This paper introduces a new capability index <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>C</m:mi> <m:mi>QT</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(C_{\rm QT})} which is based on an asymmetric loss function named Quadratic-Type Squared Error Loss Function (QTSELF). To estimate the proposed index <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>C</m:mi> <m:mi>QT</m:mi> </m:msub> </m:math> {C_{\rm QT}} , we have used five different classical methods of estimation (namely maximum likelihood, least squares and weighted least squares, Cramér–von-Mises, Anderson–Darling and maximum product of spacing) and their performances are compared. Additionally, three bootstrap confidence intervals, namely standard bootstrap ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒮</m:mi> </m:math> {\mathcal{S}} -boot), percentile bootstrap ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒫</m:mi> </m:math> {\mathcal{P}} -boot) and Student’s t bootstrap ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒯</m:mi> </m:math> {\mathcal{T}} -boot) are constructed. Their performances are compared in terms of average widths and coverage probabilities through a Monte-Carlo simulation study. Finally, two datasets are taken from electronic industries to demonstrate the practical utilities of the proposed estimation techniques and confidence intervals.

The notion that the occurrence of an event is surprising has been discussed in the literature without adequate details. By definition, a surprise index is an index by which how surprising an event is may be determined. Since its inception, this index has been evaluated for univariate discrete probability models, such as the binomial, negative binomial, and Poisson probability distributions. In this article, we derive and discuss using numerical studies, in addition to the above-mentioned probability models, surprise indices for several other univariate discrete probability models, such as the zero-truncated Poisson, geometric, Hermite, and Skellam distributions, by adopting a established strategy and using the Mathematica, version 12 software. In addition, we provide symbolical expressions for the surprise index for several univariate continuous probability models, which has not been previously discussed. For illustrative purposes, we present some possible real-life applications of this index and potential challenges to extending the notion of the surprise index to bivariate and higher dimensions, which might involve ubiquitous normalizing constants.

This study presents the Kumaraswamy Marshal–Olkin Flexible Weibull Extension (KMO-FWE) distribution, a newly developed probability model designed to extend existing lifetime distributions. This extension enhances the ability to model various hazard rate patterns with greater flexibility. The study explores the fundamental properties of this distribution, including the cumulative distribution function (CDF), probability density function (PDF), survival function, hazard function, moments, moment generating function, quantile function, and order statistics. To estimate the distribution’s parameters, several statistical methods are assessed, with a primary focus on Maximum Likelihood Estimation (MLE). Additionally, alternative estimation techniques such as Maximum Product Spacing (MPS), Least Squares (LS), Weighted Least Squares (WLS), and Percentile Estimation (PE) are considered. A Monte Carlo simulation study is conducted to evaluate the accuracy and efficiency of these estimation approaches under both complete data and Type-II censoring conditions. The results indicate that MLE is highly effective for large sample sizes but may introduce bias in smaller samples. In contrast, MPS provides stable parameter estimates under more complex conditions. While LS and WLS are computationally straightforward, their performance diminishes in the presence of censoring. Meanwhile, PE proves beneficial in reliability analysis based on percentile estimation. Overall, the findings highlight the KMO-FWE distribution as a versatile and reliable model, suitable for applications in reliability engineering, survival analysis, and biomedical research.

Survival Analysis is a research method that examines the survival time of individuals or experimental units in relation to events such as death, disease, recovery, or other experiences. This study utilizes a parametric survival analysis model with a 2 parameter log logistic distribution and Maximum Likelihood Estimation (MLE) method to analyze the survival of students during their study period. The log logistic distribution is chosen due to its ability to capture early or late failure patterns. The objective of this research is to analyze type I censored survival data using the log logistic distribution applied to secondary data on student study duration. The dataset consists of 98 observations. The calculated values for the β and γ parameters of the 2 parameters log logistic distribution are 2.12831 and 0.0918891, respectively. The probability of students completing their studies by semester 8 (hazard function h(8)) is 0.370102, while the probability of students continuing their studies in semester 9 (survival function s(9)) is 0.320817.

Frequently in inference, the observed data are modeled as a sample from a continuous probability model, implying the observed data are precisely measured. Usually, the actual data available to the investigator are discrete–-either because they are rounded,meaning the exact measurement is within an interval defined by some small measurement unit related to the precision of the measuring device, or because the data are discrete, meaning the time periods until the event of interest are countable instead of continuous. This article is motivated by the common practice of testing for duration dependence (non constant hazard function) in economic and financial data using the continuous Weibull distribution when the data are discrete. A simulation study shows that biased parameter estimates and distorted hypothesis tests result when the degree of discretization is severe. When observations are rounded, as in measuring the time between stock trades, it is proper to treat them as interval-censored. When observations are discrete, as in measuring the length of stock runs, a discrete hazard function must be specified. Both cases are examined in simulation studies and demonstrated on financial data.

The exponential distribution remains a cornerstone of probability theory, particularly valued in reliability engineering, survival analysis, and risk management for its computational efficiency and distinctive memoryless property. While mathematically elegant, this distribution's constrained single-parameter framework presents significant limitations when modeling complex empirical phenomena, especially those displaying non-monotonic failure patterns. Academic literature documents numerous extensions to address these constraints, including the inverse exponential distribution and its exponentiated variants, which achieve greater modeling flexibility by introducing additional shape parameters. Building upon these developments, the current study presents an enhanced probability model through the innovative synthesis of the Inverse Exponential distribution with the Sine Topp-Leone Exponentiated-G (STLEG) family, yielding the novel Sine Topp-Leone Exponentiated Inverse Exponential (STLEIE) Distribution. Our investigation encompasses a rigorous analysis of the model's fundamental statistical properties, including its survival characteristics, hazard function behavior, moment structure, quantile properties, and order statistics. For parameter estimation, we implement and compare two established methodologies: the widely-used Maximum Likelihood Estimation (MLE) approach and the Maximum Product of Spacing (MPS) technique, both renowned for their statistical efficiency. Comprehensive Monte Carlo simulations demonstrate the consistent performance of these estimators, with both bias and Root Mean Squared Error (RMSE) metrics showing progressive improvement as sample sizes increase. Empirical validation through application to two real-world datasets reveals the STLEIE distribution's superior modeling capabilities compared to existing alternatives, as evidenced by multiple goodness-of-fit criteria. The concluding comparative analysis substantiates the practical utility of the proposed model, confirming its enhanced flexibility and improved performance across diverse data scenarios, thereby offering researchers and practitioners a more robust analytical tool for complex statistical modeling applications.

Reliability estimation of [formula omitted]-out-of-[formula omitted] system in a multicomponent stress–strength dependent model based on copula function

The accelerated life testing is the key methodology of evaluating product reliability rapidly. This paper presents statistical inference of Gompertz distribution based on unified hybrid censored data under constant-stress partially accelerated life test (CSPALT) model. We apply the stochastic expectation-maximization algorithm to estimate the CSPALT parameters and to reduce computational complexity. It is shown that the maximum likelihood estimates exist uniquely. Asymptotic confidence intervals and confidence intervals using bootstrap-p and bootstrap-t methods are constructed. Moreover the maximum product of spacing (MPS) and maximum a posteriori (MAP) estimates of the model parameters and accelerated factor are discussed. The performances of the various estimators of the CSPALT parameters are compared through the simulation study. In summary, the MAP estimates perform superior than MLEs (or MPSs) with respect to the smallest MSE values.

A new extended rayleigh distribution with applications of COVID-19 data

A novel two-parameter Quasi-XLindley distribution that extends the current XLindley distribution is proposed in this paper. The probability density function of this distribution can be skewed to the left and has symmetric shapes. Mathematical properties of this new distribution are derived including moments and associated shape measures along with reliability properties and actuarial measures. Parameters of this distribution are estimated using different estimation strategies e.g. maximum likelihood, Anderson Darling, maximum product spacing and Cramér-von Mises, Least-squares and weighted Least-squares. A comprehensive simulation study is given to appraise the performance of the suggested estimators. Some real-life data applications of this distribution are also furnished at the end including the data of cancer patients.

This paper discussed robust estimation for point estimation of the shape and scale parameters for generalized exponential (GE) distribution using a complete dataset in the presence of various percentages of outliers. In the case of outliers, it is known that classical methods such as maximum likelihood estimation (MLE), least square (LS) and maximum product spacing (MPS) in case of outliers cannot reach the best estimator. To confirm this fact, these classical methods were applied to the data of this study and compared with non-classical estimation methods. The non-classical (Robust) methods such as least absolute deviations (LAD), and M-estimation (using M. Huber (MH) weight and M. Bisquare (MB) weight) had been introduced to obtain the best estimation method for the parameters of the GE distribution. The comparison was done numerically by using the Monte Carlo simulation study. The two real datasets application confirmed that the M-estimation method is very much suitable for estimating the GE parameters. We concluded that the M-estimation method using Huber object function is a suitable estimation method in estimating the parameters of the GE distribution for a complete dataset in the presence of various percentages of outliers.

ABSTRACTWe consider the generalized extended exponential-Weibull distribution, which can be very useful for modelling non-monotonic failure rate function, and provide a detailed study of its structural properties. This model includes at least thirteen sub-models and some of them are very known such as the Weibull, generalized Weibull, exponential-Weibull, and exponentiated generalized linear exponential distributions. We also consider different estimation procedures for estimating the model parameters, namely: maximum likelihood, least-square, weighted least-square, maximum product of spacings, Cramér-von-Mises, and Anderson-Darling methods. We also conduct Monte Carlo simulation experiments to assess the finite sample properties of the proposed estimation methods. The usefulness of the distribution is illustrated by means of two real data sets to prove its versatility in practical applications.

A simple unified formulation for generating an estimator of a univariate distribution (not necessarily continuous) is proposed. It is a variant of the maximum likelihood method and also a generalization of the maximum product of spacings (MPS) method for estimating continuous univariate distributions. The general formulation is then applied to monotone hazard rate families which arise frequently in applications. It is shown that any asymptotic MPS estimator for any family of distributions with monotone hazard rate is always strongly consistent (whether the setting is parametric or not). The MPS estimator of a distribution function with a monotone hazard rate can be derived explicitly and is asymptotically minimax for the Kolmogorov-Smirnov type loss functions.

The residential minimum water demand characterisation is of fundamental importance for water distribution system management. During the minimum consumption, indeed, maximum pressures are on network pipes, and at the same time, tank levels rise. The water consumption analysis during the period of low demand and high pressure is thus of great interest for leakage estimation due to the increase in water loss with pressure. In order to contribute to the study of and forecast the daily minimum residential water demand, some probability distributions were tested by means of statistical inferences on a data set collected from different District Metering Areas (DMAs), showing that the stochastic minimum flow demand is defined by the Log-Normal (LN), Gumbel (Gu) and Log-Logistic (LL) distributions, as an extreme minimum value. With reference to the analysed DMAs, the parameters of such statistical distributions were estimated and the relationships are provided as a function only of the supplied users for different DMAs. The data were analysed with 1 h intervals of discretisation, with the aim of providing a useful guide to water utilities, which usually manage water distribution system data with such a resolution time. Indeed, once the minimum residential flow consumption at a 1 h interval was estimated as a function of the user number, by subtracting it to the inflow measured, it is possible to estimate the leakages rate at the DMA.