The Kumaraswamy Distribution: Statistical Properties and Application

Industrial revolution leads to the manufacturing of multicomponent products; to guarantee the sufficiency of the product and consumer satisfaction, the producer has to study the lifetime of the products. This leads to the use of bivariate and multivariate lifetime distributions in reliability engineering. The most popular and applicable is Marshall–Olkin family of distributions. In this paper, a new bivariate lifetime distribution which is the bivariate inverted Kumaraswamy (BIK) distribution is found and its properties are illustrated. Estimation using both maximum likelihood and Bayesian approaches is accomplished. Using different selection criteria, it is found that BIK provides the best performance compared with other bivariate distributions like bivariate exponential and bivariate inverse Weibull distributions. As a generalization, the multivariate inverted Kumaraswamy (MIK) distribution is derived. Few studies have been conducted on the multivariate Marshall–Olkin lifetime distributions. To the best of our knowledge, none of them handle estimation process. In this paper, we developed an algorithm to show how to estimate the unknown parameters of MIK using both maximum likelihood and Bayesian approaches. This algorithm could be applied in estimating other Marshall–Olkin multivariate lifetime distributions.

In this article, a new generalization of the Topp-Leone distribution with a unit interval, namely Mixed Topp-Leone-Kumaraswamy distribution is defined and studied. The mathematical properties of this mixing distribution are described. Moments, quantile function, R?nyi entropy, incomplete moments and moments of residual are obtained for the new Mixed Topp-Leone - Kumaraswamy distribution. The maximum likelihood (MLE), Crans (CM) , Percentile (PM) and Particle Swarm Optimization(PSO) estimators of the parameters are derived. The percentile Method is more efficient method as compred to the others. Two real data sets are used to illustrate an application and superiority of the proposed distribution.

This paper studies estimation of the parameters of the generalized Gompertz distribution based on ranked-set sample (RSS). Maximum likelihood (ML) and Bayesian approaches are considered. Approximate confidence intervals for the unknown parameters are constructed using both the normal approximation to the asymptotic distribution of the ML estimators and bootstrapping methods. Bayes estimates and credible intervals of the unknown parameters are obtained using differential evolution Markov chain Monte Carlo and Lindley’s methods. The proposed methods are compared via Monte Carlo simulations studies and an example employing real data. The performance of both ML and Bayes estimates is improved under RSS compared with simple random sample (SRS) regardless of the sample size. Bayes estimates outperform the ML estimates for small samples, while it is the other way around for moderate and large samples.

The goal of the present work is to estimate the nonlinear correlation between two random variables when the sample is drawn from a Farlie-Gumbel-Morgenstern (FGM)-type bivariate gamma distribution.In the context of estimating the dependence parameter, a maximum likelihood (ML) methodology is used.Thus, the present work offers ML estimators based on simple random sampling (SRS) and ranked set sampling (RSS).Additionally, we consider generalized modified RSS (GMRSS), which only requires a single rank to obtain a sample.Using GMRSS, we aim to observe the effect of the rth order statistic and its concomitant on the ML estimator.According to the Monte Carlo simulation, it is clearly seen that RSS provides an ML estimator as efficient as the ML estimator based on SRS.On the other hand, it appears that the ML estimator based on GMRSS (with minimum or maximum ranked pairs) is the best option among the studied ML estimators.Moreover, these findings are made even more meaningful by the fact that GMRSS is easier to obtain than SRS and RSS.

The ranked set sampling (RSS) is an effective scheme popularly used to produce more precisely estimators. Despite its popularity, RSS suffers from some drawbacks which includes high sensitivity to outliers and it cannot sometimes be applicable when the population is relatively small. To overcome these limitations, varied L ranked set sampling (VLRSS) is recently introduced. It is shown that VLRSS scheme enjoys with many interesting properties over RSS and also encompasses several existing RSS schemes. In addition, it is also helpful for providing precise estimates of several population parameters. To fill this gap, this article extends the work and address the estimation of based ℛ on VLRSS when the strength and stress both follow exponential distribution. Maximum likelihood approach as well as Bayesian method are considered for estimating ℛ. The Bayes estimators are obtained by using gamma distribution under general entropy loss function and LINEX loss function. The performance of the estimators based on VLRSS are investigated by a simulation study as well as a real dataset relevant to industrial field. The results reveal that the proposed estimators are more efficient relative to their analogues estimators under L ranked set sampling given that the quality of ranking is fairly good.

In this paper, we introduce and study the size-biased form of Kumaraswamy\ndistribution. The Kumaraswamy distribution which has drawn considerable\nattention in hydrology and related areas was proposed by Kumarswamy. The new\ndistribution is derived under size-biased probability of sampling taking the\nweights as the variate values. Various distributional and characterizing\nproperties of the model are studied. The methods of maximum likelihood and\nmatching quantiles estimation are employed to estimate the parameters of the\nproposed model. Finally, we apply the proposed model to simulated and real data\nsets.\n

Bayesian estimation and credible interval of stress–strength reliability for two-parameter bathtub-shaped lifetime (TPBL) distribution are considered based on simple random sampling (SRS), ranked set sampling (RSS) and maximum ranked set sampling with unequal samples (MRSSU). Monte Carlo simulations are used to compare the performances of different proposed methods for Bayesian inferences on R. The results of simulation show that the estimator of the stress–strength reliability based on MRSSU and RSS are more efficient than SRS. Also, we make a comparison between the performance of the proposed methods in this paper and Tarvirdizade and Ahmadpour's [Estimation of the stress–strength reliability for the two-parameter bathtub-shaped lifetime distribution based on upper record values. Stat Methodol. 2016;31:58–72] method, which is based on upper record values. Finally, a real data set has been analyzed for illustrative purposes.

This article discusses the problem of estimation with step stress partially accelerated life tests using Type-II progressively censored samples. The lifetime of items under use condition follows the two-parameters inverted Kumaraswamy distribution. The maximum likelihood estimates for the unknown parameters are computed numerically. Using the property of asymptotic distributions for maximum likelihood estimation, we constructed asymptotic interval estimates. The Bayes procedure is used to calculate estimates of the unknown parameters from symmetrical and asymmetric loss functions. The Bayes estimates cannot be obtained explicitly, therefor the Lindley's approximation and the Markov chain Monte Carlo technique are used to obtaining the Bayes estimates. Furthermore, the highest posterior density credible intervals for the unknown parameters are calculated. An example is presented to illustrate the methods of inference. Finally, a numerical example of March precipitation (in inches) in Minneapolis failure times in the real world is provided to illustrate how the approaches will perform in practice.

ABSTRACTThe minimum variance unbiased estimators (MVUEs) of the parameters for various distributions are extensively studied under ranked set sampling (RSS). However, the results in existing literatures are only locally MVUEs, i.e. the MVUE in a class of some unbiased estimators is obtained. In this paper, the global MVUE of the parameter in a truncated parameter family is obtained, that is to say, it is the MVUE in the class of all unbiased estimators. Firstly we find the optimal RSS according to the character of a truncated parameter family, i.e. arrange RSS based on complete and sufficient statistics of independent and identically distributed samples. Then under this RSS, the global MVUE of the parameter in a truncated parameter family is found. Numerical simulations for some usual distributions in this family fully support the result from the above two-step optimizations. A real data set is used for illustration.

ABSTRACTRecently, a hybrid ranked set sampling (HRSS) scheme has been proposed in the literature. The HRSS scheme encompasses several existing ranked set sampling (RSS) schemes, and it is a cost-effective alternative to the classical RSS and double RSS schemes. In this paper, we propose an improved estimator for estimating the cumulative distribution function (CDF) using HRSS. It is shown, both theoretically and numerically, that the CDF estimator under HRSS scheme is unbiased and its variance is always less than the variance of the CDF estimator with simple random sampling (SRS). An unbiased estimator of the variance of CDF estimator using HRSS is also derived. Using Monte Carlo simulations, we also study the performances of the proposed and existing CDF estimators under both perfect and imperfect rankings. It turns out that the proposed CDF estimator is by far a superior alternative to the existing CDF estimators with SRS, RSS and L-RSS schemes. For a practical application, a real data set is considered on the bilirubin level of babies in neonatal intensive care.

In 2020, El-Morshedy et al. introduced a bivariate extension of the Burr type X generator (BBX-G) of distributions, and Muhammed presented a bivariate generalized inverted Kumaraswamy (BGIK) distribution. In this paper, we propose a more flexible generator of bivariate distributions based on the maximization process from an arbitrary three-dimensional baseline distribution vector, which is of interest for maintenance and stress models, and expands the BBX-G and BGIK distributions, among others. This proposed generator allows one to generate new bivariate distributions by combining non-identically distributed baseline components. The bivariate distributions belonging to the proposed family have a singular part due to the latent component which makes them suitable for modeling two-dimensional data sets with ties. Several distributional and stochastic properties are studied for such bivariate models, as well as for its marginals, conditional distributions, and order statistics. Furthermore, we analyze its copula representation and some related association measures. The EM algorithm is proposed to compute the maximum likelihood estimations of the unknown parameters, which is illustrated by using two particular distributions of this bivariate family for modeling two real data sets.

The ranked set sampling (RSS) technique has been shown to be superior to classical simple random sampling (SRS) in the sense that it always provides a more precise estimator of the population mean. However, it is quite often that some measurements are below the limit of detection and hence become censored. In such situations, the superiority of RSS over SRS may no longer be held. In this article we consider the problem of estimating the population mean and standard deviation based on a ranked set sample with some data being censored. Maximum likelihood estimators are proposed when the data are assumed to follow a lognormal distribution. In the case where the distribution is unknown, a variant of the Kaplan–Meier estimator is proposed in the estimation of the population mean. A simulation study is conducted to compare the performance of the proposed RSS estimators with the corresponding SRS estimators. The impact of imperfect judgment ranking is also discussed. The proposed methods are applied to a real data set on mercury concentration in swordfish. Copyright © 2002 John Wiley & Sons, Ltd.

A ranked set sample consists entirely of independently distributed order statistics and can occur naturally in many experimental settings, including problems in reliability. When each ranked set from which an order statistic is drawn is of the same size, and when the statistic of each fixed order is sampled the same number of times, the ranked set sample is said to be balanced. Stokes and Sager have shown that the edf F n of a balanced ranked set sample from the cdf F is an unbiased estimator of F and is more precise than the edf of a simple random sample of the same size. The nonparametric maximum likelihood estimator (MLE) F of F is studied in this article. Its existence and uniqueness is demonstrated, and a general numerical procedure is presented and is shown to converge to F. If the ranked set sample is balanced, it is shown that the EM algorithm, with F n as a seed, converges to the unique solution (F) of the problem's self-consistency equations; the consistency of every iterate of the EM a...

ABSTRACTIn this paper, ranked set sampling is proposed for constructing a non-parametric quality control chart based on sign statistic for monitoring the process center when the units are difficult and costly to measure. The characteristics of run length such as average, median and standard deviation are used as criteria to compare two sampling schemes: ranked set sampling and simple random sampling. Cauchy, double exponential and normal distributions are applied to investigate the performance of sign control chart based on studied sampling schemes. The proposed control chart is illustrated with a real data set. The results indicate that when the process starts to get out of control, the chart based on ranked set sampling reduces the run length characteristics values.

In surveys when measuring units is expensive, ranked set sampling (RSS) is a popular and economical sampling technique. The RSS algorithm selected units using a ranking procedure. Either eye inspection or an auxiliary variable is used for ranking. In the present paper, ‘stratified dual-rank ranked set sampling’ (SDuRSS) method is suggested to estimate population mean. The proposed design used dual ranking instead of traditional ranking method. The mean and variance of the suggested scheme is derived. The performance of the mean estimator of proposed scheme is investigated by relative efficiency (RE) of the estimator. A simulation study is conducted for computing such relative efficiency which shows that the proposed design is more efficient than stratified ranked set sampling (SRSS) and stratified extreme ranked set sampling (SERSS). The proposed scheme is illustrated with real data set where it also shows superiority on the SRSS and SERSS.