Distribution Of Order Statistics Research Articles

Inverted Power Rama Distribution with Applications to Life Time Data

In this paper, we introduced the Inverted Power Rama distribution as an extension of the Inverted Rama distribution. This new distribution is capable of modeling real life data with upside down bathtub shape and heavy tails. Mathematical and statistical characteristics such as the quantile function, mode, moments and moment generating function, entropy measure, stochastic ordering and distribution of order statistics have been derived. Furthermore, reliability measures like survival function, hazard function and odds function have been derived. The method of maximum likelihood was used for estimating the parameters of the distribution. To demonstrate the applicability of the distribution, a numerical example was given. Based on the results, the proposed distribution performed better than the competing distributions.

Some Properties of Univariate and Multivariate Exponential Power Distributions and Related Topics

In the paper, a survey of the main results concerning univariate and multivariate exponential power (EP) distributions is given, with main attention paid to mixture representations of these laws. The properties of mixing distributions are considered and some asymptotic results based on mixture representations for EP and related distributions are proved. Unlike the conventional analytical approach, here the presentation follows the lines of a kind of arithmetical approach in the space of random variables or vectors. Here the operation of scale mixing in the space of distributions is replaced with the operation of multiplication in the space of random vectors/variables under the assumption that the multipliers are independent. By doing so, the reasoning becomes much simpler, the proofs become shorter and some general features of the distributions under consideration become more vivid. The first part of the paper concerns the univariate case. Some known results are discussed and simple alternative proofs for some of them are presented as well as several new results concerning both EP distributions and some related topics including an extension of Gleser’s theorem on representability of the gamma distribution as a mixture of exponential laws and limit theorems on convergence of the distributions of maximum and minimum random sums to one-sided EP distributions and convergence of the distributions of extreme order statistics in samples with random sizes to the one-sided EP and gamma distributions. The results obtained here open the way to deal with natural multivariate analogs of EP distributions. In the second part of the paper, we discuss the conventionally defined multivariate EP distributions and introduce the notion of projective EP (PEP) distributions. The properties of multivariate EP and PEP distributions are considered as well as limit theorems establishing the conditions for the convergence of multivariate statistics constructed from samples with random sizes (including random sums of random vectors) to multivariate elliptically contoured EP and projective EP laws. The results obtained here give additional theoretical grounds for the applicability of EP and PEP distributions as asymptotic approximations for the statistical regularities observed in data in many fields.

Generalised Odd Frechet Family of Distributions: Properties and Applications

Abstract A new distribution called Generalized Odd Fréchet (GOF) distribution is presented and its properties explored. Some structural properties of the proposed distribution, including the shapes of the hazard rate function, moments, conditional moments, moment generating function, skewness, and kurtosis are presented. Mean deviations, Lorenz and Bonferroni curves, Rényi entropy, and the distribution of order statistics are given. The maximum likelihood estimation technique is used to estimate the model parameters, and finally applications of the model to a real data set are presented to illustrate the usefulness of the proposed distribution.

All in order: Distribution of serially correlated order statistics with applications to hydrological extremes

Classic extreme value theory provides asymptotic distributions of block maxima (BM) Y or peaks over threshold (POT). However, as BM and POT are relatively small subsets of the values assumed by the parent process Z, alternative approaches have been proposed to make inferences on extremes by using intermediate values and non-asymptotic models. In this study, we investigate the finite sample theory of extremes based on order statistics, and present a set of results enabling the analysis of the properties of non-asymptotic distributions of BM in finite-size blocks of data under the assumption that the parent process Z has stationary temporal dependence. In particular, we suggest the beta-binomial distribution (FβB) as a suitable approximation of the marginal distribution of order statistics and BM under dependence, thus generalizing the theoretical results available under independence. We demonstrate the usefulness of the FβB distribution in three conceptual applications of hydrological interest. Firstly, we review the so-called Complete Time-series Analysis (CTA) framework, showing that the differences between FY and FZ are due to the inherent theoretical nature of the processes Y and Z and the different probabilistic failure scenarios described by FY and FZ. Secondly, we provide a theoretical rule to select the number of the largest maxima (LM) and over-threshold exceedances (OT) required to approximate the desired portion of the upper tail of FZ with specified accuracy. Finally, we discuss how the FβB distribution offers an interpretation and generalization of the so-called metastatistical extreme value (MEV) framework and its simplified version (SMEV), avoiding the use of high-dimensional joint distributions and preliminary data thresholding and declustering. All methodological results are validated by Monte Carlo simulations involving widely used stochastic processes with persistence. Real-world stream flow data are also analyzed as proof of concept.

A New Odd Lindley-Gompertz Distribution: Its Properties and Applications

This article presents a comprehensive study of an odd Lindley-Gompertz distribution which has already been proposed in the literature but without any properties. The present study unlike the previous one has considered the derivation of several properties of the odd Lindley-Gompertz distribution with their graphical representations and discussions which has not been done in the first proposition of the distribution. The study looks at properties such as survival (or reliability) function, the hazard function, the cumulative hazard function, the reverse hazard function, the odds function, quantile function, moments, moment generating function, characteristic function, cumulant generating function, distribution of order statistics and maximum likelihood estimation of the distribution’s parameters none of which was treated by the previous author of the model. An illustration to evaluate the goodness-of-fit of the odd Lindley-Gompertz distribution has also been done using two real life datasets and the results show that the model fits the datasets better than the five other distributions considered in this present study.

Some Further Properties and Bayesian Inference for Inverse xgamma Distribution Under Progressive Type-II Censored Scheme

Inverse xgamma distribution is recently proposed by Yadav et al. (J Ind Prod Eng 35(1):48–55, 2018) as an inverted version of xgamma distribution. In the present article, some more statistical properties (such as, characteristic and generating functions, distributions of extreme order statistics, important entropy measures) and some additional survival and/or reliability characteristics (such as, conditional moments, mean deviation, Bonferroni and Lorenz curves, entropy, ageing intensity) of inverse xgamma distribution have been studied in detail. Classical and Bayesian inferential procedures to estimate the unknown parameter, reliability function, hazard rate function under progressively censored schemes have been investigated. Further, asymptotic confidence interval (ACI), bootstrap confidence interval (BCI) and highest posterior density (HPD) credible interval for the parameter have also been calculated. A Monte-Carlo simulation study has been performed to compare the performances of classical and Bayesian estimators of reliability function and hazard rate function. The performances of ACI, BCI and HPD credible intervals have been compared in terms of estimated average widths and coverage probabilities for the parameter. Lastly, a data set is analyzed for illustrating the proposed methodology.

Competing Pre-match Investments Revisited: A Characterization of Monotone Bayes-Nash Equilibria in Large Markets

We solve an open problem pertaining to the relationship between competitive and non-cooperative models of pre-match investment. We study an incomplete information version of Peters and Siow's (2002) model of competing pre-marital investments and NTU matching, with finitely many agents and i.i.d. types. Our main result is a precise characterization of side-symmetric, strictly monotone Bayes-Nash equilibrium (SSMBNE) behavior when the market grows large and the empirical type distributions converge to those of an unbalanced continuum economy a la Peters and Siow (2002). We find that the limits of SSMBNE strategies always differ from competitive (hedonic) equilibrium strategies, and that there is always inefficient overinvestment on both sides of the market. Our analysis relies on a novel way of using results from the theory of approximate distributions of order statistics that allows us to characterize equilibrium behavior even though the limit strategies must be discontinuous and the size of the discontinuities is determined by a complex, two-sided interaction. These techniques should be useful for the analysis of other large Bayesian games with discontinuous payoffs and interacting distributions of outcomes.

A New Mixture of Exponential-Gamma Distribution

A new distribution called New Mixture of Exponential-Gamma Distribution is presented in this paper. This new distribution contains exponential and standardized Lindley distributions as sub models. Some of the structural properties of the proposed distribution which include the survival function, hazard rate function, moments, moment generating function, quantile function, distribution of order statistics and Renyi entropy are obtained. The maximum likelihood method of estimation was used to estimate the parameters of the distribution. A Simulation study was carried out to examine the performance and accuracy of the maximum likelihood estimates of the proposed distribution. An application of the proposed distribution to two real lifetime datasets is presented to illustrate its usefulness and superiority over some existing related models.

Beta transmuted Lomax distribution with applications

Abstract In this paper we propose and test a composite generalizer of the Lomax distribution .The genesis of the beta distribution and transmuted map is used to develop the so-called beta transmuted Lomax (BTL) distribution. The properties of the distribution are discussed and explicit expressions are derived for the moments, mean deviations, quantiles, distribution of order statistics and reliability. The maximum likelihood method is used for estimating the model parameters, and the finite sample performance of the estimators is assessed by simulation. Finally, the authors demonstrate the usefulness of the new distribution in analysing positive data.

Weighted Cumulative Residual (Past) Inaccuracy For Minimum (Maximum) of Order Statistics

In this paper, we propose a measure of weighted cumulative residual inaccuracy between survival function of the first-order statistic and parent survival function $\bar{F}$. We also consider weighted cumulative inaccuracy measure between distribution of the last- order statistic and parent distribution $F$. For these concepts, we obtain some reliability properties and characterization results such as relationships with other functions, bounds, stochastic ordering and effect of linear transformation. Dynamic versions of these weighted measures are considered.

A New Family of Distributions Based on a Poly-Exponential Transformation

AbstractIn this article, we propose a new generator of distributions based on a polynomial-exponential transformation of an existing cumulative distribution function. It naturally arises when we deal with parallel-series systems. We study some of its mathematical properties, including moments, a moment generating function, some measures of uncertainty, and a distribution of order statistics. Maximum likelihood estimation along with an extensive simulation study is discussed. The applicability of some submodels of the family is illustrated by means of two hydrologic data sets over existing distributions.

Weighted Suja distribution with application to ball bearings data

The Suja distribution (SD) is one of the important distributions from lifetime models. The SD is modified in this paper using weighted method to introduce a new distribution called weighted Suja distribution (WSD). Its moments, moment generating function, the coefficient of skewness, coefficient of variation, coefficient of kurtosis, distribution of order statistics, and reliability analysis are presented. The maximum likelihood estimation is used to estimate the WSD parameters with some simulations. Also, the Fisher’s information, the generalized entropy and Renyi entropy, the mode, and the stochastic ordering are derived. The harmonic mean, mean and median deviations, and the stress–strength reliability of the WSD are presented. The Gini index, Bonferroni and Lorenz curves of the WSD are derived. A real data application of 23 ball bearings is presented to demonstrate the performance of the new model.

Efficient calculation of the joint distribution of order statistics

The problem of computing the joint distribution of order statistics of stochastically independent random variables in one- and two-group models is considered. While recursive formulae for evaluating the joint cumulative distribution function of such order statistics exist, their numerical implementation remains a challenging task. This task is tackled by presenting novel generalizations of known recursions. They are utilized to obtain exact results (calculated in rational arithmetic) as well as faithfully rounded results. Finally, some applications in goodness-of-fit testing, step-wise multiple hypothesis testing, and sample size calculation for studies with multiple endpoints are discussed.

Exponential intervened Poisson distribution

We introduce a new family of distributions using intervened power series distribution. It is a rich family containing distributions such as Marshall–Olkin extended families of distributions, families of distributions generated through truncated negative binomial, and families of distributions generated through truncated binomial distribution. Also, the proposed distribution is a generalization of the families of distributions generated through zero truncated Poisson distribution. As a particular case, we study exponential intervened Poisson (EIP) distribution. This is a generalization of exponential-Poisson distribution. The shape properties of the pdf and hazard rate are proved. The model identifiability is established. The expression for moments, mean deviation, distribution of order statistics, and residual life function of EIP distribution are derived. The unknown parameters of the distribution are estimated using the method of maximum likelihood. The existence and uniqueness of maximum likelihood estimates (MLEs) are proved. Simulation study is carried out to show the performance of MLEs. EIP distribution is fitted to two real data sets and it is shown that the distribution is more appropriate than other competitive models. The adequacy of the EIP model for the data set is established using parametric bootstrap approach.

Power Length-Biased Suja Distribution: Properties and Application

In this paper, a new distribution called weighted size biased two-parameterAkash distribution (WSBTPAD) is proposed. The WSBTPAD is a newmodication of the size biased two-parameter Akash distribution. The mainstatistical properties of the WSBTPAD are derived and proved. These prop-erties include the moments, particularly the rth moment, moment generatingfunction, harmonic mean, Bonferroni and Lorenz curves as well as the Giniindex. Also, the mean deviations of the population mean and median andthe Renyi entropy are presented. The reliability analysis of the random vari-able following WSBTPAD random variable are discussed. The method ofmaximum likelihood estimation is considered for estimating the parametersof the distribution. The distribution of order statistics from the WSBTPADare provided.

Darna Distribution: Properties and Application

In this paper, Darna distribution (DD) is suggested as a new continuousprobability density function. The statistical properties of the DD as the moments,shapes of the distribution, measures of skewness, kurtosis, coefficientof variation are presented as well as some calculations are provided. Also,the maximum likelihood estimators, the Bonferroni and Lorenz curves, andGini Index are obtained. The Stress - Strength Reliability, R´enyi entropy,mean and median deviations are derived and proved. The distribution oforder statistics are presented. The reliability analysis including hazard, reliability,odds, and reverse hazard functions are presented. An application ofWheaton River data is considered.

A New Generalized Family of Odd Lindley-G Distributions With Application

A new family of distributions, namely the Kumaraswamy Odd Lindley-G distribution is developed. The new density function can be expressed as a linear combination of exponentiated-G densities. Statistical properties of the new family including hazard rate and quantile functions, moments and incomplete moments, Bonferroni and Lorenz curves, distribution of order statistics and R´enyi entropy are derived. Some special cases are presented. We conduct some Monte Carlo simulations to examine the consistency of the maximum likelihood estimates. We provide an application of KOL-LLo distribution to a real data set.

On the joint distribution of order statistics from independent non-identical bivariate distributions

In this note, the exact joint probability density function (jpdf) of bivariate order statistics from independent non-identical bivariate distributions is obtained. Furthermore, this result is applied to derive the joint distribution of a new sample rank obtained from the rth order statistics of the first component and the sth order statistics of the second component.

A generalization of Basu-Dhar’s bivariate geometric distribution to the trivariate case

In this paper, it is introduced a new parametric distribution to be used in multivariate lifetime data analysis as an alternative for the use of some existing multivariate parametric models as the popular multivariate normal distribution which is the most widely used model assumed in the analysis of continuous multivariate data. Although the normal multivariate distribution has univariate marginal normal probability distributions and simple interpretations for all their parameters, it may not be well fitted by many data sets, especially in survival data applications, usually considering logarithm transformed data. In many cases the use of parametric multivariate discrete models could be more appropriate for the data analysis. In this paper, it is introduced a generalization of the bivariate Basu-Dhar geometric distribution to a trivariate case applied to count data. Some properties of this trivariate geometric distribution, including its marginal probability distributions, order statistics distributions, the probability generating function and some simulation studies are presented. It is also presented some discussion on an extension of the trivariate case for the multivariate case. Classical and Bayesian inferences are presented assuming censored or uncensored observations. To illustrate the proposed methodology, two applications with real lifetime data are considered as examples.

Extended Exponentiated Inverse Lindley Distribution: Model, Properties and Applications

In this article, a four-parameter generalization of inverse Lindley distribution is obtained, with the purpose of obtaining a more flexible model relative to the behavior of hazard rate functions. Various statistical properties such as density, hazard rate functions, moments, moment generating functions, stochastic ordering, Renyi entropy, distribution of kth order statistics has been derived. The method of maximum likelihood estimation has been used to estimate parameters. Further confidence intervals are also obtained. Finally, applicability of the proposed model to the real data is analyzed. A comparison has also been made with some existing distributions.