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Mazucheli et al. (J Appl Stat 46(4):700–714, 2019) introduced unit Lindley distribution by transforming Lindley (J Roy Stat Soc Ser B Stat Methodol 20(1):102–107, 1958) distribution for modelling proportion data. In this paper, we consider an exponential version of unit Lindley distribution. Various statistical and structural properties of the new distribution are discussed such as moments, hazard rate function, inequality measures, entropy etc. Different estimation methods are used to estimate the parameters of the model and their performances are demonstrated by Monte Carlo simulation. Finally the dominance of proposed distribution is embellished through real life data sets by comparing with some other unit distributions available in literature. [ABSTRACT FROM AUTHOR]

This paper addresses certain distributional aspects of a scale-free scatter matrix denoted by R that is stemming from a matrix-variate gamma distribution having a positive definite scale parameter matrix B. Under the assumption that B is a diagonal matrix, a structural representation of the determinant of R is derived; the exact density functions of products and ratios of determinants of matrices possessing such a structure are obtained; a closed form expression is given for the density function of R. Moreover, a novel procedure is utilized to establish that certain functions of the determinant of the sample scatter matrix are asymptotically distributed as chi-square or normal random variables. Then, representations of the density function of R that respectively involve multiple integrals, multiple series and Gauss' hypergeometric function are provided for the general case of a positive definite scale parameter matrix, and an illustrative numerical example is presented. Cutting-edge mathematical techniques have been employed to derive the results. Naturally, they also apply to the conventional sample correlation matrix which is encountered in various multivariate inference contexts. [ABSTRACT FROM AUTHOR]

In this paper, we study properties of a series–parallel system. The component lifetimes may be dependent and non-identically distributed (DNID) discrete random variables. We consider the number of failed components upon system failure. We derive the probability mass function and the expected value of this quantity. In addition, we find the conditional probabilities corresponding to this variate given some partial information about the system failure. We also provide a numerical example to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

We consider the problem of approximating a continuous random variable, characterized by a cumulative distribution function (cdf) F(x), by means of k points, x 1 < x 2 < ⋯ < x k , with probabilities p i , i = 1 , ⋯ , k . For a given k, a criterion for determining the x i and p i of the approximating k-point discrete distribution can be the minimization of some distance to the original distribution. Here we consider the weighted Cramér-von Mises distance between the original cdf F(x) and the step-wise cdf F ^ (x) of the approximating discrete distribution, characterized by a non-negative weighting function w(x). This problem has been already solved analytically when w(x) corresponds to the probability density function of the continuous random variable, w (x) = F ′ (x) , and when w(x) is a piece-wise constant function, through a numerical iterative procedure based on a homotopy continuation approach. In this paper, we propose and implement a solution to the problem for different choices of the weighting function w(x), highlighting how the results are affected by w(x) itself and by the number of approximating points k, in addition to F(x); although an analytic solution is not usually available, yet the problem can be numerically solved through an iterative method, which alternately updates the two sub-sets of k unknowns, the x i 's (or a transformation thereof) and the p i 's, till convergence. The main apparent advantage of these discrete approximations is their universality, since they can be applied to most continuous distributions, whether they possess or not the first moments. In order to shed some light on the proposed approaches, applications to several well-known continuous distributions (among them, the normal and the exponential) and to a practical problem where discretization is a useful tool are also illustrated. [ABSTRACT FROM AUTHOR]

In this paper, we prove a local limit theorem and a refined continuity correction for the negative binomial distribution. We present two applications of the results. First, we find the asymptotics of the median for a NegativeBinomial (r , p) random variable jittered by a Uniform (0 , 1) , which answers a problem left open in Coeurjolly and Trépanier (Metrika 83(7):837–851, 2020). This is used to construct a simple, robust and consistent estimator of the parameter p, when r > 0 is known. The case where r is unknown is also briefly covered. Second, we find an upper bound on the Le Cam distance between negative binomial and normal experiments. [ABSTRACT FROM AUTHOR]

Within the theory of non-negative integer valued multivariate infinitely divisible distributions, the multivariate Poisson distribution plays a key role. As in the univariate case, any non-negative integer valued infinitely divisible multivariate distribution can be approximated by a multivariate distribution belonging to the compound Poisson family. The multivariate Poisson distribution is an important member of this family. In recent years, the multivariate Poisson distributions also has gained practical importance, since they serve as models to describe counting data having a positive covariance structure. However, due to the computational complexity of computing the multivariate Poisson probability mass function (pmf) and its corresponding cumulative distribution function (cdf), their use within these counting models is limited. Since most of the theoretical properties of the multivariate Poisson probability distribution seem already to be known, the main focus of this paper is on proposing more efficient algorithms to compute this pmf. Using a well known property of a Poisson multivariate distributed random vector, we propose in this paper a direct approach to calculate this pmf based on finding all solutions of a system of linear Diophantine equations. This new approach complements an already existing procedure depending on the use of recurrence relations existing for the pmf. We compare our new approach with this already existing approach applied to a slightly different set of recurrence relations which are easier to evaluate. A proof of this new set of recurrence relations is also given. As a result, several algorithms are proposed where some of them are based on the new approach and some use the recurrence relations. To test these algorithms, we provide an extensive analysis in the computational section. Based on the experiments in this section, we conclude that the approach finding all solutions of a set of linear Diophantine equations is computationally more efficient than the approach using the recurrence relations to evaluate the pmf of a multivariate Poisson distributed random vector. [ABSTRACT FROM AUTHOR]

Data including significant losses are a pervasive issue in general insurance. The computation of premiums and reinsurance premiums, using deductibles, in situations of heavy right tail for the empirical distribution, is crucial. In this paper, we propose a mixture model obtained by compounding the Birnbaum–Saunders and gamma distributions to describe actuarial data related to financial losses. Closed-form credibility and limited expected value premiums are obtained. Moment estimators are utilized as starting values in the non-linear search procedure to derive the maximum-likelihood estimators and the asymptotic variance–covariance matrix for these estimators is determined. In comparison to other competing models commonly employed in the actuarial literature, the new mixture distribution provides a satisfactory fit to empirical data across the entire range of their distribution. The right tail of the empirical distribution is essential in the modeling and computation of reinsurance premiums. In addition, in this paper, to make advantage of all available data, we create a regression structure based on the compound distribution. Then, the response variable is explained as a function of a set of covariates using this structure. [ABSTRACT FROM AUTHOR]

Based on the design and potential application of wind-solar storage intelligent power generation systems in engineering practice, this paper develops a novel reliability model of k-out-of-n: G mixed standby retrial system with failure dependency and J-vacation policy. The working components in the system have redundant dependencies. When any component of the system fails and the repairman is working or on vacation, the failed component goes into the retrial space. If the retrial space has no failed components, the idle repairman goes on vacation, which may last for up to J consecutive vacations, until at a minimum one failed component appears in the retrial space on a vacation return. Firstly, the performance indexes of the system under steady state are analyzed based on the Markov process theory. Secondly, an algorithm for modelling the failure process of the proposed model is developed through a Monte Carlo method, and numerical solutions for the reliability function and mean time to first failure (MTTFF) are presented. Then, some numerical examples are provided to demonstrate the influence of different parameters on the system reliability indexes. Finally, a system cost optimization model based on availability control is developed, and the optimal component configuration schemes for systems with no vacations and different maximum numbers of vacations J are compared and analyzed by genetic algorithm (GA). [ABSTRACT FROM AUTHOR]

In this paper, a new truncated (0, b)-F family of lifetime distributions is introduced. Its main interest is to provide simple and flexible probability density functions with a bounded support without introducing non-explicit normalization constants, unlike the family based on the classical truncation technique. A detailed examination of a subdistribution known as the (0, b)-exponential distribution is carried out. Among its features, this distribution has unimodal, left-skewed, and decreasing shapes for the probability density function, while it has increasing and convex shapes for the corresponding hazard rate function. The distribution's properties are studied in depth, including moments, quantiles, and first-order stochastic dominance. Maximum likelihood estimates of the distribution parameters are obtained. Furthermore, a simulation study is conducted to gain some information on the performance of these estimates. Fitting two reliability data sets demonstrates the model's utility among alternative models via some measures of goodness-of-fit. [ABSTRACT FROM AUTHOR]

In this paper, the Bahadur representation of the empirical and Bernstein polynomials estimators of the quantile function based on associated sequences are investigated. The rate of approximation depends on the rate of decay in covariances, and it converges to the optimal rate observed under independence when the covariances quickly approach zero. As an application, we establish a Berry-Esseen bound with the rate O (n - 1 / 3) assuming polynomial decay of covariances. All these results are established under fairly general conditions on the underlying distributions. Additionally, we perform Monte Carlo simulations to evaluate the finite sample performance of the suggested estimators, utilizing an associated and non-mixing model. [ABSTRACT FROM AUTHOR]

In earlier studies of circular data, the corresponding probability distributions considered were mostly assumed to be symmetric. However, the assumption of symmetry may not be meaningful for some data. Thus there has been increased interest, more recently, in developing skewed circular distributions. In this article we introduce three skewed circular models based on inverse stereographic projection (ISP), originally introduced by Minh and Farnum (Comput. Stat.–Theory Methods, 32, 1–9, 2003), by considering three different versions of skewed-t distribution on real line considered in the literature, namely skewed-t by Azzalini (Scand. J. Stat., 12, 171–178, 1985), two-piece skewed-t, (seemingly first considered in Gibbons and Mylroie Appl. Phys. Lett., 22, 568–569, 1973 and later by Fernández and Steel J. Amer. Statist. Assoc., 93, 359–371 1998) and skewed-t by Jones and Faddy (J. R. Stat. Soc. Ser. B (Stat. Methodol.), 65, 159–174, 2003). Unimodality and skewness of the resulting distributions are addressed in this paper. Further, real data sets are used to illustrate the application of the new models. It is found that under certain condition on the original scaling parameter, the resulting distributions may be unimodal. Furthermore, the study in this paper concludes that ISP circular distributions obtained from skewed distributions on the real line may provide an attractive alternative to other asymmetric unimodal circular distributions, especially when combined with a mixture of uniform circular distribution. [ABSTRACT FROM AUTHOR]

Regression model is a popular and well-acknowledged technique for finding a relationship between random phenomena. In this context, the normality assumption on the error terms is one of the challenging issues in practical studies because many phenomena in the real world are prone to skewness, peakness, and outliers. In this paper, a new generalization of nonlinear regression models is postulated by considering asymmetric Laplace distribution on the error terms to cover the drawbacks of the normal-based model in accommodating skew data with peakness and mild outliers. We present three stochastic representations of the model which enables us to develop an expectation–maximization (EM) algorithm to computationally obtain the maximum-likelihood (ML) parameter estimates. The observed information matrix is computed by adopting an information-based approach. The Bayesian analysis for the proposed model is also investigated. Finally, experiments on the simulation and real-world datasets illustrate some computational and robust aspects of the proposed model. [ABSTRACT FROM AUTHOR]

Inference for correlation is central in statistics. From a Bayesian viewpoint, the final most complete outcome of inference for the correlation is the posterior distribution. An explicit formula for the posterior density for the correlation for the binormal is derived. This posterior is an optimal confidence distribution and corresponds to a standard objective prior. It coincides with the fiducial introduced by R.A. Fisher in 1930 in his first paper on fiducial inference. C.R. Rao derived an explicit elegant formula for this fiducial density, but the new formula using hypergeometric functions is better suited for numerical calculations. Several examples on real data are presented for illustration. A brief review of the connections between confidence distributions and Bayesian and fiducial inference is given in an Appendix. [ABSTRACT FROM AUTHOR]

In the present paper, two pivotal statistics are suggested to construct prediction intervals of future observations from the exponential and Pareto distributions in the context of ordered ranked set sample. Our study encompasses two cases. The first case, when the sample size is assumed to be fixed and the second case when the sample size is assumed to be a positive integer-valued random variable. In addition to deriving explicit forms for the distribution functions of the two pivotal statistics, we consider some special cases for the random size of the sample. Moreover, a simulation study is carried out to assess the efficiency of the suggested methods. Finally, an example representing lifetime data is analyzed. [ABSTRACT FROM AUTHOR]

A nonparametric test labelled 'Rao Spacing-frequencies test' is explored and developed for testing whether two circular samples come from the same population. Its exact distribution and performance relative to comparable tests such as the Wheeler-Watson test and the Dixon test in small samples, are discussed. Although this test statistic is shown to be asymptotically normal, as one would expect, this large sample distribution does not provide satisfactory approximations for small to moderate samples. Exact critical values for small samples are obtained and tables provided here, using combinatorial techniques, and asymptotic critical regions are assessed against these. For moderate sample sizes in-between i.e. when the samples are too large making combinatorial techniques computationally prohibitive but yet asymptotic regions do not provide a good approximation, we provide a simple Monte Carlo procedure that gives very accurate critical values. As is well-known, the large number of usual rank-based tests are not applicable in the context of circular data since the values of such ranks depend on the arbitrary choice of origin and the sense of rotation used (clockwise or anti-clockwise). Tests that are invariant under the group of rotations, depend on the data through the so-called 'spacing frequencies', the frequencies of one sample that fall in between the spacings (or gaps) made by the other. The Wheeler-Watson, Dixon, and the proposed Rao tests are of this form and are explicitly useful for circular data, but they also have the added advantage of being valid and useful for comparing any two samples on the real line. Our study and simulations establish the 'Rao spacing-frequencies test' as a desirable, and indeed preferable test in a wide variety of contexts for comparing two circular samples, and as a viable competitor even for data on the real line. Computational help for implementing any of these tests, is made available online "TwoCircles" R package and is part of this paper. [ABSTRACT FROM AUTHOR]

This paper addresses the exact Bayesian analysis of the difference between two independent Poisson distributions with means μ1 and μ2 respectively, known as the Skellam distribution with parameters (μ1, μ2). We develop a closed form for the posterior predictive distribution of the future distribution under the order constraint of μ1 > μ2. This kind of constraint is quite common and useful in applications specially in sports data analysis. We show that the proposed distribution estimator outperforms other types of distribution estimators in the literature. We use a simulation study with an example regarding the prediction in soccer games to show the performance of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, the first-order non-negative integer-valued autoregressive process with Poisson-transmuted exponential innovations is introduced. Three estimation methods, namely, the conditional maximum likelihood, conditional least squares and Yule-Walker estimation methods are discussed to estimate the unknown parameters of the proposed process. Additionally, the simulation study is presented to assess the efficiencies of these estimation methods. Applications to two real-life data sets illustrate the usefulness of the proposed process. [ABSTRACT FROM AUTHOR]

We have reviewed the development of the exact distribution of the difference of two independently and identically distributed chi-square variables and presented derivations based on moment generating function and also by convolution method. We have also developed general moment results of the difference of two independently and identically distributed random variables and applied these to chi-square random variables. Some properties of the distribution, namely moment generating function, mean centered moments, coefficient of skewness and kurtosis, and cumulative distribution function, have been derived. The graph of the density functions has been presented. We conclude the paper with directions for further research along the line. [ABSTRACT FROM AUTHOR]

Several extensions of the familiar Dirichlet process have been widely investigated to nonparametric Bayesian model fittings parallel with appealing subsequent studies on their particular properties. This paper presents an explicit form for the joint distribution of drawn samples from the beta two-parameter process using an extension of stick-breaking construction. In particular, we evaluate the joint distribution of a random sequence for a specific process case and compare it with the Blackwell-MacQueen process. We obtain moments of the beta two-parameter process and present a formula for the number of distinct values in the sample. We establish the precision ratio and explore its effect on this number. [ABSTRACT FROM AUTHOR]

Mixture models provide a logical approach to modeling complex data. The use of Lindley distribution to model different types of lifetime variables has made it attractive for further studies. The discovery of a fractional mixture of the Lindley distribution by time scale theory, and applications of the obtained distribution can be an exciting discussion. This work presents two fractional extensions of the mixed distribution that have applications in count data. For both parent and mixture distributions, the nabla and delta types of fractional distributions are considered: mixed nabla Poisson–Lindley distribution, referring to the situation where applied time scale for both parent and mixture distributions are nabla; and mixed delta Poisson–Lindley distribution, where both parent and mixture distributions are obtained using delta calculus. The parent distributions are special cases of nabla and delta power series distributions defined in this paper, and we call them nabla and delta Poisson distributions, and the mixture distributions are the nabla and delta fractional Lindley distributions. Some mathematical properties are obtained for nabla Poisson and nabla fractional Lindley distributions. Some estimation methods for one of model the parameters are studied. Extensive simulation studies are revealed to observe the performances of the estimation methods. Finally, a practical data example is carried out and it was concluded that the best model was nabla Poisson–Lindley distribution. [ABSTRACT FROM AUTHOR]

In this paper, we study the number of failed components of a coherent system. We consider the case when the component lifetimes are discrete random variables that may be dependent and non-identically distributed. Firstly, we compute the probability that there are exactly i, i = 0 , ... , n - k , failures in a k-out-of-n system under the condition that it is operating at time t. Next, we extend this result to other coherent systems. In addition, we show that, in the most popular model of independent and identically distributed component lifetimes, the obtained probability corresponds to the respective one derived in the continuous case and existing in the literature. [ABSTRACT FROM AUTHOR]

This paper considers the problem of finding the exact distribution and exact moments of the median of a sample of even size drawn from a jump type distribution with bounded domain in a computable form. The results which are obtained with the help of integral calculus are then applied to two-piece uniform distribution. The exact variances are calculated for both odd and even sample sizes, and efficiency comparisons are made. [ABSTRACT FROM AUTHOR]

The main object of this paper is to study an extension of the half normal distribution defined by adding a positive truncation to it. The new model is more flexible than the half-normal distribution and contains the half normal distribution as a special case. Properties of this distribution, such as moments, hazard function and entropy are studied and parameters estimation is dealt with by using moments and maximum likelihood. A real data application indicates good fit performance of the new model when compared to other competitors in literatures. [ABSTRACT FROM AUTHOR]

In the present paper we extend the definition of r-weak runs in sequences of binary trials so that such a run contains both a minimum and a maximum number of successes and we study the distribution of the statistic enumerating such (r-weak) runs. We are also investigating the distribution of the total number of successes in all the (r-weak) runs that are enumerated. The new introduced distributions may be of great applicability in scientific fields such as Agriculture. Our study requires an appropriate generalization of the Markov chain imbedding technique. More specifically, we introduce and study the family of Markov chain imbeddable variables of returnable-polynomial type, which generalizes the families of Markov chain imbeddable variables of binomial type, returnable type and polynomial type. The new family allows a unified approach for the study of the distributions of the statistics of current interest and other binomial-type distributions related to weak runs and typical runs. [ABSTRACT FROM AUTHOR]

In this paper, we modify the Burr-XII distribution through the inverse exponential scheme to obtain a new two-parameter distribution on the unit interval called the unit Burr-XII distribution. The basic statistical properties of the newly defined distribution are studied. Parameters estimation is dealt and different estimation methods are assessed through two simulation studies. A new quantile regression model based on the proposed distribution is introduced. Applications of the proposed distribution and its regression model to real data sets show that the proposed models have better modeling capabilities than competing models. [ABSTRACT FROM AUTHOR]

Coefficient of variation (CV) plays an important role in statistical practice; however, its sampling distribution may not be easy to compute. In this paper, the distributional properties of the sample CV from an inverse Gaussian distribution are investigated through transformations. Specifically, the symmetrizing transformation as outlined in Chaubey and Mudholkar (1983), that requires numerical techniques, is contrasted with the explicitly available variance stabilizing transformation (VST). The symmetrizing transformation scores very high as compared to the VST, especially in a power family. The usefulness of the resulting approximation is illustrated through a numerical example. [ABSTRACT FROM AUTHOR]

The hazard rate function plays a crucial role to extract some informations of several random life phenomena. In this paper, we investigate important properties satisfied by the hazard rate function for a general power series class of distributions, as monotonicity properties, sharp bounds and convexity properties. In particular, we highlight how these properties are related to those of the baseline distribution and the corresponding probability-generating function of discrete power series distributions. [ABSTRACT FROM AUTHOR]

Methods to obtain discrete analogs of continuous distributions have been widely applied in recent years. In general, the discretization process provides probability mass functions that can be competitive with traditional models used in the analysis of count data. The discretization procedure also avoids the use of continuous distribution to model strictly discrete data. In this paper, we propose two discrete analogs for the quasi xgamma distribution as alternatives to model under- and overdispersed datasets. The methods of infinite series and survival function have been considered to derive the models and, despite the difference between the methods, the resulting distributions are interchangeable. Several statistical properties of the proposed models have been derived. The maximum likelihood theory has been considered for estimation and asymptotic inference concerns. An intensive simulation study has been carried out in order to evaluate the main properties of the maximum likelihood estimators. The usefulness of the proposed models has been assessed by using two real datasets provided by literature. A general comparison of the proposed models with some well-known discrete distributions has been provided. [ABSTRACT FROM AUTHOR]

An accelerated life test (ALT) is commonly used in reliability studies of high quality products. A step-stress experiment is one such ALT that is often used in practice. In this paper, we develop exact inference based on likelihood-ratio test for a simple step-stress model with exponential lifetimes under Type-II censoring. We consider both unrestricted and order-restricted maximum likelihood estimators (MLEs) for this purpose. We also derive the exact power function of this test, which we then use for designing an optimal step-stress testing experiment. By using a similar approach, we extend these results to other forms of censoring. Monte Carlo simulations are then carried out to evaluate the performance of the inferential methods developed here. Finally, an illustrative example is presented for demonstrating all the inferential results. [ABSTRACT FROM AUTHOR]

Pareto distributions are very flexible probability models with various forms and kinds. In this paper, a new bivariate Pseudo-Pareto distribution and its properties are presented and discussed. Main variables, order statistics and concomitants of this distribution are studied and their importance for risk and reliability analysis is explained. Joint and marginal distributions, complementing cumulative distributions and hazard functions of the variables are derived. Numerical illustrations, graphical displays and interpretations for the obtained distributions and derived functions are provided. An implementation example on defaultable bonds is performed. [ABSTRACT FROM AUTHOR]

The likelihood ratio test is one of the commonly used procedures for hypothesis testing. Several results on likelihood ratio test have been discussed for testing the scale parameter of an exponential distribution under complete and censored data; however, all of them are based on approximations of the involved null distributions. In this paper, we first derive the exact distribution of the likelihood ratio statistic for testing the scale parameter of an exponential distribution based on a time-constrained life-testing experiment. We also obtain the asymptotic distribution, which is useful in the case of a large sample size. We then discuss the derivation of its power function. Next, we consider the likelihood ratio test of θ 2 = γ 0 θ 1 when data are obtained from two exponential distributions based on a time-constrained life-testing experiment. Here again, we derive both exact and asymptotic distributions of the likelihood ratio statistic and then use them to determine the reject region as well as the power function. Monte Carlo simulations are then carried out to evaluate the performance of the inferential methods developed here. Finally, some examples are used to illustrate all the inferential results. [ABSTRACT FROM AUTHOR]

Discrete tempered stable distributions are a large and flexible class of models for heavy tailed and overdispersed count data. In this paper we derive various properties of these distributions and develop an exact simulation method based on rejection sampling and a compound Poisson representation. We extend this method to exact simulation of the corresponding bilateral distributions and Lévy processes. [ABSTRACT FROM AUTHOR]

In this paper, we consider a random vector X = X 1 , X 2 following a multivariate Skew Normal distribution and we provide an explicit formula for the expected value of X conditioned to the event X ≤ X ¯ , with X ¯ ∈ R 2 . Such a conditional expectation has an intuitive interpretation in the context of risk measures. [ABSTRACT FROM AUTHOR]

In this paper the authors study the problem of testing the hypothesis of a doubly exchangeable covariance matrix for three-level multivariate observations, taken on m variables over u sites and over v time/space points. Through the decomposition of the main hypothesis into a set of three sub-hypotheses, the likelihood ratio test statistic is defined, its exact moments are determined, and its exact distribution is studied. Because this distribution is very much intricate, a very precise near-exact distribution is developed. Numerical studies conducted to evaluate the closeness between this near-exact distribution and the exact distribution show the very good performance of this approximation even for very small sample sizes. A simulation study is also conducted and two real-data examples are presented. [ABSTRACT FROM AUTHOR]

The current paper introduces a new flexible probability distribution model called transmuted XGamma distribution which pullulates from the XGamma distribution and possesses the characteristics of XGamma distribution in special cases. In the paper, we obtain the explicit expressions for some important statistical properties of the introduced distribution such as hazard rate and survival functions, mean residual life, moment-generating function, moments, skewness, kurtosis, distribution of its order statistics, Lorenz and Bonferroni curves. Besides obtaining the various effective estimators for the parameters of the distribution, estimation performances of these estimators are comparatively examined with a series of Monte Carlo simulations. Furthermore, to demonstrate the modeling ability of the proposed distribution on real-world phenomena, an illustrative example is performed by using an actual data set in connection with the field of the lifetime. [ABSTRACT FROM AUTHOR]

The researchers, engineers, and physical experimenters may face difficulty to get a distribution that fits the failure data arising from certain systems. So, in this paper, a new distribution is introduced, named Poisson–logarithmic half-logistic distribution, based on a parallel–series system's failure times. Specific statistical properties are investigated for the introduced distribution. Also, two real data sets are considered to compare the introduced distribution with some other distributions. The progressive-stress accelerated life test is applied using an increasing exponential function of time to units whose lifetimes are expected to follow the new distribution at normal stress conditions. Different estimation methods, such as maximum likelihood, percentile, least squares, and weighted least square methods, are considered on the basis of adaptive type-II progressive hybrid censoring. To assess the efficiency of the estimation methods, a simulation study is conducted, as well as numerical calculations. [ABSTRACT FROM AUTHOR]

Discrete mixture models are one of the most successful approaches for density estimation. Under a Bayesian nonparametric framework, Dirichlet process location-scale mixture of Gaussian kernels is the golden standard, both having nice theoretical properties and computational tractability. In this paper we explore the use of the skew-normal kernel, which can naturally accommodate several degrees of skewness by the use of a third parameter. The choice of this kernel function allows us to formulate nonparametric location-scale-shape mixture prior with desirable theoretical properties and good performance in different applications. Efficient Gibbs sampling algorithms are also discussed and the performance of the methods are tested through simulations and applications to galaxy velocity and fertility data. Extensions to accommodate discrete data are also discussed. [ABSTRACT FROM AUTHOR]

Geometric distribution of order $$k$$ as one of the generalization of well known geometric distribution is the distribution of the number of trials until the first $$k$$ consecutive successes in Bernoulli trials with success probability $$p$$ . In this paper, it is shown that this generalized distribution can be represented as a discrete phase-type distribution. Using this representation along with closure properties of phase-type distributions, the distributions of sum, minima and maxima of two independent random variables having geometric distribution of order $$k$$ are obtained. Numerical results are presented to illustrate the computational details. [ABSTRACT FROM AUTHOR]

The paper deals with three generalized dependent setups arising from a sequence of Bernoulli trials. Various distributional properties, such as probability generating function, probability mass function and moments are discussed for these setups and their waiting time. Also, explicit forms of probability generating function and probability mass function are obtained. Finally, two applications to demonstrate the relevance of the results are given. [ABSTRACT FROM AUTHOR]

The Dagum distribution plays an important role both in statistical theory and in economics as a model for income distribution. The main goal of this paper was to develop a bivariate extension of the Dagum distribution and study its distributional properties. Also we will study its dependence properties using copula function. Also we derive the cumulative distribution function and probability density function of the maximum and minimum of the bivariate Dagum order statistics. We estimate the parameters of the distribution and demonstrate it on a real data example. [ABSTRACT FROM AUTHOR]

In this paper we develop two pivotal quantities to construct prediction intervals for future two-parameter exponential lifetimes based on a random number of generalized order statistics (gOs) under a general set-up including progressive type II censored order statistics (pOs) with general scheme. Moreover, the expected value as well as the mean square error for the upper limit of predictive confidence interval (PCI) are obtained. Furthermore, a maximum likelihood predictor (MLP) for future exponential lifetimes is derived when the sample size is deterministic or random. Finally, a simulation study and an application to real data illustrate and corroborate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper considers the ratio estimator in a finite population setting in a ranked set sampling (RSS) design, where the sample is constructed either with or without replacement policies. It is shown that the proposed ratio estimator is slightly biased, but the amount of bias is smaller than the amount of bias of a simple random sample (SRS) ratio estimator. For the proposed ratio estimator, the paper provides explicit expressions for its mean square error and precision relative to the other competing estimators. It is shown that the new estimator has a substantial amount of improvement in efficiency with respect to SRS estimator. The proposed estimator is applied to two different finite population settings to estimate population mean. [ABSTRACT FROM AUTHOR]

In this paper, nonparametric methods are proposed to construct prediction intervals for the lifetime of a coherent system with known signatures. An explicit expression for the coverage probability of the prediction intervals is presented based on Samaniego's signature. The existence and optimality of these intervals are discussed. In our derivation, we also obtain an exact expression for the marginal distribution of the $$i$$ th order statistic from a pooled sample. [ABSTRACT FROM AUTHOR]

We tackle the problem of obtaining the consistency factors of robust S-estimators of location and scale both in regression and multivariate analysis. We provide theoretical results, proving new formulae for their calculation and shedding light on the relationship between these factors and important statistical properties of S-estimators. We also give computational advances, suggesting efficient procedures so that hardly any time is lost for their calculation when computing S-estimates. In addition, when the purpose is to fix the efficiency of the scale estimator, we are able to quantify to what extent the approximate algorithms which are currently available provide an acceptable solution, and when it is necessary to resort to the exact formulae. Finally, even if this paper concentrates on S-estimates and Tukey's Biweight and optimal loss functions, the main results can be easily extended to calculate the tuning consistency factors for other popular loss function and other robust estimators. [ABSTRACT FROM AUTHOR]

This paper is concerned with the distribution of runs associated with claim indicators in a compound binomial risk model. We study the total number of claims, the longest run without claim, the shortest run without claim and the total number of runs up to a fixed period before the occurrence of a ruin. These quantities are potentially useful for an investment strategy of an insurance company and for understanding the behavior of a specific portfolio over time. We obtain recursive equations for the exact distributions of these random variables. We also illustrate the theoretical results with numerical computations. [ABSTRACT FROM AUTHOR]

In this paper we deduce the probability mass function for a random variable which follows the hypergeometric—(binomial and right truncated geometric) mixtures distribution. Also, we derive the probability mass function for a random sum of the hypergeometric—(binomial and right truncated geometric) mixtures, where the two parameters of the hypergeometric distribution vary randomly. Moreover, we investigate the mean and the variance of this random sum when the number of terms is also a random variable. [ABSTRACT FROM AUTHOR]

In this paper, we derive an exact test for a column of the covariance matrix. The test statistic is calculated by using a single observation. The exact distributions of the test statistic are derived under both the null and alternative hypotheses. We also obtain an analytical expression of the power function of the test for the equality of a column of the covariance matrix to a given vector. It is shown that the information contained in a single vector is large enough to ensure a good performance of the test. Moreover, the suggested test can be applied for time-dependent multivariate Gaussian processes. [ABSTRACT FROM AUTHOR]

For a fixed unit vector $${a = (a_1, a_2,..., a_n) \in S^{n-1}}$$ , that is, $${\sum^n_{i=1} a^2_1 = 1}$$ , we consider the 2 signed vectors $${\varepsilon = (\varepsilon_1, \varepsilon_2,..., \varepsilon_n) \in \{-1, 1\}^n}$$ and the corresponding scalar products $${a \cdot \varepsilon = \sum^n_{i=1} a_i \varepsilon_i}$$ . In [3] the following old conjecture has been reformulated. It states that among the 2 sums of the form $${\sum \pm a_i}$$ there are not more with $${|\sum^n_{i=1} \pm a_i| > 1}$$ than there are with $${|\sum^n_{i=1} \pm a_i| \leq 1}$$ . The result is of interest in itself, but has also an appealing reformulation in probability theory and in geometry. In this paper we will solve an extension of this problem in the uniform case where $${a_1 = a_2 = \cdot\cdot\cdot = a_n = n^{-1/2}}$$ . More precisely, for S being a sum of n independent Rademacher random variables, we will give, for several values of $${\xi}$$ , precise lower bounds for the probabilities or equivalently for where $${T_n}$$ is a standardized binomial random variable with parameters n and $${p = 1/2}$$ . These lower bounds are sharp and much better than for instance the bound that can be obtained from application of the Chebyshev inequality. In case $${\xi = 1}$$ Van Zuijlen solved this problem in [5]. We remark that our bound will have nice applications in probability theory and especially in random walk theory (cf. [1, 2]). [ABSTRACT FROM AUTHOR]