Rényi entropy of m-generalized order statistics

Abstract

Characterizing relations via Rényi entropy of m-generalized order statistics are considered along with examples and related stochastic orderings. Previous results for common order statistics are included.     

Simulation of Balakrishnan skew-normal order statistics

ABSTRACT

The novel Balakrishnan skew-normal distribution introduced in 2008 has received considerable interest. Here, we derive stochastic representations for simulating order statistics of the novel Balakrishnan skew-normal distribution. The resulting algorithms are more efficient than the ordinary sorting algorithm.     

The Weibull–Dagum distribution: Properties and applications

ABSTRACT

In this article, we define a new lifetime model called the Weibull–Dagum distribution. The proposed model is based on the Weibull–G class. It can also be defined by a simple transformation of the Weibull random variable. Its density function is very flexible and can be symmetrical, left-skewed, right-skewed, and reversed-J shaped. It has constant, increasing, decreasing, upside-down bathtub, bathtub, and reversed-J shaped hazard rate. Various structural properties are derived including explicit expressions for the quantile function, ordinary and incomplete moments, and probability weighted moments. We also provide explicit expressions for the Rényi and q-entropies. We derive the density function of the order statistics as a mixture of Dagum densities. We use maximum likelihood to estimate the model parameters and illustrate the potentiality of the new model by means of a simulation study and two applications to real data. In fact, the proposed model outperforms the beta-Dagum, McDonald–Dagum, and Dagum models in these applications.     

Stochastic comparisons on sample extremes of dependent and heterogenous observations

Stochastic comparison on order statistics from heterogeneous-dependent observations has been paid lots of attention recently. This paper devotes to investigating the ordering properties of order statistics from dependent observations. We derive the usual stochastic order for sample minimums and the second smallest order statistic, the dispersive order and the star order for minimums of samples having proportional hazards and Archimedean survival copulas. Similar ordering results are also obtained for maximums and the second largest order statistic of samples having proportional reversed hazards and Archimedean copulas. Several examples illustrating the main results are presented as well.     

Prediction intervals for ordinary and dual generalized order statistics from two independent sequences

ABSTRACT

Based on the observed dual generalized order statistics drawn from an arbitrary unknown distribution, nonparametric two-sided prediction intervals as well as prediction upper and lower bounds for an ordinary and a dual generalized order statistic from another iid sequence with the same distribution are developed. The prediction intervals for dual generalized order statistics based on the observed ordinary generalized order statistics are also developed. The coverage probabilities of these prediction intervals are exact and free of the parent distribution, F. Finally, numerical computations and real examples of the coverage probabilities are presented for choosing the appropriate limits of the prediction.     

Stochastic comparisons of series and parallel systems with independent heterogeneous lower-truncated Weibull components

Abstract

In this paper, we discuss stochastic comparisons of series and parallel systems with independent heterogeneous lower-truncated Weibull components. When a system with possibly different shape and scale parameters and its matrix of parameters changes to another matrix in a certain mathematical sense, we study the hazard rate order of lifetimes of series systems and the usual stochastic order of lifetimes of parallel systems.     

Generalized order statistic processes and Pfeifer records

We introduce a uniform generalized order statistic process. It is a simple Markov process whose initial segment can be identified with a set of uniform generalized order statistics. A standard marginal transformation leads to a generalized order statistic process related to non-uniform generalized order statistics. It is then demonstrated that the nth variable in such a process has the same distribution as an nth Pfeifer record value. This process representation of Pfeifer records facilitates discussion of the possible limit laws for Pfeifer records and, in some cases, of sums thereof. Because of the close relationship between Pfeifer records and generalized order statistics, the results shed some light on the problem of determining the nature of the possible limiting distributions of the largest generalized order statistic.     

A NOTE ON MOMENTS OF PROGRESSIVELY TYPE II CENSORED ORDER STATISTICS

ABSTRACT

Upper and lower bounds for moments of progressively Type II censored order statistics in terms of moments of (progressively Type II censored) order statistics are derived. In particular, this yields conditions for the existence of moments of progressively Type II censored order statistics based on an absolutely continuous distribution function.     

Domains of attraction of asymptotic distributions of extreme generalized order statistics

Abstract

In extreme value theory for ordinary order statistics, there are many results that characterize the domains of attraction of the three extreme value distributions. In this article, we consider a subclass of generalized order statistics for which also three types of limit distributions occur. We characterize the domains of attraction of these limit distributions by means of necessary and/or sufficient conditions for an underlying distribution function to belong to the respective domain of attraction. Moreover, we compare the domains of attraction of the limit distributions for extreme generalized order statistics with the domains of attraction of the extreme value distributions.     

Comparison among non parametric prediction intervals of order statistics

Abstract

In this article, we are interested in conducting a comparison study between different non parametric prediction intervals of order statistics from a future sample based on an observed order statistics. Typically, coverage probabilities of well-known non parametric prediction intervals may not reach the preassigned probability levels. Moreover, prediction intervals for predicting future order statistics are no longer available in some cases. For this, we propose different methods involving random indices and fractional order statistics. In each case, we find the optimal prediction intervals. Numerical computations are presented to assess the performances of the so-obtained intervals. Finally, a real-life data set is presented and analyzed for illustrative purposes.     

Sharp Bounds for Generalized Order Statistics via Logarithmic Moments

ABSTRACT

We present sharp bounds for expectations of generalized order statistics with random indices. The bounds are expressed in terms of logarithmic moments E X ^a (log max {1, X}) ^b of the underlying observation X. They are attainable and provide characterizations of some non trivial distributions. No restrictions are imposed on the parameters of the generalized order statistics model.     

Concomitants in a Sequence of Independent Nonidentically Distributed Random Vectors

ABSTRACT

Concomitants of order statistics are considered for the situation in which the random vectors (X ₁, Y ₁), (X ₂, Y ₂),…, (X _n , Y _n ) are independent but otherwise arbitrarily distributed. The joint and marginal distributions of the concomitants of order statistics and stochastic comparisons among the concomitants of order statistics are studied in this situation.     

On stochastic behaviors of load-sharing parallel systems

Abstract

This paper mainly investigates a general load-sharing parallel system having two units. First, we construct some comparisons among a load standby system, a warm standby system, a hot standby system and a cold standby system. Moreover, some stochastic comparisons between the load-sharing parallel system and one of its two components are obtained in the sense of the usual stochastic order. Finally, the residual life of this system and its properties are examined.     

Dependence Structure of a Class of Symmetric Distributions

Abstract

In this article, dependence structure of a class of symmetric distributions is considered. Let X and Y be two n-dimensional random vectors having such distributions. We investigate conditions on the generators of densities of X and Y such that X is MTP₂, and X and Y can be compared in the multivariate likelihood ratio order. Nonnegativity of the covariance between functions of two adjacent order statistics of X is also given.     

On exact strong laws of large numbers for ratios of random variables and their applications

Abstract

We study the almost sure convergence of weighted sums of ratios of independent random variables satisfying some general, mild conditions. The obtained results are applied to exact laws for order statistics. An exact law for independent random variables which are nonidentically distributed is also proved and applied to ratios of adjacent order statistics for a sample of uniformly distributed random variables.     

Stochastic comparisons of order statistics,with applications in relibility

This paper reviews recent developments in the stochastic comparison of order statistics. The results discussed are basically: (l) Stochastic comparisons of linear combinations of order statistics from distributions F and G where G^?1 F is convex or starshaped. (2) Stochastic comparisons of individual order statistics and of vectors of order statistics from underlying heterogeneous distributions by the use of majorization and Schur function theory. (3) Stochastic comparison of random processes. Applications to reliability problems are presented illustrating the use and value of the theoretical results described     

Diagrams for Set Theory & Probability Problems of Four or More Variables

It is pointed out that in many one-sided testing situations for a real-valued parameter θ, the monotonicity of the power function hinges on the stochastic order of the underlying family of distributions [F_θ] rather than on the stronger property of monotone likelihood ratio of the family. An elementary proof, accessible to students of introductory probability and statistics, is presented.     

The optimal allocation of active redundancies to k-out-of-n systems with respect to hazard rate ordering

This paper deals with the allocation of active redundancies to a k-out-of-n system with independent and identically distributed (i.i.d.) components in the sense of the hazard rate order. It is shown that the system's hazard rate may be decreased by balancing the allocation of active redundancies. This generalizes the main result of Singh and Singh (1997) and improves the corresponding one of Hu and Wang (2009) as well. As an application, we build the reversed hazard rate order on order statistics from sample having proportional hazard rates, which strengthens the usual stochastic order in Theorem 2.1 of Pledger and Proschan (1971) to the reversed hazard order in the situation that all components are of (rational) proportional hazard rates.     

Comparisons of order statistics from heterogeneous negative binomial variables with applications

This paper studies the ordering properties of extreme order statistics arising from independent negative binomial random variables. Employing the useful tool of majorization-type orders, sufficient conditions are given for comparing extreme negative binomial order statistics according to the usual stochastic order. Some numerical examples are provided to illustrate the theoretical results. Applications in Poisson-Gamma shock model in reliability engineering and claim frequency in insurance are presented to show the practicability of our results as well.     

Preservation properties of stochastic orders by transformation to the transmuted-G model

Abstract

The transmuted-G model is a useful technique to construct some new distributions by adding a parameter. This paper considers stochastic comparisons in the transmuted-G family with different parameters and different baseline distributions in the sense of the usual stochastic, shifted stochastic, proportional stochastic and shifted proportional stochastic orders. Also, we present a necessary and sufficient condition for existence of the moments of the transmuted-G model and then we obtain some bounds for the survival and aging intensity functions of the transmuted-G model conditioned on its parameter and its baseline distribution.