On generalized order statistics from linear exponential distribution and its characterization

In this paper, recurrence relations for single and product moments of generalized order statistics (gOSs) from linear exponential distribution (LE) are derived and characterizations of this distribution based on the conditional moments of the gOSs are given.     

Characterizations of geometric distribution through progressively Type-II right-censored order statistics

In this paper, we consider characterizations of geometric distribution based on some properties of progressively Type-II right-censored order statistics. Specifically, we establish characterizations through conditional expectation, identical distribution, and independence of functions of progressively Type-II right-censored order statistics. Moreover, extensions of these results to generalized order statistics are also sketched. These generalize the corresponding results known for the case of ordinary order statistics.     

Extended tables for moments of gamma distribution order statistics

Apart from having intrinsic mathematical interest, order statistics are also useful in the solution of many applied sampling and analysis problems. For a general review of the properties and uses of order statistics, see David (1981). This paper provides tabulations of means and variances of certain order statistics from the gamma distribution, for parameter values not previously available. The work was motivated by a particular quota sampling problem, for which existing tables are not adequate. The solution to this sampling problem actually requires the moments of the highest order statistic within a given set; however the calculation algorithm used involves a recurrence relation, which causes all the lower order statistics to be calculated first. Therefore we took the opportunity to develop more extensive tables for the gamma order statistic moments in general. Our tables provide values for the order statistic moments which were not available in previous tables, notably those for higher values of m, the gamma distribution shape parameter. However we have also retained the corresponding statistics for lower values of m, first to allow for checking accuracy of the computtions agtainst previous tables, and second to provide an integrated presentation of our new results with the previously known values in a consistent format     

Generalized order statistics from arbitrary distributions and the Markov chain property

The joint and marginal distributions of generalized order statistics based on an arbitrary distribution function are established in terms of the lexicographic distribution function. Furthermore, we show that generalized order statistics and the corresponding number of ties form a two-dimensional Markov chain.     

Some identities among moments of order statistics

Some new identities among the m oments of order statistics are derived. These are more general in nature and are applicable when moments of Some extreme order statistics do not exist.     

Inequalities for generalized order statistics from some restricted family of distributions

The Steffensen inequality is applied to derive quantile bounds for the expectations of generalized order statistics from a distribution belonging to a particular subclass of distributions. The subclass consists of F having the property that F^?1(0+)=x₀>0 and that x →[1? F(x)]x^z is nonincreasing for all x > X₀ and some z > 0.     

Moments of order statistics of the standard two-sided power distribution

The standard two-sided power distribution is a flexible distribution having uniform, power function and triangular as subdistributions, and it is a reasonable alternative to the Laplace distribution in some cases. In this work, computationally efficient expressions for moments of order statistics, expressions for L-moments, and asymptotic results for sample extrema are derived. Then a simulation study is performed for the location-scale estimation problem of a small data set by considering the maximum likelihood estimation method and the best linear unbiased estimation method based on the moments of order statistics.     

Approximations of Variances and Covariances for Order Statistics from the Standard Extreme Value Distribution

We consider simple approximations of variances and covariances for order statistics from the standard extreme value distribution. Exact values and simulation results of the variances and covariances for certain sample sizes are used to determine the validity of the suggested approximations.     

Moments of order statistics from a non-overlapping mixture model with applications to truncated laplace distribution

We employ two different approaches to derive single and product moments of order statistics from a truncated Laplace distribution. A direct evaluation method establishes recurrence relations whereas the more general non-overlapping mixture model incorporates the truncated Laplace distribution as a special case. The results are thereafter applied to estimate location and scale parameters of such distributions.     

Regression via order statistics and their concomitants

We consider a five-dimensional normal distribution and derive the exact joint distribution one variable, linear combinations of order statistics from two other variables, and linear combinations of the corresponding concomitants of these order statistics. We show that this joint distribution is a mixture of trivariate unified skew-normal distributions. This mixture representation enables us to predict one variable based on linear combinations of order statistics from two other variables and linear combinations of the corresponding concomitants. We finally illustrate the usefulness of these results by using a real data.     

Fitting the generalized Pareto distribution to data based on transformations of order statistics

Generalized Pareto distribution (GPD) has been widely used to model exceedances over thresholds. In this article we propose a new method called weighted nonlinear least squares (WNLS) to estimate the parameters of the GPD. The WNLS estimators always exist and are simple to compute. Some asymptotic results of the proposed method are provided. The simulation results indicate that the proposed method performs well compared to existing methods in terms of mean squared error and bias. Its advantages are further illustrated through the analysis of two real data sets.     

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.     

Some distributional problems connected with order statistics

Two characterizations of distributions symmetric about zero are given. These are based on the distributional properties of the squates of the order statistics from a random sample from these distributions. A result explering the relation between the distribution funcitons of two unordered (not necessarily independent) variables and those of their order statistics is presented. This has some interesting applications.     

On order statistics from thelog-logistic distribution and their properties

The aim of this paper is to derive the exact forms of the p.d.f. and the moments of the rth order statistics in a sample of size n from the Log-logistic (L_l ) distribution. Measures of skewness and kurtosis are tabulated. The recurrence relations between the moments of all order statistics and an expression of the covariance between any two order statistics, x_i and x_jand the distribution of the ratio of X_i to x_j are derived.     

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.     

Two-sample Bayesian prediction for sequential order statistics from exponential distribution based on multiply Type-II censored samples

In this paper, the problem of predicting the future sequential order statistics based on observed multiply Type-II censored samples of sequential order statistics from one- and two-parameter exponential distributions is addressed. Using the Bayesian approach, the predictive and survival functions are derived and then the point and interval predictions are obtained. Finally, two numerical examples are presented for illustration.     

Nonlinear regression in a trivariate elliptical distribution via a linear combination of order statistics

In this paper, by assuming that (X, Y ₁, Y ₂)^T has a trivariate elliptical distribution, we derive the exact joint distribution of X and a linear combination of order statistics from (Y ₁, Y ₂)^T and show that it is a mixture of unified bivariate skew-elliptical distributions. We then derive the corresponding marginal and conditional distributions for the special case of t kernel. We also present these results for an exchangeable case with t kernel and illustrate the established results with an air-pollution data.     

Outliers and order statistics

Outliers are to be found among the extremes of a data set. Extremes are examples of order statistics. It is thus relevant to ask to what extent the statistical methods (and probabilistic properties) of outliers and of order statistics coincide and depend on each other. Whilst clear overlap is identifiable, aims and procedures are often quite distinct and each topic plays its own important role in the panoply of statistical principles and methodology.     

A note on moments of order statistics from exchangeable variates

Joshi (1973) and Balakrishnan and Malik (1985) have derived some some identities for the moments of order statistics from independent and identically distributed random variables. In this paper, we make use of a basic result due to David and Joshi (1968) and show that these identities for the moments also hold when the order statistics arise from exchangeable variables.