Order statistics from non-identical power function random variables

In this paper, we derive some recurrence relations satisfied by the single and the product moments of order statistics arising from n independent and non-identically distributed power function random variables. These recurrence relations will enable one to compute all the single and the product moments of all order statistics in a simple recursive manner. The results for the multiple-outlier model are deduced as special cases. The results are further generalized to the case of truncated power function random variables.     

Order statistics from non-identical right-truncated Lomax random variables with applications

In this paper, we derive some recurrence relations for the single and the product moments of order statistics from n independent and non-identically distributed Lomax and right-truncated Lomax random variables. These recurrence relations are simple in nature and could be used systematically in order to compute all the single and product moments of all order statistics in a simple recursive manner. The results for order statistics from the multiple-outlier model (with a slippage of p observations) are deduced as special cases. We then apply these results by examining the robustness of censored BLUE's to the presence of multiple outliers. Received: November 30, 1998; revised version: March 8, 2000     

RECURRENCE RELATIONS FOR SINGLE AND PRODUCT MOMENTS OF GENERALIZED ORDER STATISTICS FROM PARETO,GENERALIZED PARETO,AND BURR DISTRIBUTIONS

We give recurrence relations for single and product moments of generalized order statistics under the concept of Kamps from Pareto, generalized Pareto and Burr distributions. The results include as particular cases the above relations for moments of k–th record values.     

Order statistics from discrete distributions

Several recurrence relations and identities available for single and product moments of order1 statistics in a sample size n from an arbitrary continuous distribution are extended for the discrete case,, Making use of these recurrence relations it is shown that it is sufficient to evaluate just two single moments and (n-l)/2 product moments when n is odd and two single moments and {n-2)/2 product moments when n is even, in order to evaluate the first, second and product moments of order statistics in a sample of size n drawn from an arbitrary discrete distribution, given these moments in samples of sizes n-1 and less.. A series representation for the product moments of order statistics is derived.. Besides enabling us to obtain an exact and explicit expression for the product moments of order statistics from the geometric distribution, it. makes the computation of the product moments of order statistics from other discrete distributions easy too.     

On order statistics from non-identical right-truncated exponential random variables and some applications

By considering order statistics arising from n independent non-identically distributed right-truncated exponential random variables, we derive in this paper several recurrence relations for the single and the product moments of order statistics. These recurrence relations are simple in nature and could be used systematically in order to compute all the single and the product moments of order statistics for all sample sizes in a simple recursive manner. The results for order statistics from a multiple-outlier model (with a slippage of p observations) from a right-truncated exponential population are deduced as special cases. These results will be useful in assessing robustness properties of any linear estimator of the unknown parameter of the right-truncated exponential distribution, in the presence of one or more outliers in the sample. These results generalize those for the order statistics arising from an i.i.d. sample from a right-truncated exponential population established by Joshi (1978, 1982).     

Recurrence relations for single and product moments of progressively Type-II censored order statistics from a generalized half-logistic distribution with application to inference

In this paper, we establish several recurrence relations for the single and product moments of progressively Type-II right-censored order statistics from a generalized half-logistic distribution. The use of these relations in a systematic recursive manner enables the computation of all the means, variances, and covariances of progressively Type-II right-censored order statistics from the generalized half-logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R ₁, …, R _m ). The results established here generalize the corresponding results for the usual order statistics due to Balakrishnan and Sandhu [Recurrence relations for single and product moments of order statistics from a generalized half-logistic distribution with applications to inference, J. Stat. Comput. Simul. 52 (1995), pp. 385–398.]. The moments so determined are then utilized to derive the best linear unbiased estimators of the scale and location–scale parameters of the generalized half-logistic distribution. The best linear unbiased predictors of censored failure times are discussed briefly. Finally, a numerical example is presented to illustrate the inferential method developed here.     

Order statistics from the linear-exponential distribution,part I: increasing hazard rate case

For the linear-exponential distribution with increasing hazard rate, exact and explicit expressions for means, product moments and percentage points of order statistics are obtained. Some recurrence relations for both single and product moments of order statistics are also derived. These recurrence relations would enable one to obtain all the higher order moments of order statistics for all sample sizes from those of the lower order     

Relations and identities for the moments of order statistics from a sample containing a single outlier

In this paper, we derive several new recurrence relations and indentities satisfied by the single and the product moments of order statistics from a sample of size n in the presence of an outlier. These recurrence relations involve the first two single moments and the product moments in samples of sized n?1 and less. By making use of these recurrence relations we show that it is sufficient to evaluate at most two single rerents and (n?2)/2 product moment when n is even and two single moments and (n?2)/2 product moments when n is odd, in order to evaluate the first and second single moments and product moments of all order statistics in a sample of size n comprising an outlier, given these moments for the all sample of size less than n. These generalize the results of Govindarajulu (1963), Joshi (1971), and Joshi and Balakrishnan (1982) to the case when the sample includes a single outlier. We also establish some simple identitites involving linear combination of convariances of order statistics. These results can be effectively used to reduce the amount of numerical computation considerably and also to check the accuracy of the computations while evaluating means, variances and covariances of order statistics from an outlier model.     

Recurrence relations for moments of progressive type-II right censored order statstics from burr distribution

In this paper some recurrence relations between moments of progressive Type-II right censored order statistics from doubly truncated Burr distribution are established. These recurrence relations would enable one to obtain all the single and product moments of Burr progressive Type-II right censored order statistics in a simple recursive manner.     

Recurrence Relations And Identities For Moments Of Order Statistics,II; Specific Continuous Distributions

In an earlier paper, Malik et al. (1987) have reviewed several recurrence relations and identities available for the single and product moments of order statistics from an arbitrary continuous distribution. In this paper, we review several such relations and identities established for both single and product moments of order statistics from some specific continuous distributions. We also mention some important applications or. these results.     

Recurrence relations for single and product moments of order statistics from a generalized half logistic distribution with applications to inference

In this paper, we derive several recurrence relations satisfied by the single and product moments of order statistics from a generalized half logistic distribution. These generalize the corresponding results for the half logistic distribution established by Balakrishnan (1985). The relations established in this paper will enable one to compute the single and product moments of all order statistics for all sample sizes in a simple recursive manner; this may be done for any choice of the shape parameter k. These moments can then be used to determine the best linear unbiased estimators of location and scale parameters from complete as well as Type-I1 censored samples.     

Recurrence relations mid identities for moments of order statistics,i: arbitrary continuous distribution

In this paper, we review several recurrence relations and identities established for the single and product moments of order statistics from an arbitrary continuous distribution. We point out the interrelationships between many of these recurrence relations. We discuss the results giving the bounds for the number of single and double integrals needed to be evaluated in order to compute the first, second and product moments of order statistics in a sample of size n from an arbitrary continuous distribution, given these moments in samples of sizes n-1 and less. Improvements of these bounds for the case of symmetric continuous distributions are also discussed     

Generalized exponential distribution: Moments of order statistics

In this paper, we consider the generalized exponential distribution (GED) with shape parameter α. We establish several recurrence relations satisfied by the single and the product moments for order statistics from the GED. The relationships can be written in terms of polygamma and hypergeometric functions and used in a simple recursive manner in order to compute the single and the product moments of all order statistics for all sample sizes.     

Recurrence relations for single and product moments of progressively Type-II censored order statistics from generalized logistic distribution with applications to inference

In this article, we establish several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from a generalized logistic distribution. The use of these relations in a systematic manner allow us to compute all the means, variances, and covariances of progressively Type-II right censored order statistics from the generalized logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R₁, …, R_m). These moments are then utilized to derive best linear unbiased estimators of the scale and location-scale parameters of the generalized logistic distribution. A comparison of these estimators with the maximum likelihood estimates is then made through Monte Carlo simulations. Finally, the best linear unbiased predictors of censored failure times is discussed briefly.     

Inferences for generalized exponential distribution based on record statistics

In this paper we consider three parameter generalized exponential distribution. Exact expressions for single and product moments of record statistics are derived. These expressions are written in terms of Riemann zeta and polygamma functions. Recurrence relations for single and product moments of record statistics are also obtained. These relations can be used to obtain the higher order moments from those of the lower order. The means, variances and covariances of the record statistics are computed for various values of the shape parameter and for some record statistics. These values are used to compute the coefficients of the best linear unbiased estimators of the location and scale parameters. The variances of these estimators are also presented. The predictors of the future record statistics are also discussed.     

Progressive type II censored order statistics from exponential distributions

In the model of progressive type II censoring, point and interval estimation as well as relations for single and product moments are considered. Based on two-parameter exponential distributions, maximum likelihood estimators (MLEs), uniformly minimum variance unbiased estimators (UMVUEs) and best linear unbiased estimators (BLUEs) are derived for both location and scale parameters. Some properties of these estimators are shown. Moreover, results for single and product moments of progressive type II censored order statistics are presented to obtain recurrence relations from exponential and truncated exponential distributions. These relations may then be used to compute all the means, variances and covariances of progressive type II censored order statistics based on exponential distributions for arbitrary censoring schemes. The presented recurrence relations simplify those given by Aggarwala and Balakrishnan (1996)     

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.     

On moments of dual generalized order statistics from Topp-Leone distribution

In this paper, we have derived exact and explicit expressions for the ratio and inverse moments of dual generalized order statistics from Topp-Leone distribution. This result includes the single and product moments of order statistics and lower records . Further, based on n dual generalized order statistics, we have deduced the expression for Maximum likelihood estimator (MLE) and Uniformly minimum variance unbiased estimator (UMVUE) for the shape parameter of Topp-Leone distribution. Finally, based on order statistics and lower records, a simulation study is being carried out to check the efficiency of these estimators.     

Moments of order statistics from doubly truncated continuous distributions and characterizations

In this paper, recurrence relations from a general class of doubly truncated continuous distributions which are satisfied by single as well as product moments of order statistics are obtained. Recurrence relations from doubly truncated generalized Weibull, exponential, Raleigh and logistic distributions have been derived as special cases of our result, Some previous results for doubly truncated Weibull, standard exponential, power function and Burr type XII distributions are obtained as special cases. The general recurrence relation of single moments has been used in the case of the left and right truncation to characterize the Weibull, Burr type XII and Pareto distributions.     

Recursive computation of the single and product moments of order statistics from the complementary exponential–geometric distribution

The complementary exponential–geometric distribution has been proposed recently as a simple and useful reliability model for analysing lifetime data. For this distribution, some recurrence relations are established for the single and product moments of order statistics. These recurrence relations enable the computation of the means, variances and covariances of all order statistics for all sample sizes in a simple and efficient recursive manner. By using these relations, we have tabulated the means, variances and covariances of order statistics from samples of sizes up to 10 for various values of the shape parameter θ. These values are in turn used to determine the best linear unbiased estimator of the scale parameter β based on complete and Type-II right-censored samples.