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.     

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.     

Recurrence relations between product moments of order statistics

A general result for obtaining recurrence relations between product moments of order statistics is established and this result is used to determine the recurrence relations between product moments of some doubly truncated distributions. The examples considered are Weibull, exponential, Pareto, power function and Cauchy distributions.     

Higher order moments of order statistics from INID exponential random variables

In this paper we develop recurrence relations for the third and fourth order moments of order statistics from I.NI.D exponential random variables. Recurrence relations for the p-outlier model (with a slippage of p observations) are derived as a special case. Applications of these results will also be described.     

Higher order moments of order statistics from INID symmetric random variables

In this paper we present analogues of Balakrishnan's (1989) relations that relate the triple and quadruple moments of order statistics from independent and nonidentically distributed (I.NI.D.) random variables from a symmetric distribution to those of the folded distribution. We then apply these results, along with the corresponding recurrence relations for the exponential distribution derived recently by Childs (2003), to study the robustness of the Winsorized variance.     

Computing the moments of order statistics from nonidentical random variables

In this paper, we establish new representations, identities and recurrence relations of order statistics (o.s.) arising from general independent nonidentically distributed random variables (r.v.s). These recurrence relations will enable one to compute all moments of all o.s. in a simple manner. Applications for some known distributions are given.     

Higher order moments of order statistics from the Lindley distribution and associated inference

ABSTRACT

The Lindley distribution is an important distribution for analysing the stress–strength reliability models and lifetime data. In many ways, the Lindley distribution is a better model than that based on the exponential distribution. Order statistics arise naturally in many of such applications. In this paper, we derive the exact explicit expressions for the single, double (product), triple and quadruple moments of order statistics from the Lindley distribution. Then, we use these moments to obtain the best linear unbiased estimates (BLUEs) of the location and scale parameters based on Type-II right-censored samples. Next, we use these results to determine the mean, variance, and coefficients of skewness and kurtosis of some certain linear functions of order statistics to develop Edgeworth approximate confidence intervals of the location and scale Lindley parameters. In addition, we carry out some numerical illustrations through Monte Carlo simulations to show the usefulness of the findings. Finally, we apply the findings of the paper to some real data set.     

Expected sample moments of concomitants of selected order statistics

In this paper, the task of determining expected values of sample moments, where the sample members have been selected based on noisy information, is considered. This task is a recurring problem in the theory of evolution strategies. Exact expressions for expected values of sums of products of concomitants of selected order statistics are derived. Then, using Edgeworth and Cornish-Fisher approximations, explicit results that depend on coefficients that can be determined numerically are obtained. While the results are exact only for normal populations, it is shown experimentally that including skewness and kurtosis in the calculations can yield greatly improved results for other distributions.     

Alternating sums of sub-diagonal product moments of order statistics

A relation satisfied by the product moments of order statistics from an arbitrary continuous distribution symmetric about the origin has been established in this paper. We have shown that in such a situation, for samples of even size, sum of the sub-diagonal product moments of order statistics with alternating plus and minus signs assumes a compact form.     

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     

Computing the moments of order statistics from nonidentically distributed Erlang variables

Erlang distribution has wide applications in the field of reliability models, stochastic activity networks and many other fields. In this paper a recurrence relation for computing all moments of all order statistics arising from independent nonidentically distributed Erlang variables is established.     

Role of concomitants of order statistics in determining parent bivariate distributions

In this paper, we establish the role of concomitants of order statistics in the unique identification of the parent bivariate distribution. From the results developed, we have illustrated by examples the process of determination of the parent bivariate distribution using a marginal pdf and the pdf of either of the concomitant of largest or smallest order statistic on the other variable. An application of the results derived in modeling of a bivariate distribution for data sets drawn from a population as well is discussed.     

Limit theory of bivariate dual generalized order statistics with random index

The class of limit distribution functions of bivariate extreme, intermediate and central dual generalized order statistics from independent and identically distributed random variables with random sample size is fully characterized. Two cases are considered. The first case is when the random sample size is assumed to be independent of all basic random variables. The second case is when the interrelation of the random size and the basic random variables is not restricted.     

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.     

Recurrence relations for the moments of order statistics from a beta distribution

Certain recurrence relations for the single and product moments of the order statistics of a random sample of sizen arising from a beta distribution are derived. The usefulness of these relations in evaluating the single and product moments of beta order statistics is also discussed.     

Computing the moments of order statistics from independent nonidentically distributed Beta random variables

Recently Thomas and Samuel (Stat Pap 49:139–146, 2008) derived recurrence relations for the moments of order statistics arising form independent identically distributed Beta variables. In this paper the case of moments of independent nonidentically distributed Beta random variables is discussed. Another proof to the independent identical case is also presented.     

On the Markov property of order statistics

The order statistics from a sample of size n≥3 from a discrete distribution form a Markov chain if and only if the parent distribution is supported by one or two points. More generally, a necessary and sufficient condition for the order statistics to form a Markov chain for (n≥3) is that there does not exist any atom x₀ of the parent distribution F satisfying F(x₀-)>0 and F(x₀)<1. To derive this result a formula for the joint distribution of order statistics is proved, which is of an interest on its own. Many exponential characterizations implicitly assume the Markov property. The corresponding putative geometric characterizations cannot then be reasonably expected to obtain. Some illustrative geometric characterizations are discussed.     

An improved form of a recurrence relation on the product moments of order statistics

In this work, an improved version of an existing recurrence relation on the product moments of order statistics of a random sample of size n arising from an arbitrary distribution is derived.     

On the generation of subsets of order statistics

A modification of Schucany's (1972) method for generating order statistics is given which allows for efficient generation of certain subsets of order statistics.