Bounds for moments through a general orthogonal expansion in a pre-hilbert space-II

In an earlier article [Canad. J. Statist., Vol, 3, No. 1, 1975, 13–34] bounds are obtained for the product moments of an arbitrary finite number of ordered random variables. These bounds are obtained with the help of a representation of an arbitrary function in terms of a complete orthonormal system in a pre-Hilbert space of square integrable functions defined in a k-dimensional unit cube. These results are extended to symmetric cases in this article.     

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.     

Moments of Order Statistics—From Even to Odd Sample Sizes

Abstract

A method is demonstrated to compute the complete set of first moments of order statistics for an arbitrary distribution, given only the first moments of the maximal order statistics either for all even sample sizes, or for all odd samples sizes.     

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.     

Moments for matrix normal variables

In order to obtain moments for matrix normally distributed variables the moment generating function is differentiated by aid of matrix derivatives. Moments of arbitrary order as well as a recursive relation are obtained. Further, some more details are given for the first four moments     

Moments of order statistics from one truncation parameter densities

Analogs of some classical recurrence relations for moments of order statistics for a one truncation parameter densities are derived. These relate the k^thorder moment of the r^thorder statistic X_r:nto the k^thorder moment of X_l:ior X_r:rvia an integral operator. Similar results are obtained for product moments. An application is also given.     

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.     

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.     

Estimation of long-run inefficiency levels: a dynamic frontier approach

Cornwell, Schmidt, and Sickles (1990) and Kumbhakar (1990), among others, developed stochasticfrontier production models which allow firm specific inefficiency levels to change over time. These studies assumed arbitrary restrictions on the short-run dynamics of efficiency levels which have little theoretical justification. Further, the models are inappropriate for estimation of long-run efficiencies. We consider estimation of an alternative frontier model in which firmspecific technical inefficiency levels are autoregressive. This model is particularly useful to examine a potential dynamic link between technical innovations and production inefficiency levels. We apply our methodology to a panel of US airlines.     

Estimation of long-run inefficiency levels: a dynamic frontier approach

Cornwell, Schmidt, and Sickles (1990) and Kumbhakar (1990), among others, developed stochasticfrontier production models which allow firm specific inefficiency levels to change over time. These studies assumed arbitrary restrictions on the short-run dynamics of efficiency levels which have little theoretical justification. Further, the models are inappropriate for estimation of long-run efficiencies. We consider estimation of an alternative frontier model in which firmspecific technical inefficiency levels are autoregressive. This model is particularly useful to examine a potential dynamic link between technical innovations and production inefficiency levels. We apply our methodology to a panel of US airlines.     

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.     

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.     

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.     

Two identities involving order statistics in the presence of an outlier

In this note, we derive two simple identities involving order statistics from a sample of size n in the presence of an outlier. These generalize the results of Joshi (1973). These identities will be quite useful in checking the computation of the single moments of order statistics from an outlier model.     

Joint distribution of rises,falls, and number of runs in random sequences

By using the matrix formulation of the two-step approach to the distributions of runs, a recursive relation and an explicit expression are derived for the generating function of the joint distribution of rises and falls for multivariate random sequences in terms of generating functions of individual letters, from which the generating functions of the joint distribution of rises, falls, and number of runs are obtained. An explicit formula for the joint distribution of rises and falls with arbitrary specification is also obtained.     

Unbiased Estimation of the Distribution Function of a Two-Parameter Exponential Population Using Order Statistics

In this article, we consider the problem of unbiased estimation of the distribution function of a two-parameter exponential population using order statistics based on a random sample from the population. We give necessary and sufficient conditions for the existence of an unbiased estimator based on an arbitrary set of order statistics and suggest unbiased estimators in some situations where unbiased estimators exist. A few properties of the suggested estimators for some special cases have also been discussed.     

Conditioning of Marked Point Processes within a Bayesian Framework

Shale units with low permeability create barriers to fluid flow in a sandstone reservoir. A spatial stochastic model for the location of shale units in a reservoir is defined. The model is based on a marked point process formulation, where the marks are parameterized by random functions for the shape of a shale unit. This extends the traditional formulation in the sense that conditioning on the actual observations of the shale units is allowed in an arbitrary number of wells penetrating the reservoir. The marked point process for the shale units includes spatial interaction of units and allows a random number of units to be present. The model is defined in a Bayesian setting with prior pdfs assigned to size–shape parameters of shale units. The observations of shales in wells are associated with a likelihood function. The posterior pdf of the marked point process can only partially be developed analytically; the final solution must be determined by sampling using the Metropolis–Hastings algorithm. An example is presented, demonstrating the consequences of increasing the number of wells in which observations are made.     

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)     

Rank statistics expressible as integrals under P–P-plots and receiver operating characteristic curves

Given a probability measure on the unit square, the measure of the region under an empirical P – P -plot defines a two-sample rank statistic. Instances include trimmed and censored versions of the Mann–Whitney–Wilcoxon statistic and a class of statistics with applications in the analysis of receiver operating characteristic (ROC) curves. A large sample distribution for such a statistic is obtained, which is valid under sampling from general populations. Explicit results are presented for comparing arbitrary quantile segments of two populations. The results are not restricted to continuous data and incorporate adjustments for tied values in the discrete case. A multivariate version of the large sample distribution extends the class of tractable statistics in ROC analysis and facilitates the use of methods based on partial areas when the data are discrete.     

Optimal bounds on l-statistics based on samples drawn with replacement from finite populations

We describe a method of establishing optimal bounds on the expectations of arbitrary linear combinations of order statistics based on iid samples drawn with replacement from finite populations of a fixed size. The bounds are expressed in terms of the population size, mean, central absolute moments, and coefficients of the combination. The bounds are precisely determined for the trimmed means and their differences, and single order statistics and their differences in particular. We also show that with increase in population size, our bounds approach the respective universal ones for arbitrary iid samples.