Best bounds for order statistics and their expectations in range and mean units with applications

Best bounds for the order statistics are obtained in terras of the sample range and, for non-negative samples, in terms of the sample mean. Best bounds for the differences of two order statistics are found in this case also. Corresponding bounds on the expectation of order statistics and their differences are also found. Several applications of these bounds are considered     

On Order Statistics from Sine Distribution

We derive closed form expressions for the first two moments of order statistics from the sine distribution. For the higher moments, a recurrence relation is given. We also give a recurrence relation for the product moments. These relations will be useful for moment computations based on ordered data.     

Moment Information for Probability Distributions,Without Solving the Moment Problem. I: Where is the Mode?

How much information does a small number of moments carry about the unknown distribution function? Is it possible to explicitly obtain from these moments some useful information, e.g., about the support, the modality, the general shape, or the tails of a distribution, without going into a detailed numerical solution of the moment problem? In this paper a theoretical result of Johnson and Rogers is generalized to be valid for all moment problems and is exploited to demonstrate that a few moments are able to provide us with valuable information about the position of the mode of an unknown (unimodal) distribution.     

The 3 F 2 with Complex Parameters as Generating Function of Discrete Distribution

A new discrete family of probability distributions that are generated by the ₃ F ₂ function with complex parameters is presented. Some of the properties of this new family are studied as well as methods of estimation for its parameters. It affords considerable flexibility of shape which turns the distribution into an appropriate candidate for modeling data that cannot be adequately fitted by classical families with fewer parameters. Finally, three examples in the fields of Agriculture and Education are included in order to show the versatility and utility of this distribution.     

Moments for a ratio of correlated gamma variates

This paper considers the finite integral moments for the ratio, R = X/Y, where X and Y re correlated gamma distributed variables. An analytical and numerical comparison is given for two classes of underlying bivariate gamma distributions. It is shown that the two bivariate gamma structures provide indentical experessions for the mth unadjussted moment, E(R^m), if and only if either of the following conditions hold : 1) X and Y are uncorrelated of 2) m=1. A numerical evaluation is performed to determine the extent that the two methods differ whenever the variables are correlated     

Construction of orthogonal factorial designs controlling interaction efficiencies

This paper employs some variants of the usual Kronecker product to construct orthogonal factorial designs controlling the interaction efficiencies. The methods suggested have a fairly wide coverage and the resulting designs involve a small number of replicates.     

Estimating the mean of the selected uniform population

Let π_i(i=1,2,…K) be independent U(0,?_i) populations. Let Y_i denote the largest observation based on a random sample of size n from the i-th population. for selecting the best populaton, that is the one associated with the largest ?_i, we consider the natural selection rule, according to which the population corresponding to the largest Y_i is selected. In this paper, the estimation of M. the mean of the selected population is considered. The natural estimator is positively biased. The UMVUE (uniformly minimum variance unbiased estimator) of M is derived using the (U,V)-method of Robbins (1987) and its asymptotic distribution is found. We obtain a minimax estimator of M for K≤4 and a class of admissible estimators among those of the form cY_max. For the case K = 2, the UMVUE is improved using the Brewster-Zidek (1974) Technique with respect to the squared error loss function L₁ and the scale-invariant loss function L₂. For the case K = 2, the MSE'S of all the estimators are compared for selected values of n and ρ=?₁/(?₁+?₂).     

On shrinkage m-estimators of location parameters

For a general class of continuous ( and marginally symmetric ) inultivariate distributions, based on suitable M-statistics ( involving bounded but possibly discontinuous score generating functions), shrinkage estimators of location are considered. These estimators are based on the James-Stein type rule and incorporates the idea of preliminary test estimation too. The main emphasis is laid on the study of asymptotic tdistributional ) risk properties of these est-innators, and asymptotic tin-) adraissibility results are also studied under fairly general regularity conditions.     

LIMIT THEOREMS FOR ASYMPTOTICALLY MINIMAX ESTIMATION OF A DISTRIBUTION WITH INCREASING FAILURE RATE UNDER A RANDOM MIXED CENSORSHIP/TRUNCATION MODEL

ABSTRACT

The search for optimal non-parametric estimates of the cumulative distribution and hazard functions under order constraints inspired at least two earlier classic papers in mathematical statistics: those of Kiefer and Wolfowitz[1] Kiefer, J. and Wolfowitz, J. 1976. Asymptotically Minimax Estimation of Concave and Convex Distribution Functions. Z. Wahrsch. Verw. Gebiete, 34: 73–85. [Crossref], [Web of Science ®] , [Google Scholar] and Grenander[2] Grenander, U. 1956. On the Theory of Mortality Measurement. Part II. Scand. Aktuarietidskrift J., 39: 125–153.  [Google Scholar] respectively. In both cases, either the greatest convex minorant or the least concave majorant played a fundamental role. Based on Kiefer and Wolfowitz's work, Wang3-4 Wang, J.L. 1986. Asymptotically Minimax Estimators for Distributions with Increasing Failure Rate. Ann. Statist., 14: 1113–1131. Wang, J.L. 1987. Estimators of a Distribution Function with Increasing Failure Rate Average. J. Statist. Plann. Inference, 16: 415–427.   found asymptotically minimax estimates of the distribution function F and its cumulative hazard function Λ in the class of all increasing failure rate (IFR) and all increasing failure rate average (IFRA) distributions. In this paper, we will prove limit theorems which extend Wang's asymptotic results to the mixed censorship/truncation model as well as provide some other relevant results. The methods are illustrated on the Channing House data, originally analysed by Hyde.5-6 Hyde, J. 1977. Testing Survival Under Right Censoring and Left Truncation. Biometrika, 64: 225–230. Hyde, J. 1980. “Survival Analysis with Incomplete Observations”. In Biostatistics Casebook 31–46. New York: Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics.       

On Block Skip Free Transition Matrices and First Passage Times

Starting from an abstract setting which extends the property “skip free to the left” for transition matrices to a partition of the state space, we develop bounds for the mean hitting time of a Markov chain to an arbitrary subset from an arbitrary initial law. We apply our theory to the embedded Markov chains associated with the M/G/1 and the GI/M/1 queueing systems. We also illustrate its applicability with an asymptotic analysis of a non-reversible Markovian star queueing network with losses.