Z. Govindarajulu

Characterizations of Geometric Distribution and Discrete IFR (DFR) Distributions Using Order Statistics.

Let X be a discrete random variable the set of possible values (finite or infinite) of which can be arranged as an increasing sequence of real numbers a1<a2<a3<…. In particular, ai could be equal to i for all i. Let X1n≦X2n≦⋯≦Xnn denote the order statistics in a random sample of size n drawn from the distribution of X, where n is a fixed integer ≧2. Then, we show that for some arbitrary fixed k(2≦k≦n), independence of the event {Xkn=X1n} and X1n is equivalent to X being either degenerate or geometric. We also show that the montonicity in i of P{Xkn = X1n | X1n = ai} is equivalent to X having the IFR (DFR) property. Let ai = i and G(i) = P(X≧i), i = 1, 2, …. We prove that the independence of {X2n − X1n ∈B} and X1n for all i is equivalent to X being geometric, where B = {m} (B = {m,m+1,…}), provided G(i) = qi−1, 1≦i≦m+2 (1≦i≦m+1), where 0<q<1.

Characterization of the geometric distribution using properties of order statistics

Abstract Using certain properties of order statistics, the geometric distribution has been characterized when the components are independent and identically distributed. When the components are independent, the geometric distribution has been characterized in the class of either IFR or DFR discrete distributions. In particular, Fergusons (1967) characterization theorem for independent components in a sample of size two has been extended in several directions.

Corrected confidence intervals following a sequential adaptive clinical trial with binary responses

A sequential clinical trial model is considered in which two treatments are compared and the responses are binary. One of the properties of the model is that, for a given stopping boundary, the error probability is insensitive to the allocation rule. It is shown how a corrected confidence interval may be determined for the log odds ratio at the end of the trial. Simulation is used to assess the accuracy of the method for a selection of stopping boundaries and data-dependent allocation rules.

An asymptotically distribution-free test for ordered alternatives in two-way layouts

Abstract An asymptotically distribution-free test is proposed for unequally spaced ordered alternatives in two-way layouts. The test statistic is a linear function of the ranks of residuals when the nuisance parameters are estimated. We show that the limiting distribution of the test statistic, when properly standardized, is normal. The asymptotic relative efficiency comparisons (in Pitman sense) with respect to the likelihood derivative test and nonparametric tests for randomized complete blocks show that our procedure is generally more powerful.

Sequential estimation of the variance of an unknown distribution

Sequential fixed-width and risk-efficient estimation of the variance of an unspecified distribution is considered. The second-order asymptotic properties of the sequential rules are studied. Extensive simulation studies are carried out in order to study the small sample behavior of the sequential rules for some frequently used distributions.

Stopping times of one-sample rank order sequential probability ratio tests

Abstract Weed, Bradley and Grovindarajulu (1974) propose one-sample probability ratio tests based on Lehmann alternatives. They also study the finite sure termination of the stopping times. Motivated by Steins proof of (1946) of the termination of a sequential probability ratio test (SPRT) in the case of independent and identically distributed (i.i.d.) random variables and the work of Sethuraman (1970) for the two- sample rank order SPRT, we obtain a very mild condition (namely, that a certain random variable U ( Z ) is not identically zero) for the finite sure termination of the existence of the moment generating function (m.g.f.) for the stopping time of one- sample rank order SPRTs.

Sequential determination of the number of bootstrap samples

The primary purpose of this paper is that of developing a sequential Monte Carlo approximation to an ideal bootstrap estimate of the parameter of interest. Using the concept of fixed-precision approximation, we construct a sequential stopping rule for determining the number of bootstrap samples to be taken in order to achieve a specified precision of the Monte Carlo approximation. It is shown that the sequential Monte Carlo approximation is asymptotically efficient in the problems of estimation of the bias and standard error of a given statistic. Efficient bootstrap resampling is discussed and a numerical study is carried out for illustrating the obtained theoretical results.

The 'demon' problem of Youden: Exponential case

We consider the problem of finding the probability of a sample mean falling above the (n - k)th-order statistic in a random sample of size n. Explicit expressions are obtained for the exponential distribution. Some applications that pertain to testing for outliers and goodness of fit are given.

A brief review of the role of lattices in rank order statistics

Abstract Partial ordering of the rank order probabilities arising in one-sample, two-sample and c -sample cases is essential in constructing most powerful rank order statistics. Savage (1956, 1957, 1959, 1960, 1964), Narayana (1978) and Narayana et al. (1978) spent considerable time in exploring such partial orderings by applying lattice theory methods. In this article we review some of these results and also indicate how certain recursive schemes are essential in computing the distribution of rank tests.