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Given N points independently drawn from the uniform distribution on (0, 1), let p[sub n], be the size of the smallest interval that contains n out of the N points; let n[sub p], be the largest number of points to be found in any subinterval of (0, 1) of length p. This paper uses a result of Karlin, McGregor, Barton and Mallows to determine the distribution of n[sub p] for p = 1/k, k an integer. The paper gives simple determinations for the expectations and variances of p[sub n], for all fixed n > (N + 1)/2, and of n[sub 1/2]. The distribution and expectation of n[sub p] are estimated and tabulated for the cases p = 0.1(0.1)0.9, N =2(1)10. [ABSTRACT FROM AUTHOR]

Maximum likelihood and other BAN estimators have been shown to possess certain optimal asymptotic properties in estimating the parameters of probability distributions satisfying specific regularity conditions. The subject of non-regular estimation is concerned with problems in which these conditions do not hold. In many such problems, classical-lower bounds on the variance of unbiased estimators, such as the Cramer-Rao bound, lead to the trivial result V(t) * (These characters cannot be converted in ASCII text) 0, where t is any unbiased estimator. A number of alternative bounds for application in the non-regular case have been derived. In this paper previous results of this type are reviewed and an additional bound is given. The specific applications of interest involve estimation of a location parameter. Applications of the bounds to the exponential, uniform and Pearson Type III distributions are investigated. [ABSTRACT FROM AUTHOR]

This paper presents three upper bounds for the variance of an estimator, based on observations selected with varying probabilities from a finite population, the elements of which are ranked with respect to the Y values. Accordingly, the usefulness of these bounds relates to the pre-enumeration analysis where one may well know the intended probabilities and joint probabilities corresponding to the sampling scheme but does not know the Y values. If, however, one can make a conservative guess at the largest Y value, one can use these bounds. Some examples are included to illustrate the theory. [ABSTRACT FROM AUTHOR]

The paper presents the results of an inquiry concerning the small sample relative efficiency of maximum likelihood and best unbiased estimators of reliability functions of one-unit systems. Three cases are considered: The Poisson, exponential, and normal (standard deviation known). Two kinds of relative efficiency functions are studied. The first kind consists of the common ratio of the Cramer-Rao lower bound of the variances of unbiased estimators, to the mean-square-error of the considered estimator. The second kind is a new type of a relative efficiency function, which is called 'the closeness relative efficiency function.' This function is defined as the ratio of the probabilities that the maximum likelihood and the beat unbiased estimators yield estimates in a prescribed neighborhood of the unknown reliability value. A substantial part of the study is devoted to the derivation of the required moments of the estimators. [ABSTRACT FROM AUTHOR]