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The article discusses moments of the distribution of sample size, N in a sequential probability ratio test (SPRT). The present paper provides variance, the third and the fourth moments of N. The details are worked out in five common applications of the SPRIT. The relation of the variance of N to the truncation of a SPRT is discussed is also discussed in the paper. Scholar A. Wald indicated in passing how one can obtain the moments of N, but the only published literature where the author encountered a general expression for the variance of N. However, their expression is incorrect. Using scholar J. Wolfowitz's results, which they do, or differentiating Wald's, fundamental identity twice one gets provided. In many practical applications of the SPRT, μ and moments in an equation derived are differentiable functions of a real-valued parameter. The limiting expressions for the moments can then be determined by standard methods of mathematical analysis. However, for the third and fourth moments the actual technique may involve an excessive amount of arithmetic.

The principal part of this paper is devoted to the study of the distribution and density functions of the ratio of two normal random variables. It gives several representations of the distribution function in terms of the vivariate normal distribution and Nicholson's V function, both of which have been extensively studied, and for which tables and computational procedures are readily available. One of these representations leads to an easy derivation of the density function in terms of the Cauchy density and the normal density and integral. A number of graphs of the possible shapes of the density are given, together with an indication of when the density is unimodal or bimodal. The last part of the paper discusses the distribution of the ratio (u[sub 1] + ... + u[sub n])/(v[sub I] + ... + v[sub m]) where the u's and v's are independent, uniform variables. The exact distribution for all n and m is given, and some approximations discussed. [ABSTRACT FROM AUTHOR]

In this paper methods for obtaining approximations and bounds for the moments of order statistics from a continuous parent distribution are discussed. These bounds and approximations depend on the distribution function only through certain moments of order statistics in small samples. It is shown that for the Cauchy distribution bounds and approximations of all finite moments can be obtained. Some numerical calculations for normal and Cauchy distributions are also given. [ABSTRACT FROM AUTHOR]

This paper describes and derives the asymptotic behavior of the new Normal approximation introduced in Part I and of a family of Normal approximations based on roots, including the square root approximations of Fisher, and Freeman and Tukey, and the cube root approximations of Wilson and Hilferty, Camp, and Paulson. Various asymptotic comparisons are made, all of which rank the new approximation first, the cube root approximations second, and the other root approximations (and the ordinary Normal approximation) third. For instance, in the binomial case, if the tail probability is fixed as n arrow right infinity, the errors resulting from the foregoing approximations are generally of order n[sup -3/2], n[sup -1], and n[sup -1/2] respectively, while for a tail probability approaching 0, the relative error in approximating it approaches 0 if the corresponding standard Normal deviate is of smaller order than n[sup 1/2] n[sup 1/4], or n[sup 1/6] respectively, but not generally otherwise. These comparisons are deduced from far more detailed results which are also given. [ABSTRACT FROM AUTHOR]

It is desired to set confidence limits for the probability of successful operation at least until time x[sub 0] of a series system of k dissimilar components. The components follow exponential failure laws, and data are available on failure times of components of each type. Exact confidence limits for k=2 have previously been given by Lentner and Buehler. The present paper deals with a large-sample approximation to the exact solution for arbitrary k. [ABSTRACT FROM AUTHOR]

This paper gives analytic expressions for the moments and recurrence relations for the first four raw moments of a doubly noncentral t-distribution with v degrees of freedom and noncentrality parameters δ and λ. Table I provides numerical values of these moments for v = 2 (I) 20, λ = 2 (2) 8 (4) 20 and any suitable σ. Two approximations to the t"-distribution, involving the moments, are considered. [ABSTRACT FROM AUTHOR]

A general model for the evaluation of partial transfer evidence is developed in this article. This model is less restrictive than were the original models presented in an earlier paper, thus allowing it to be applied to a more general class of situations. A Bayesian approach to the problem of evidence evaluation is explored, and the probability of error calculated in the Bayesian model is shown to be a reasonable approximation to the probability of possible error calculated in the general model when these probabilities are small. [ABSTRACT FROM AUTHOR]

In this paper the first two moments of the ratio (treatment sum of squares)/(treatment sum of squares + error sum of squares)over all possible random assignment of treatments to the experimental plots, for a class of 2 associate PBIBD has been obtained. These two moments are compared with the corresponding moments of a continuous beta distribution to settle the question of approximating the randomization test by the usual F-test. It has been shown that a reasonable approximation to the randomization test based on the statistic F is equivalent to modifying the normal theory test by multiplying the numbers of d.f. of the F-distribution by a factor depending on the heterogeneity of the blocks. [ABSTRACT FROM AUTHOR]