The purpose of this paper is to investigate a certain probability of a large deviation for a sequence of random variables $\{W_n\}$ which have moment-generating functions. We will assume that the mean of $W_n$ is given by $n\mu_n$ and the variance by $n\sigma_n^2$, where $\{\mu_n\}$ and $\{\sigma_n^2\}$ are covergent sequences. We seek the limit, as $n \rightarrow \infty$, of the expression $n^{-1} \ln P\lbrack W_n > na_n \rbrack,$ where $\{a_n\}$ is a convergent sequence with $\lim a_n > \lim \mu_n$. It is shown that, if the moment-generating function of $W_n$ satisfies certain limiting conditions, the above expression has a limit which depends on certain limits of this moment-generating function and its derivative. This result can be used in the computation of exact slopes for test statistics whose moment-generating function is known under the null hypothesis. Some applications are given.

The $k$-sample problem is studied. Given a set of independent observations from $k$ populations, the pooled sample is ordered, and certain predetermined order statistics are chosen to form the endpoints of random intervals. A random vector is formed, whose components represents the number of observations from each sample in each of these intervals. The exact and limiting distributions of this vector are derived, both under the null-hypothesis of a single underlying distribution and under the alternative of unequal distributions for the $k$ populations. This leads to the definition of a test-statistic for the null-hypothesis, and its limiting null-distribution and limiting distribution under a suitable sequence of alternative hypotheses are obtained. Hence a consistent test of the null-hypothesis is determined and shown to be an extension of the Mood-Brown test. Asymptotic efficiencies are calculated for this test relative to a family of common tests for the $k$-sample problem. To this end, a method is suggested for comparing efficiencies of tests whose limiting distributions under an alternative sequence are non-central chi-square, with unequal degrees of freedom. Some consideration is given to the problem of design for high relative efficiency.

In the multiple linear regression model a class of confidence regions based on rank statistics is constructed. The asymptotic behavior of the center of gravity of a region corresponding to the Wilcoxon type rank statistic is considered. Section 1 consists of assumptions, introduction and notation. Section 2 consists of a monotonicity lemma. A uniform continuity theorem for Wilcoxon type linear rank statistics is proved in Section 3. This and the monotonicity lemma are used to show that the regions are bounded for large sample size, with large probability. Section 4 gives the asymptotic normality of the above mentioned quantity and defines a consistent estimator of $(\int f^2(x))$. Finally, the Appendix contains some results on relative compactness of some stochastic processes which are used in Section 3.

This is a straightforward continuation of Hajek (1968). We provide a further extension of the Chernoff-Savage (1958) limit theorem. The requirements concerning the scores-generating function are relaxed to a minimum: we assume that this function is a difference of two non-decreasing and square integrable functions. Thus, in contradistinction to Hajek (1968), we dropped the assumption of absolute continuity. The main results are accumulated in Section 2 without proofs. The proofs are given in Sections 4 through 7. Section 3 contains auxiliary results.

The purpose of this paper is to compare a modified likelihood ratio test (Bartlett [2]) with the asymptotically UMP invariant test (Lehmann [8]) for testing homogeneity of variances of $k$ normal populations. We denote these tests by the "$M$ test" and "$L$ test," respectively. The $M$ test has been investigated by many authors, whereas the $L$ test has not. Fitting beta type distributions, Mahalanobis [9] and Nayer [11] computed the percentage points of $M$, when the numbers of observations in each sample are the same. Nayer's results were confirmed by Bishop and Nair [3], using the exact null distribution of $M$ in a form of infinite series derived by Nair [10]. Asymptotic series expansion of the null distribution of $M$ was obtained by Hartley [6], using Mellin inversion formula, from which tables for percentage points were calculated by Thompson and Merrington [16], without assuming the equality of $k$-sample sizes. Later in a more general formulation, Box [4] derived the asymptotic series expansions of the null distributions of many test statistics, including that of the $M$ test, by using the characteristic function. Recently Pearson [12] obtained some approximate powers of the $M$ test both by fitting a gamma type distribution to the inverse of the modified likelihood ratio statistic and by using the Monte Carlo method. No attempt was made, however, to investigate the asymptotic non-null distribution of $M$. Sugiura [18] has shown the limiting distribution of $M$ in multivariate case under fixed alternative hypothesis to be normal. In Section 2 of this paper we shall show that the $L$ test is not unbiased, though the $M$ test is known to be unbiased (Pitman [14], Sugiura and Nagao [19]). In Section 3, we shall derive the limiting distributions of $L$ and $M$ under sequences of alternative hypothesis with arbitrary rate of convergence to the null hypothesis as sample sizes tend to infinity. Limiting distributions are characterized by $\chi^2$, noncentral $\chi^2$, and normal distributions, according to the rate of convergence of the sequence. In Section 4, asymptotic expansion of the null distribution of $L$ is given in terms of $\chi^2$-distributions, and asymptotic formulas for the percentage points of $L$ and $M$ are obtained by using the general inverse expansion formula of Hill and Davis [7], with some numerical examples. In Section 5, asymptotic expansions of the non-null distributions of $L$ and $M$ under a fixed alternative hypothesis are derived in terms of normal distribution function and its derivatives, from which approximate powers are computed. It may be remarked that the limiting non-null distributions of $L$ and $M$ degenerate at the null hypothesis, by which asymptotic null distributions cannot be derived.

The only non-parametric multivariate paired comparison tests presently available for testing the hypothesis of no difference among several treatments are (i) the Sen-David (1968) test and (ii) the Davidson-Bradley (1969) test. Both these tests are applicable to situations which involve the preferences of each individual comparison. Both these tests are the generalizations of the one-sample multivariate sign tests [1]. As such their A.R.E.'s (Asymptotic Relative Efficiencies) with respect to the normal theory $\mathfrak{F}$-test are not expected to be high. In fact the A.R.E. of the Sen-David (1968) test with respect to the normal theory $\mathfrak{F}$-test can be as low as zero (under normality). The purpose of this paper is to develop test procedures which could be considered as competitors to the Sen-David (1968) and to the Davidson-Bradley (1968) tests. The proposed procedures are based on the ranks of the observed comparison differences, and include as special cases the multivariate normal scores and the multivariate rank sum paired comparison tests. For convenience of presentation we develop the theory when the paired comparisons involve paired characteristics. Under suitable regularity conditions the limiting distributions of the proposed test statistics are derived under the null as well as non-null hypotheses, and their large sample properties are studied. It is shown that for various situations of interest the proposed procedures have considerable efficiency improvements over the Sen-David (1968) and the normal theory procedures.

In this paper, loss is taken to be proportional to squared error with the constant of proportionality equal to the square of the inverse of a scale parameter, and an invariant estimator is defined to be one with risk invariant under transformations of location and scale. For certain classes of estimators, best (minimum-mean-squared-error) invariant estimators are found for specified linear functions of an unknown scale parameter and one or more unknown location parameters. Even when the specified function is equal to a single location parameter, the best invariant estimator is not equal to the best unbiased estimator in the class except for complete samples from certain distributions such as the Gaussian.

The present paper is partly expository and does not assume any previous knowledge of information theory or coding theory. It is meant to be the first in a series of papers on coding theory for noisy channels, this series replacing the report [12] which was widely circulated. A few of the results were reported in [11], [27] and [28]. In Sections 2 and 3 we present certain refinements and generalizations of known methods of Shannon, Fano, Feinstein and Gallager. A discussion of some other methods may be found in the surveys by Kotz [15] and Wolfowitz [28] such as those of Khintchine [14], McMillan [18] and Wolfowitz [25], [26]. Section 4 contains a number of applications. Most of this section is devoted to a certain memoryless channel with additive noise. Some of the proofs have been collected in Section 5. Finally, Section 6 describes some new results on the relative entropy $H(\mu_1\mid\mu_2),$ of one measure $\mu_1$ relative to another measure $\mu_2$, and its relation with the total variation $\|\mu_1 - \mu_2\|$.

Schwartz and Root [5] used Mehler's identity to obtain a bound for the integral of the absolute difference between the bivariate Gaussian density function and the product of its corresponding marginal densities. The result was also extended to the case of two dependent Gaussian vectors. The bounds were given in terms of the correlation coefficient in the bivariate case and canonical correlations in the two vector case. In this note an information-theoretic inequality is applied to derive a better bound than reached in [5] and to extend the result to the case of $m > 2$ dependent gaussian vectors. No series expansion is required as in [5].

It is well known that any quadratic form in random variables whose joint distribution is nondegenerate multivariate normal is distributed as a linear combination of independent noncentral chi-square variables. Thus, when normality holds, quadratic estimators of the components of variance associated with Eisenhart's [3] Model II are so distributed, even when the design is unbalanced. The problem considered here is that of determining the appropriate linear combinations for given estimators and designs. The expressing of quadratic variance-component estimators as linear combinations of independent noncentral chi-squares is useful, for several reasons, in studying the distributions of the estimators: (i) it permits application of results like those of Press [5] on the distributions of such linear combinations; (ii) it leads to Monte-Carlo techniques for approximating the distributions of these estimators which are more efficient than those heretofore used; and (iii) it may give some insight into the ways in which various types of imbalance affect the distributions of the estimators. We now define the transformation to be used in expressing quadratic forms as linear combinations of independent noncentral chi-squares. Let $\beta$ represent an $n \times n$ real symmetric matrix. Take $\mathbf{y}$ to be an $n \times 1$ random vector having the multivariate normal distribution with mean $\mathbf{u}$ and symmetric, nonsingular variance-covariance matrix $\mathbf{V}$. Clearly, the $n \times 1$ random vector $\mathbf{z}$ defined by the linear transformation $\mathbf{z} = \mathbf{W}'\mathbf{S}^{-1}\mathbf{C}'\mathbf{y}$; where $\mathbf{C}$ is an $n \times n$ orthogonal matrix whose columns are eigenvectors of $\mathbf{V}, \mathbf{S}$ is an $n \times n$ diagonal matrix whose $i$th diagonal element is the square root of the eigenvalue corresponding to the eigenvector represented by the $i$th column of $\mathbf{C}$, and $\mathbf{W}$ is an $n \times n$ orthogonal matrix whose columns are eigenvectors of the necessarily symmetric matrix $\mathbf{P} = \mathbf{SC}'\beta\mathbf{CS}$; has the multivariate normal distribution with \begin{equation*} \tag{1} E\{\mathbf{z}\} = \mathbf{W}'\mathbf{S}^{-1}\mathbf{C}'\mathbf{u}\end{equation*} and variance-covariance matrix $\mathbf{I}$ (the identity matrix). Then the distribution of the quadratic form $\mathbf{y}'\beta\mathbf{y}$ is the same as that of $\mathbf{z}'\mathbf{Dz}$, where $\mathbf{D} = \mathbf{W}'\mathbf{PW}$ is an $n \times n$ diagonal matrix whose $i$th diagonal element is the eigenvalue of $\mathbf{P}$ corresponding to the eigenvector represented by the $i$th column of $\mathbf{W}$. It is clear from the above that, for any specified $\mathbf{V}$ and $\beta$, the expression of $\mathbf{y}'\beta\mathbf{y}$ as a linear combination of independent noncentral chi-squares can be accomplished by determining $\mathbf{C}, \mathbf{S}$, and $\mathbf{W}$. The linear combination so obtained can be readily converted into a linear combination of independent noncentral chi-squares having distinct coefficients. It follows from the "only if" part of Theorem 1 of Baldessari [1] that this latter expression is unique; i.e., for the specified $\mathbf{V}, \beta$ pair, there exists no other linear combination of independent noncentral chi-squares, with distinct coefficients, having the same distribution as $\mathbf{y}'\beta\mathbf{y}$. In Sections 2-4, we study the determination of the $\mathbf{C, S}$, and $\mathbf{W}$ matrices for estimators of the variance components associated with the one-way random classification. While the techniques used here for the one-way classification can also be applied to estimators of the components associated with higher classifications, the complexity of the problem is much greater for the more-complicated models. In Section 2, results that are applicable to any quadratic form in the one-way data are given. Some special results are also presented on those quadratic estimators having a certain invariance property to be called `$\mu$-invariance.' It is shown that an estimator exhibiting this property is distributed as a linear combination of independent central chi-squares, and consequently it is not necessary to find a $\mathbf{W}$ matrix for such an estimator. In Section 3, attention is restricted to a certain subclass of quadratic estimators which includes all those in common usage, and, in Section 4, we further restrict ourselves to some cases where the subclass numbers display a very simple type of imbalance.

The purpose of this paper is to compare the asymptotic expected sample sizes of two sequential procedures for ranking $k$ normal populations with known variance and unknown means for the cases (i) $\mu_1 \leqq \mu_2 \leqq \cdots \leqq \mu_{k-1} < \mu_k$ and (ii) $\mu_k - \mu_{k-1} = \delta^\ast > 0$. The procedures are: (1) the Bechhofer-Kiefer-Sobel (BKS) sequential procedure [1], and (2) Paulson's (P) sequential procedure [2].

Let $X_1,\cdots, X_n$ be $n$ independent random variables with a common continuous distribution function (df) $F$. Let $F_n$ denote the sample df of the $X$'s. Let $\mathfrak{F}$ be the class of all continuous df's and $\mathfrak{F}_1$ denote the df's in $\mathfrak{F}$ which are symmetric about zero. To test the hypothesis $F \varepsilon \mathfrak{F}_1$ a common test is a weighted sign test of the form $\sum a_k \operatorname{sgn} X_k$ which has been studied by van Eeden and Bernard (1957), Hajek (1962), and Hajek and Sidak (1967). Usually the test included in nonparametric texts is for $a_k$ equal to the rank of $|X_k|$ and is known as the Wilcoxon one-sample or signed rank test. This test is consistent against certain alternatives including the case when $F$ is symmetric about some $\mu \neq 0$. That the test is not consistent against all alternatives in $\mathfrak{F} - \mathfrak{F}_1$ is evident from a discussion of its properties in Noether (1967). In this paper a test statistic for the hypothesis $F \varepsilon \mathfrak{F}_1$ is constructed in the spirit of the Kolmogorov-Smirnov statistics and shown to be consistent against all alternatives in $\mathfrak{F} - \mathfrak{F}_1$. Its $\operatorname{df}$ for both the finite and asymptotic cases are included along with the $\operatorname{df}$'s of two closely associated "one-sided" test statistics.