It is known (Lehmann (1959), p. 187, and Bell, Moser and Thompson (1966), p. 134) that for the two-sample problem, the closer the two samples are together stochastically, the smaller is the power of monotone rank tests. Here it is shown that if one uses the ideas of van Zwet (1964) to define "skewness" and "heavy tails," then the more skew the distributions of the two samples are, the smaller is the power of monotone rank tests; and heavy tails similarly leads to smaller power of monotone rank tests. Skewness and heavy tails are defined using convex and star-shaped transformations of random variables. These are the same transformations used in reliability theory (Barlow and Proschan (1965), Birnbaum, Esary and Marshall (1966), and others) to describe the concept of "wear-out." Thus if $X$ is a random variable that represents "time to failure," and if failure is caused by wear-out or by the environment, then there exists a convex or starshaped function $g$ such that $Z = g(X)$ is an exponential $(1 - \exp \lbrack -\lambda z \rbrack)$ random variable. The distributions of these variables $X$ are called increasing failure rate (IFR) distributions when $g$ is convex and (IFRA) distributions when $g$ is starshaped. It turns out that if one restricts attention to such distributions, then the results of this paper can be used to construct a simple optimality theory for rank tests. This is done in a later paper [6]. The power inequalities related to skewness and heavy tails readily extends to sequential rank tests. It is shown (Example 5.1) that the sequential probability ratio test based on ranks for exponential scale alternatives (e.g. [11] and [12]) also is valid for the class of IFRA scale alternatives.

Let $X_1, X_2, \cdots$ be independent identically distributed random variables having a common probability density function $f$. After a so-called kernel class of estimates $f_n$ of $f$ based on $X_1, \cdots, X_n$ was introduced by Rosenblatt [7], various convergence properties of these estimates have been studied. The strongest result in this direction was due to Nadaraya [5] who proved that if $f$ is uniformly continuous then for a large class of kernels the estimates $f_n$ converges uniformly on the real line to $f$ with probability one. For a very general class of kernels, we will show that the above assumptions on $f$ are necessary for this type of convergence. That is, if $f_n$ converges uniformly to a function $g$ with probability one, then $g$ must be uniformly continuous and the distribution $F$ from which we are sampling must be absolutely continuous with $F' (x) = g(x)$ everywhere. When in addition to the conditions mentioned above, it is assumed that $f$ and its first $r + 1$ derivatives are bounded, we are able to show how to construct estimates $f_n$ such that $f^{(s)}_n$ converges uniformly to $f^{(s)}$ at a given rate with probability one for $s = 0, 1, \cdots, r$.

Let $(x_1, x_2, \cdots, x_n)$ be independent realizations of a random variable taking values on a circle $C$ of unit circumference, and let $T_n = n^{-1} \int^1_0 \lbrack \sum^n_{j=1} f(x + x_j) - n \rbrack^2 dx,$ where $f(x)$ is a probability density on $C, f \varepsilon L_2\lbrack 0, 1 \rbrack$, and the addition $x + x_j$ is performed modulo 1. $T_n$ is used to test whether the observations are uniformly distributed on $C$. It includes as special cases several other statistics previously proposed for this purpose by Ajne, Rayleigh and Watson. The main results of the paper are the asymptotic distributions of $T_n$ under fixed alternatives to uniformity and under sequences of local alternatives to uniformity. A characterization is found for those alternatives against which $T_n$, with specified $f(x)$, gives a consistent test. The approximate Bahadur slope of $T_n$ is calculated from the asymptotic null distribution; however, an example indicates that this slope may not always reflect the power of $T_n$ reliably. A Monte Carlo simulation for a special case of $T_n$ suggests that a fair approximation to the power of $T_n$ may be obtained from its mean and variance under the alternative.

Wald [8] treated the problem of testing $H_0:\theta = \theta_0$ against one-sided alternatives by giving conditions under which tests based on the maximum likelihood estimator are asymptotically most powerful. He defines a sequence of tests $\{\lambda_n\}$ to be an asymptotically most powerful test of $H_0$ against $H_1:\theta > \theta_0$, on level of significance $\alpha$, if for any other sequence of tests $\{\omega_n\}$ of level $\alpha$, \begin{equation*} \tag{1.1} \lim \sup \lbrack \sup (\varepsilon_\theta\omega_n - \varepsilon_\theta\lambda_n; \theta > \theta_0, \theta \varepsilon \Theta) \rbrack = 0,\end{equation*} where $\Theta$ is the parameter space. A similar expression holds for testing $H_0$ against $H_2:\theta < \theta_0$. Wald's regularity conditions on the population density are quite strong. The maximum likelihood estimator, being the value which maximizes the likelihood function, is required to be a consistent estimator in the probability sense and the consistency must be uniform over certain intervals in $\Theta$. Also, his conditions imply that the centered and scaled version converges uniformly in law, over certain intervals in $\Theta$, to the standard normal. Wald formulates tests in terms of the maximum likelihood estimate. In the present work, we extend Wald's results in two directions. First, the regularity conditions are substantially weakened through use of the techniques of LeCam and the tests need not be based on the maximum likelihood estimate. Secondly, tests concerning the parameter in the joint distribution of the random variables involved are shown to be asymptotically most powerful when the observations arise from a stationary Markov process. In order that an $\alpha$-level test exist for any $\alpha$, it was necessary to consider tests which are possibly randomized. Section 2 contains the basic assumptions on the Markov process and the preliminary results appear in Section 3. The results through Section 3 hold for a $k$-dimensional parameter space and are presented in this general formulation. The main results are presented as Theorems 4.1 and 4.2 in Section 4. The following section treats the special case of independent identically distributed random variables. Four examples are presented in Section 6. In order to avoid unnecessary repetition in this paper, all limits will be taken as the sequence $\{n\}$ of positive integers, or a subsequence, converges to infinity. The present authors hope to be able to report soon on results concerning $k$-dimensional parameter versions of the main theorems in this paper.

Let $F$ be a distribution with density $f$, and let $q(t) = f(t)/\lbrack 1 - F(t) \rbrack$ be the failure rate of $F$. Tests for constant versus monotone increasing failure rate based on the ranks of the normalized spacings between the ordered observations have been considered by Proschan and Pyke (1967). They show that these statistics are asymptotically normally distributed for fixed alternatives $F$ and compute the ratios of the efficacies of one of their rank tests to the best statistics for Weibull and Gamma alternatives. In this paper, it is shown that asymptotic normality holds also for sequences of alternatives $\{F_{\theta_n}\}$ that approach the $H_0$ distribution $1 - \exp (-\lambda t), t \geqq 0,$ as $n \rightarrow \infty$; and that the above mentioned ratios of efficacies are in fact Pitman efficiencies. Let $R_1, \cdots, R_n$ be the ranks of the normalized spacings, $T_1 = \sum iR_i$ and $T_2 = - \sum i \log \lbrack 1 - R_i/(n + 1) \rbrack$. Then $T_1$ is asymptotically equivalent to the Proschan Pyke statistic. It is shown that the Pitman efficiency satisfies \begin{equation*} \tag{1.1} e(T_1, T_2) \equiv \frac{3}{4}\end{equation*} for all sequences of alternatives $\{F_{\theta_n}\}$ and thus $T_1$ is asymptotically inadmissible. Statistics that are linear in the normalized spacings and asymptotically most powerful for parametric alternatives $\{F_{\theta_n}\}$ if the scale parameter $\lambda$ is known, are derived, and it is shown that the rank statistics that are asymptotically most powerful in the class of linear rank tests, are nowhere most powerful in the class of all tests, when $\lambda$ is known. If $\lambda$ is unknown, studentizing of the linear normalized spacing tests which are asymptotically most powerful for $\lambda$ known leads to procedures which have only the same asymptotic power as the most powerful linear rank tests. Unbiasedness is shown for tests that are monotone in the normalized spacings, and Monte Carlo power estimates are used to compare the various statistics with the likelihood ratio tests considered by Barlow (1967).

A problem which has received considerable attention in recent years is that of estimating the tail probability in a distribution belonging to a specified family. Kolmogorov (1950), Lieberman and Resnikoff (1955), and Healy (1956) obtained the unique minimum variance unbiased (UMVU) estimator of $P\{x > c\}$, where $x$ has a normal distribution with unknown mean and variance; they used the Rao-Blackwell theorem for this purpose. Washio, Morimoto and Ikeda (1956) used integral transform theory to study the Koopman-Pitman family. Barton (1961) provided UMVU estimators of the normal, Poisson and binomial distribution functions. The most extensive study is that by Tate (1959) who obtained the UMVU estimators for the cdf (and other functionals) for several probability densities; this was accomplished by using transform theory. In the context of reliability theory, Glasser (1962) estimated the tail of the exponential distribution and Basu (1964) that of the gamma distribution; other related work in the area of reliability theory is that of Rutemiller (1966), and Zacks and Even (1966). Non-parametric estimation of probability densities has been treated by Rosenblatt (1956), Parzen (1962), Lead-better and Watson (1963), and Weiss and Wolfowitz (1967). The more general problem of estimating $Ef(x)$, where $f$ is any given function, was considered by Neyman and Scott (1960) and Schmetterer (1961) in the case that $x$ has a normal distribution with unknown mean and variance. Schmetterer's solution really consists in obtaining the UMVU estimator of the density and then integrating the product of this estimate and $f$. That this procedure is of wider applicability is seen from the following result.

This paper shows that the epsilon entropy of any mean-continuous Gaussian process on $L_2\lbrack 0, 1 \rbrack$ is finite for all positive $\epsilon$. The epsilon entropy of such a process is defined as the infimum of the entropies of all partitions of $L_2\lbrack 0, 1 \rbrack$ by measurable sets of diameter at most $\epsilon$, where the probability measure on $L_2$ is the one induced by the process. Fairly tight upper and lower bounds are found as $\epsilon \rightarrow 0$ for the epsilon entropy in terms of the eigenvalues of the process.

For a general multivariate linear hypothesis testing problem, a class of permutationally (conditionally) distribution-free tests is proposed and studied. The asymptotic distribution theory of the proposed class of test statistics is studied along with a generalization of the elegant results of Hajek (1968) to the multistatistics and multivariate situations. Asymptotic power and optimality of the proposed tests are established and a characterization of the multivariate multisample location problem [cf. Puri and Sen (1966)] in terms of the proposed linear hypothesis is also considered.

In fractional replication, the problem is to estimate a subvector, $\alpha$, of $S = p^s$ pre-assigned parameters in the presence of an additional subvector of nuisance parameters. Estimation procedures for randomized fractional replicates in $2^m$ factorial systems have been treated by Ehrenfeld and Zacks [1], [2] and Zacks [6], [7] under an orthogonalized form of the statistical model. Extension to systems with general prime $p$ has recently been studied under a "fully orthogonalized" model by Lentner [4]; this is discussed in Section 3. The generalized inverse operator is then applied to the study of estimation procedures in general $N = p^m$ fractional replication. The present work differs from that of Zacks [7] where the problem of estimating the entire vector of $N$ parameters was considered. The class of all type-$g$ (generalized inverse) solutions for $\alpha$ is given and the class is investigated with respect to unbiasedness and optimality. Specifically, it is shown that there is a unique unbiased type-$g$ estimator which coincides with the classical estimator under the assumption of zero nuisance parameters. Using the trace of the mean square error matrix as the risk function, it is shown that there is a coincidence of Bayes, minimax, admissible, and classical procedures under certain conditions.

In 1929 Behrens introduced the concept of testing equality of two population means without assuming the homogeneity of the two population variances. Since that time there have been extensive discussions of the validity of the test and of the interpretation of the results. Fisher (1939) published a detailed paper defending Behrens' work and thus the work started by Behrens became known as the Behrens-Fisher problem. Fisher presented the test statistic (unequal variance $t$-test; Fryer, 1966) he felt was best for handling this situation. However, Fisher could only approximate the distribution of his test statistic. This approximation was tabled in Fisher and Yates (1948). Fisher's test statistic can also be approximated by the $t$-distribution, but the $t$-distribution approximation is not very good for small sample sizes (Cochran and Cox, 1957). Box (1954, a and b) wrote two extensive papers concerned with the violation of the assumptions in the analysis of variance. Box states that the one-way analysis of variance with equal sample sizes (the $t$-test is the same as a two-sample one-way analysis of variance) is robust when the variances are heterogeneous. However, little is said about the power of the test when the variances are heterogeneous. F. N. David and N. L. Johnson (1951, a and b) and Bozivich, Bancroft, and Hartley (1956) have written papers on the approximate theoretical power of the analysis of variance when the assumptions are violated. B. L. Welch (1937) and D. G. C. Gronow (1951) have written papers examining both the robustness and the power of the unequal variance $t$-test when the variances are heterogeneous. This paper examines by simulation techniques both the power and the robustness of the $t$-test and several other tests when the variances are heterogeneous, and presents a new test statistic designed for the situation where the coefficients of variation are homogeneous. This new test is more powerful than the $t$-test for certain ranges of the coefficient of variation. The assumption of homogeneous coefficients of variation is a valid assumption in many types of agricultural, biological, and psychological experimentation, because many times the treatment that yields a larger mean also has a larger standard deviation.

In [8] the problem of nonparametric estimation in Markov processes has been considered, and estimates of the initial, two-dimensional joint, and transition densities of the process, satisfying a number of optimal properties, have been obtained. In the present paper, under the same nonparametric setup, the attention is centered primarily on the transition distribution function of the process. It will be assumed that the underlying Markov process, defined on a probability space $(\Omega, \mathscr{a}, P)$ and taking values in the real line $R$, is (strictly) stationary, and has initial, two-dimensional joint, and transition densities $p(\cdot), q(\cdot, \cdot)$, and $t(\cdot \mid x), x \varepsilon R$, respectively, relative to the appropriate Lebesgue measures. Let $K$ be a probability density. On the basis of the first $n + 1$ random variables $X_j, j = 1, 2, \cdots, n + 1$ of the process, we define the random variables $p_n(x), x \varepsilon R$, and $q_n(y), y \varepsilon R \times R$ (by suppressing the random element $\omega$) by the following relations \begin{equation*} \tag{1.1} p_n(x) = (nh)^{-1} \sum^n_{j=1} K((x - X_j)h^{-1})\end{equation*} \begin{equation*} \tag{1.2} q_n(y) = q_n(x, x') = (nh)^{-1} \sum^n_{j=1} K((x - X_j)h^{-\frac{1}{2}})K((x' - X_{j+1})h^{-\frac{1}{2}}),\end{equation*} where $h = h(n)$ is a sequence of positive constants satisfying also some additional conditions. We further set \begin{equation*} \tag{1.3} t_n(x' \mid x) = q_n(x, x')/p_n(x).\end{equation*} Next, by means of $p_n(x)$ and $q_n(x, x')$, define the random variables $F_n(x) = \int^x_{-\infty} p_n(z) dz, \quad G_n(z \mid x) = \int^z_{-\infty} t_n(dx' \mid x).$ We finally let $F(\cdot)$ and $G(\cdot \mid x), x \varepsilon R$, be the initial and transition distribution functions of the process. Under suitable conditions on the function $K$, the sequence $\{h\}$, and the process, the main results of this paper are the following. The distribution function $F_n(x)$, as an estimate of $F(x)$, obeys the Glivenko-Cantelli theorem. This is Theorem 3.1. Turning now to the estimate $G_n(\cdot | x)$ of $G(\cdot | x)$, we have been able to establish in Theorem 3.2 that $\sup \{|G_n(z | x) - G(z | x)a|; z \varepsilon R\}$ converges to zero, as $n \rightarrow \infty$, but in the probability sense. This is true for all $x \varepsilon R$. In Section 4 it is assumed that the $r$th absolute moment of $X_1$ exists (for some $r = 1, 2, \cdots$), and the problem is that of gaining further information about $G(\cdot \mid x)$, by estimating its $k$th moment, to be denoted by $m(k; x)$, for $k = 1, 2, \cdots, r$. By letting the rather simple expression $m_n(k; x) = (nh)^{-1}p_n^{-1}(x) \sum^n_{j=1} X^k_{j+1}K((x - X_j)h^{-1})$ stand for an estimate of $m(k; x)$, it is shown that, as $n \rightarrow \infty, m_n(k; x) \rightarrow m(k; x)$ in probability, for $k = 1, 2, \cdots, r$ and $x \varepsilon R$. This is the content of Theorem 4.1. Finally in Section 5 we look into the problem of estimating the quantiles of $G(\cdot \mid x)$, and in connection with this, two results are derived. For some $p$ in the interval (0, 1), it is assumed that the $p$th quantile, $\xi(p, x)$, of $G(\cdot \mid x)$ is unique. By defining $\xi_n(p, x)$ as the smallest root of the equation $G_n(z \mid x) = p$ and using it as an estimate of $\xi(p, x)$, it is proved that, as $n \rightarrow \infty$, $\xi_n(p, x) \rightarrow \xi(p, x)\quad \text{in probability (Theorem 5.1)},$ and $(nh)^{\frac{1}{2}} \lbrack \xi_n(p, x) - \xi(p, x) \rbrack \rightarrow N(0, \tau^2(\xi, x))\quad \text{in law}.$ This is Theorem 5.2, and the variance $\tau^2(\xi, x)$ is explicitly given in that theorem.

Let the random variable $Y_N$ be defined by $Y_N = \sum^N_{k=1} W^2(k)/k^2,$ where $W(t)$ is the Wiener process, the Gaussian random process with mean zero and covariance $EW(s)W(t) = \min (s, t)$. Note that $EY_N \sim \log N$. We show that for $a > 1$ $\operatorname{Pr}\lbrack Y_N \geqq a \log N \rbrack = N^{-(8a)^{-1}(a-1)^2(1+\epsilon_N)},$ where $\epsilon_N \rightarrow 0$ as $N \rightarrow \infty$.

This paper is a study of statistical distributions by analyzing the structure of density functions. This work was motivated by the paper [10] in which the distribution of the product of two non-central chi-square variables is obtained. Their problem arose from a problem in the physical sciences connected with the theory of spinstabilized rockets. It is pointed out in [10] that the original problem was to obtain the distribution of the product of a central Raleigh and a non-central Raleigh variable. By examining the structure of the density function of non-central Raleigh or non-central chi-square it is apparent that the density is the product of the special cases of two special functions. In [8, chapter 2] a lengthy treatment of Raleigh and associated distributions is given. We do not know any other particular problem in the physical sciences, but from the structural property of the problem pointed out in [10] it is quite likely that there may be a number of such problems in the physical sciences or in other disciplines, which are to be tackled. Hence we will give the distribution of the product of two independent stochastic variables whose density functions can be expressed as the product of any two special functions. Products of $H$-functions are used so that almost all classical density functions (central or non-central) will be taken care of, since $H$-functions are the most generalized special functions. Since a number of types of factors can be absorbed inside the $H$-function we think that all the distributions which are frequently used in the statistical theory of distributions will be included in this problem that we discuss here, with some modifications in some cases. Several special cases are pointed out so that one can easily get the distribution of the product or ratio of independent stochastic variables whose density functions are products of special functions. The result in [10] is obtained as a special case and further, the distributions of the products of two non-central $F$ simple and multiple correlation coefficients are pointed out for the sake of mathematical interest because, structurally, the density functions in these cases belong to different categories. Since several properties of $H$-functions are available in the literature, it is easy to study other properties or to compute percentage points of the product distribution discussed in this paper. The approach of examining the structure of densities may simplify the problem of obtaining distributions of several statistics.

In [2] Doeblin obtained a central limit for discrete parameter Markov chains with discrete state space. In obtaining this theorem the principal tool is the Doeblin dissection of the sequence of partial sums of a functional of a Markov chain into a random sum of independent, identically distributed random variables. Indeed, given this dissection, the remainder of Doeblin's proof is, in essence, a proof of a random central limit theorem. Although Doeblin does not state a stable limit theorem for Markov chains, at the end of the paper he observes that the dissection should also be of use in obtaining such theorems, and comments on a possible method of obtaining theorems of this nature. In this paper three stable limit theorems for Markov chains together with the appropriate solidarity theorems are obtained depending on whether the index $\alpha$ of the limiting distribution is $< 1, = 1, or > 1$. The principal tools used in obtaining these theorems are the Doeblin dissection, a well known result concerning the rate of growth of the sequence of norming coefficients of a random variable in the domain of attraction of a stable law, and a random stable limit theorem [8], [10].

We show that the existence of a continuous minimal sufficient statistic not equivalent to the order statistics, for $n \geqq 2$ independent observations, is not a sufficient condition for the family of densities, assumed to be Lipschitz, to be an exponential family. This result is intended to be compared with a theorem of Dynkin (p. 24 of [3]) which asserts that the existence of a sufficient statistic not equivalent to the order statistics implies that the family of densities is an exponential family, provided that the densities possess continuous derivatives.

This self-contained note extends and generalizes smoothing procedures proposed by Whittle (1957), (1958) and Dickey (1968). The use of linear filters, chosen through analysis of the data to be smoothed, is advocated. Hence, smoothing is nonlinear. The setting can be viewed as a multivariate extension of empirical Bayes settings in which, usually, there are strong independence assumptions.

Motivated by Loynes' (1969) treatment of (weak) consistency of sequential estimators, we establish here some allied results on strong consistency. The strengthened conclusion is achieved by imposing further restrictions, so that our results are not as broadly applicable as Loynes'. The reader is referred to Loynes' paper for additional motivation and discussion. As in that paper, we are concerned with estimators that improve a given one by taking conditional expectations with respect to a sufficient statistic, in the case where the sample size may itself be a random variable. $x_1, x_2, \cdots$ will denote the data sequence, random variables defined on a measurable space $(\Omega, \mathscr{a})$. All probability measures considered on $(\Omega, \mathscr{a})$ will render the sequence iid. It is seen below (Theorem 3.4) that a given sequence $\{t_i\}$ of stopping times for the data sequence leads to a strongly consistent sequence of estimators if $\lim t_i = + \infty$ a.s., and the $t_i$ are $C$-ordered (Definition 2.3). This entails that the $t_i$ increase monotonically to $+ \infty$ a.s., but requires additional structure as well. The specific considerations are in Section 3; Section 2 presents some general notions.

Let $X_1, \cdots, X_n$ be independent random variables identically distributed with absolutely continuous distribution function $F$ and density function $f$. Loftsgaarden and Quesenberry [3] propose a consistent nonparametric point estimator $\hat{f}_n(z)$ of $f(z)$ which is quite easy to compute in practice. In this note we introduce a step-function approximation $f_n^\ast$ to $\hat{f}_n$, and show that both $\hat{f}_n$ and $\hat{f}_n^\ast$ converge uniformly (in probability) to $f$, assuming that $f$ is positive and uniformly continuous in $(-\infty, \infty)$. For more general $f$, uniform convergence over any compact interval where $f$ is positive and continuous follows. Uniform convergence is useful for estimation of the mode of $f$, for it follows from our theorem (see [4], section 3) that a mode of either $\hat{f}_n$ or $f_n^\ast$ is a consistent estimator of the mode of $f$. The mode of $f_n^\ast$ is particularly tractable; it is applied in [2] to some problems in pattern recognition. From the point of view of mode estimation, we thus obtain two new estimates which are similar in conception to those proposed by some previous authors. Let $k(n)$ be an appropriate sequence of numbers in each case. Chernoff [1] estimates the mode as the center of the interval of length $2k(n)$ containing the most observations. Venter [5] estimates the mode as the center (or endpoint) of the shortest interval containing $k(n)$ observations. The estimate based on $\hat{f}_n$ is that $z$ such that the distance from $z$ to the $k(n)$th closest observation is least. Finally, the estimate from $f_n^\ast$ is that observation such that the distance from it to the $k(n)$th closest observation is least.

Let $\{X_n, n \geqq 1\}$ be a sequence of random variables and let $S_n = \sum^n_{i=1} X_i$. Under the condition that $\{S_n\}$ forms a martingale sequence, it was shown in [2] that, for $\nu \geqq 2$, $E(|S_n)^\nu) \leqq C_\nu n^{(\nu/2)-1} \sum^n_{i=1} E|X_i|^\nu,$ where \begin{equation*} \tag{(1)} C_\nu = \lbrack 8(\nu - 1) \max (1, 2^{\nu-3}) \rbrack^\nu.\end{equation*} The purpose of this paper is to show that the constant $C_\nu$ can be replaced by a much smaller constant in the following two cases: (i) $\nu$ is an even integer and the martingale dependence condition is replaced by one which is more explicit in terms of moments (Theorem 1); (ii) the $X_n$'s are independent with zero means (Theorem 2). For case (i) we give for $E(|S_n|^\nu)$ a bound which is a polynomial in $n$. This last bound does not appear to be too exhorbitant because, as shown by an example, it is not valid for all martingales $\{S_n\}$.*