Chernoff and Rubin [1] and Rubin [5] investigated the problem of estimation of the location of a discontinuity in density. They have shown that the maximum likelihood estimator (MLE) is hyper-efficient under some regularity conditions on the density and that asymptotically the estimation problem is equivalent to that of a non-stationary process with unknown center of non-stationarity. We have obtained here similar results for a family of densities $f(x, \vartheta)$ which are continuous with a cusp at the point $\vartheta$. In this connection, it is worth noting that Daniels [2] has obtained a modified MLE for the family of densities $f(x, \vartheta) = C(\lambda) \exp \{-|x - \vartheta|^\lambda\}$, for $\lambda$ such that $\frac{1}{2} < \lambda < 1$, where $C(\lambda)$ is a constant depending on $\lambda$ and he has shown that this estimator is asymptotically efficient. In this paper, we shall show that the MLE of $\vartheta$ is hyper-efficient for the family of densities $f(x, \vartheta)$ given by \begin{equation*}\begin{align*}\tag{1.1}\log f(x, \vartheta) = \epsilon(x, \vartheta)|x - \vartheta|^\lambda + g(x, \vartheta) \text{for} |x| \leqq A \\ = g(x,\vartheta)\quad\text{for} |x| > A \\ \end{align*}\end{equation*} where $A$ is a constant greater than zero, \begin{equation*}\begin{align*}\tag{1.2}\epsilon (x,\vartheta) = \beta(\vartheta) \text{if} x < \vartheta \\ = \gamma(\vartheta) \text{if} x > \vartheta, \\ \end{align*}\end{equation*}\begin{equation*}\tag{1.3}0 < \lambda < \frac{1}{2} \text{and}\end{equation*}\begin{equation*}\tag{1.4}\vartheta\varepsilon (a, b) \text{where} -A < a < b < A,\end{equation*} under some regularity conditions on $f(x, \vartheta)$ and we shall derive the asymptotic distribution of the MLE implicitly. Section 2 contains the regularity conditions imposed on the family of densities $f(x, \vartheta)$. Section 3 contains some results related to the asymptotic properties of the MLE. The estimation problem is reduced to that of a stochastic process in Section 4. The asymptotic distribution of MLE is obtained in Section 5.

Let $g_1, \cdots, g_n$ and $h$ be given real-valued Borel measurable functions on a fixed measurable space $T = (T, \mathscr{A})$. We shall be interested in methods for determining the best upper and lower bound on the integral $\mu(h) = \int_Th(t)\mu(dt),$ given that $\mu$ is a probability measure on $T$ with known moments $\mu(g_j) = y_j, j = 1, \cdots, n$. More precisely, denote by $\mathscr{M}^+ = \mathscr{M}^+(T)$ the collection of all probability measures on $T$ such that $\mu(|g_j|) < \infty (j = 1, \cdots, n)$ and $\mu(|h|) < \infty$. For each $y = (y_1, \cdots, y_n) \varepsilon R^n$, consider the bounds $L(y) = L(y | h) = \inf \mu(h), U(y) = U(y | h) = \sup \mu(h),$ where $\mu$ is restricted by $\mu \varepsilon \mathscr{M}^+(T); \mu(g_1) = y_1, \cdots, \mu(g_n) = y_n.$ If there is no such measure $\mu$ we put $L(y) = + \infty, U(y) = - \infty$. In many applications, $h$ is the characteristic function (indicator function) $h = I_s$ of a given measurable subset $S$ of $T$. In that case we usually write instead $L(y | I_s) = L_s(y), U(y | I_s) = U_s(y)$. Thus, $L_s(y) \leqq \mu(S) \leqq U_s(y)$ are the best possible bounds on the probability mass $\mu(S)$ contained in $S$, given that $\mu \varepsilon \mathscr{M}^+$ and that $\mu(g) = y$. Here, $g$ denotes the mapping $g:T \rightarrow R^n$ defined by $g(t) = (g_1(t), \cdots, g_n(t))$. By $g_0$ we shall denote the function on $T$ with $g_0(t) = 1$ for all $t \varepsilon T$. The following tentative method for finding $L(y \mid h)$ may be said to go back to Markov [8] and Riesz [13], see [7]. Choose an $(n + 1)$-tuple $d^\ast = (d_0, d_1, \cdots, d_n)$ of real numbers such that $d_0 + d_1g_1(t) + \cdots + d_ng_n(t) \leqq h(t) \text{for all} t \varepsilon T,$ and define $B(d^\ast) = \{z \varepsilon R^n: z = g(t) \text{for some} t \varepsilon T \text{with} \sum^n_{j=0} d_jg_j(t) = h(t)\}.$ Then $L(y \mid h) = d_0 + \sum^n_{j=1} d_jy_j \text{for each} y \varepsilon \operatorname{conv} B(d^ast),$ ($\operatorname{conv} =$ convex hull). The main purpose of the present paper is to investigate the merits of this method and certain more general methods. It turns out (Theorem 5) that for almost all $y \epsilon R^n$ there exists at most one admissible $d^\ast$ with $y \varepsilon \operatorname{conv} B(d^\ast)$. Moreover, provided $y \varepsilon \int (V)$ where $V = \operatorname{conv} g(T)$, there exists at least one such $d^\ast$ if and only if there exists a measure $\mu \varepsilon \mathscr{M}^+$ with $\mu(g) = y$ and $\mu(h) = L(y \mid h)$. A sufficient condition for the latter would be that $T$ has a compact topology with respect to which $g$ is continuous and $h$ is lower semi-continuous. More interesting is a related method for finding $L(y \mid h)$, see Theorem 6, which will work for each $y \varepsilon \int (V)$ as soon as $g$ is bounded. The situation where $y \not\in \int (V)$ is discussed in Section 4. It appears that the assumption $y \varepsilon \int(V)$ is a rather natural one. We have chosen to develop the important special case $h = I_s$ in a partly independent manner, see the Sections 5, 6, and 7. In this case, the $(n + 1)$-tuple $d^\ast$ must satisfy \begin{align*}d_0 + \sum^n_{j=1} d_jz_j \leqq 1 \text{for all} z \varepsilon g(T),\\ \leqq 0 \text{for all} z \varepsilon g(S').\end{align*} Here, $S'$ denotes the complement of $S$ in $T$. Assuming that $d_1, \cdots, d_n$ are not all zero, let us associate to $d^\ast$ the pair of hyperplanes $H$ and $H'$ with equations $\sum^n_{j=1 d_jz_j} = 1 - d_0 \text{and} \sum^n_{j=1} d_jz_j = -d_0,$ respectively. This pair is such that $H, H'$ are distinct parallel hyperplanes with $g(S')$ and $H$ on opposite sides of $H'$ and $g(T)$ and $H'$ on the same side of $H;$ such a pair $H, H'$ will be said to be admissible. Observe that $B(d^\ast) = (g(S) \mathbf{cap} H) \mathbf{cup} (g(S') \mathbf{cap} H'),$ with $H, H'$ as the admissible pair determined by $d^\ast$. The present $(n + 1)$-tuple $d^\ast$ is useful, for determining $L_s(y) = L(y \mid I_s)$ for at least some points $y$, only when both $g(S) \mathbf{cap} H \not\equiv 0$ and $g(S') \mathbf{cap} H' \not\equiv 0$. That is, $H'$ should not only support the set $g(S')$ but even "intersect" it; similarly, $H$ and $g(S)$. Fortunately, one can usually replace "intersect" by "touch". More precisely (Corollary 13), if $H$ and $H'$ form an admissible pair as above then $L_s(y) = d_0 + \sum^n_{j=1} d_jy_j$ for each point $y$ such that both $y \varepsilon \int(V),\quad y \varepsilon \operatorname{conv}\lbrack\{H \mathbf{cap} \overline{\operatorname{conv}} g(S)\} \mathbf{cup} \{H' \mathbf{cap} \overline{\operatorname{conv}} g(S')\}\rbrack,$ a bar denoting closure. Provided $g$ is bounded the latter generalization will yield the value $L_s(y)$ for all relevant $y$, see Theorem 7. Whether or not $g$ is bounded, we have for almost all $y$ that there can be at most one admissible pair of hyperplanes $H$ and $H'$ yielding $L_s(y)$ in the above manner. A detailed discussion of the method on hand may be found in Section 6. The present method is geometrical in the following sense: (i) one only needs to know the sets $g(S)$ and $g(S')$ in $R^n;$ (ii) afterwards, one considers all the pairs $H$ and $H'$ of parallel hyperplanes touching $g(S)$ and $g(S')$ in the above manner. Each such pair yields $L_s(y)$ for certain values $y;$ varying the pair $H, H'$ one often obtains the value $L_s(y)$ for all relevant $y \varepsilon R^n$. Usually, there are many different regions in $y$-space, each with its own analytic formula for $L_s(y)$. Nevertheless, all these different formulae are derived from one and the same geometrical principle. A number of specific illustrations, all with $n = 2$, are presented in Section 7. They indicate that it is often quite easy to solve the following problem in a geometric manner. Let $X$ be a random variable taking its values in a measurable space $T$, such that $E(g_1(X)) = y_1,\quad E(g_2(X)) = y_2,$ with $g_1$ and $g_2$ as known real-valued Borel measurable functions on $T$. The problem is to determine the best possible lower bound $L_s(y)$ on $\mathrm{Pr} (X \varepsilon S)$ where $S$ is a given Borel measurable subset of $T$.

Let $\{X_n\}$ be an infinite sequence of independent non-negative random variables such that, for some regularly varying non-decreasing function $\lambda(n)$, with exponent $1/\beta, 0 < \beta < \infty$, as $n \rightarrow \infty$, $P\{X_1 + \cdots + X_n/\lambda(n) \leqq x\} \rightarrow K(x)$ at all continuity points of some d.f. $K(x)$. Let $\Lambda(x)$ be the function inverse to $\lambda(n)$, let $R(x)$ be any other regularly varying function of exponent $\alpha > 0$. Then, if $N(x)$ is the maximum $k$ for which $X_1 + \cdots + X_k \leqq x$, it is proved that, as $x \rightarrow \infty$, $\varepsilon R(N(x)) \sim I(\alpha\beta)R(\Lambda(x))$ where $I(\alpha\beta) = \int^\infty_0 u^{\alpha\beta} dK(u)$ and this latter integral may diverge.

Heller's (1965) concept of a stochastic module is examined with the purpose of seeing what are the algebraic implications of various probabilistic properties of discrete time, finite state processes. $k$th order Markovity, recurrence, stationarity, and ergodicity are given characterizations in terms of the stochastic module.

Let $X_1, \cdots, X_N$ be independent normal $p$-vectors with common mean vector $\xi = (\xi_1, \cdots, \xi_p)'$ and common non-singular covariance matrix $\Sigma$. Write $N\bar{X} = \sum^N_1 X_i, S = \sum^N_1 (X_i - \bar{X})(X_i - \bar{X})', b\lbrack i\rbrack$ for the $i$-vector consisting of the first $i$ components of a $p$-vector $b$ and $C\lbrack i\rbrack$ for the upper left-hand $i \times i$ sub-matrix of a $p \times p$ matrix $C$. Let $\delta = N\xi'\Sigma^{-1}\xi (\geqq0)$. We will consider here the problem of testing the hypothesis $H_0:\xi_1 = \cdots = \xi_p = 0$ against the alternative $H_\lambda:\xi_1 = \cdots = \xi_q = 0,\quad \delta = \lambda,$ where $\xi, \Sigma$ are unknown, $q < p$ and $\lambda > 0$ is given. The problem of testing $H_0$ against $H_\lambda$ remains invariant under the group $G$ of $p \times p$ non-singular matrices \begin{equation*}\tag{1.0}g = \begin{pmatrix}g_{11} 0 \\ g_{21} g_{22}\end{pmatrix}\end{equation*} where $g_{11}$ is a $q \times q$ sub-matrix of $g$. A maximal invariant in the space of $(\bar{X}, S)$ under $G$ is $\bar{R} = (\bar{R}_1, \bar{R}_2)$ where \begin{equation*}\tag{1.1}\bar{R}_1 + \bar{R}_2 = N\bar{X}'(S + N\bar{X}\bar{X}')^{-1}\bar{X}\end{equation*}\begin{equation*}\bar{R}_1 = N\bar{X}'_{\lbrack q\rbrack}(S_{\lbrack q\rbrack} + N\bar{X}_{\lbrack q\rbrack}\bar{X}'_{\lbrack q\rbrack})^{-1}\bar{X}_{\lbrack q\rbrack};\end{equation*} and a corresponding maximal invariant in the parametric space of $(\xi, \Sigma)$ under $G$ is $\bar{\Delta} = (\bar\delta_1, \bar\delta_2)$, where \begin{equation*}\tag{1.2}\bar\delta_1 + \bar\delta_2 = N\xi'\Sigma^{-1}\xi,\end{equation*}\begin{equation*}\bar\delta_1 = N\xi'_{\lbrack q\rbrack}\Sigma^{-1}_{\lbrack q\rbrack}\xi_{\lbrack q\rbrack}\end{equation*} (see Giri (1961)). Giri (1962) has shown that for testing $H_0$ against the alternatives \begin{equation*}\tag{1.3}H_1':\xi_1 = \cdots = \xi_q = 0,\quad\delta > 0,\end{equation*} the likelihood ratio test of significance level $\alpha$ is given by \begin{align*}\tag{1.4}\phi(X_1, \cdots, X_N) = 1, \quad \text{if} \quad Z = (1 - \bar{R}_1 - \bar{R}_2)/(1 - \bar{R}_1) \leqq C, \\ = 0, \quad \text{otherwise};\end{align*} where the constant $C$ is chosen in such a way that $E_{H_0}\phi = \alpha$ and under $H_0, Z$ has central beta distribution with parameters $(N - p)/2, (p - q)/2$. It follows from Giri (1961) (also from (2.3) in Section 2) that with respect to $G$ the likelihood ratio test for testing $H_0$ against $H'_1$ is not uniformly most powerful invariant and that there is no uniformly most powerful invariant test for this problem. However, for fixed $p$, this test is nearly optimum as the sample size becomes large (Wald (1943)). Thus if $p$ is not large, it seems likely that the sample size occurring in practice will usually be large enough for this result to be relevant. However, if the dimension $p$ of the basic multivariate distribution is large, it may be that the sample size must be extremely large for this large sample result to apply, for example, there are cases where $N/p^3$ is large. The only satisfactory property of this test procedure known to us at this writing is that the difference of the powers of this test and the best invariant test with respect to $G$ is $O(N^{-1})$ when $p, q$ are both $O(N), \delta$ is $O(N^{\frac{1}{2}})$ and $N$ becomes large (Giri (1967)). In this paper it will be shown that the likelihood ratio test for this problem is neither locally minimax as $\lambda \rightarrow 0$ nor asymptotically minimax as $\lambda \rightarrow \infty$. Thus it will be established here that it can not be minimax for every $\lambda$ for this problem. Attempts are also made to find test procedures based on $\bar{R}$ which are locally and asymptotically minimax. It is easy to see that the power function of any test procedure based on $\bar{R}$ is constant on each contour $\delta = \lambda$. It will be shown in Sections 2 and 3 that the test (hereafter called $\phi^\ast$) which rejects $H_0$ if $\bar{R}_1 + (N - q)\bar{R}_2/(p - q) \geqq C_\alpha$, where $C_\alpha$ is chosen so that $E_{H_0}\phi^\ast = \alpha$, is locally minimax as $\lambda \rightarrow 0$ but not asymptotically logarithmically (sometimes will be called asymptotically only) minimax as $\lambda \rightarrow \infty$ and the test (hereafter called Hotelling's test) which rejects $H_0$ if $\bar{R}_1 + \bar{R}_2 \geqq C_\alpha'' (C_\alpha''$ depends on the size $\alpha$ of the test) is asymptotically minimax as $\lambda \rightarrow \infty$ but not locally minimax as $\lambda \rightarrow 0$. Furthermore for this problem no invariant test (under $G$) is minimax for every $\lambda$. This includes the likelihood ratio test, Hotelling's test and $\phi^\ast$. It may be pointed out here that none of these last two test procedures is derived following after any well known statistical theory. They are derived so as to possess the required local and asymptotic property stated above.

Two statistics for testing that the mean of a normal distribution is zero are compared to show that the measures of asymptotic relative efficiency proposed by Pitman (cf., e.g., [5]), Hodges and Lehmann [4], and Bahadur [1], [2] do not always agree, even locally for alternative means near zero.

Recently Gani [3] has considered a stochastic model for the attachment of phages to bacteria. In this paper we describe a simple method of estimating the parameter $\alpha$ which occurs in this model and study the asymptotic properties of the estimate.

In the general problem of statistical estimation, it is possible to obtain from a random sample a single estimate, known as a point estimate, of a population parameter. But this estimate is not very meaningful unless associated with a measure of its reliability. One approach to this problem consists of giving a point estimate with its standard error, but this has a major drawback. In using this approach, one typically fails to make an assertion regarding the error involved in estimating the standard error. In this paper, the problem of interval estimation is considered within the framework of decision theory, in which the cost to the statistician depends on the true value of the parameter and the interval chosen. For example, if $\theta$ is the true value of the parameter and $(a, b)$ is the interval chosen, a typical loss function is \begin{align*}L(\theta, (a, b)) = h(a - \theta, b - \theta) \text{if} a < \theta < b \\ = h(a - \theta, b - \theta) + 1 \text{if} \theta < a \text{or} > b\end{align*} where $h(a, b)$ is defined on $\{(a, b); a \leqq b\}$. Thus, the statistician would like to choose the interval $(a, b)$ small to make the payment $h(a, b)$ small, and yet he wants it large enough to have a good chance of containing $\theta$ so that he will not have to pay the "extra" unit. We consider two methods for finding optimal decision rules for the problem of interval estimation. The first one uses the Bayes principal and the second uses the invariance principle. The invariance principle is available only in decision problems which are invariant under certain transformations. Here we find a form of a best invariant interval estimate for the location parameter, and give certain conditions under which the best invariant interval estimate is minimax. In analogy of Blackwell and Girshick's suggestions, we present a loss function for which a best invariant interval estimate exists but is not minimax. (See the example in Section 4) Finally, in Section 5, we show that a best invariant interval estimate for a scale parameter of a distribution can be found by transforming the scale parameter problem to the location parameter problem.

The present investigation is concerned with the proposal and study of a class of nonparametric paired comparison tests for the hypothesis of no differences among several objects with respect to a pair of characteristics.

H. Teicher [5] has initiated a valuable study of the identifiability of finite mixtures (these terms to be defined in the next section), revealing a sufficiency condition that a class of finite mixtures be identifiable and from this, establishing the identifiability of all finite mixtures of one-dimensional Gaussian distributions and all finite mixtures of gamma distributions. From other considerations, he has generalized [4] a result of Feller [1] that arbitrary (and hence finite) mixtures of Poisson distributions are identifiable, and has also shown binomial and uniform families do not generate identifiable mixtures. In this paper it is proven that a family $\mathscr{F}$ of cumulative distribution functions (cdf's) induces identifiable finite mixtures if and only if $\mathscr{F}$ is linearly independent in its span over the field of real numbers. Also we demonstrate that finite mixtures of $\mathscr{F}$ are identifiable if $\mathscr{F}$ is any of the following: the family of $n$ products of exponential distributions, the multivariate Gaussian family, the union of the last two families, the family of one-dimensional Cauchy distributions, and the non-degenerate members of the family of one-dimensional negative binomial distributions. Finally it is shown that the translation-parameter family generated by any one-dimensional cdf yields identifiable finite mixtures.

Let $\mathbf{X}$ be a $p \times f_2$ matrix variate $(p \leqq f_2)$ and $\mathbf{Y}$ a $p \times f_1$ matrix variate $(p \leqq f_1)$ and the columns be all independently normally distributed with covariance matrix $\mathbf{\Sigma}, E(\mathbf({X}) = \mathbf{M}$ and $E(\mathbf{Y}) = \mathbf{0}$. Let $0 < l_1 \leqq \cdots \leqq l_p < 1$ be the ordered characteristic roots of \begin{equation*}\tag{1.1} |\mathbf{XX}' - l(\mathbf{YY}' + \mathbf{XX}')| = \mathbf{0}\end{equation*} and $\omega_1, \cdots, \omega_p$ those of $|\mathbf{MM}' - \omega\mathbf{\Sigma}| = \mathbf{0},$ then the joint density function of $l_1, \cdots, l_p$ is given by Constantine [1], James [2] in the form \begin{equation*}\tag{1.2}\exp (-\frac{1}{2} \operatorname{tr} \mathbf{\Omega}) _1F_1(\frac{1}{2}\nu; \frac{1}{2}f_2; \frac{1}{2}\mathbf{\Omega}, \mathbf{L})f_0(l_1, \cdots, l_p),\end{equation*} where \begin{equation*}\begin{align*}{1.3}f_0(l_1, \cdots, l_p) = C(p, f_1, f_2) \mathbf{\prod}^p_{i=1} \{l_1^{\frac{1}{2}(f_2-p-1)}(1 - l_i)^{\frac{1}{2}(f_1-p-1)}\}\alpha_p (\mathbf{L}), \\ \mathbf{L} = \mathbf{X}'(\mathbf{YY}' + \mathbf{XX}')^{-1}\mathbf{X},\quad \mathbf{\Omega} = \mathbf{M}'\mathbf{\Sigma}^{-1}\mathbf{M},\quad \nu = f_1 + f_2\end{align*},\end{equation*} $_1F_1$ is the hypergeometric function of matrix argument defined in [2] given by $\sum^\infty_{k=0} \sum_\mathbf{kappa}{2}\nu)_\mathbf{kappa}C_\mathbf{kappa}(\frac{1}{2} \mathbf{\Omega})_\mathbf{kappa}(\mathbf{L})/(\frac{1}{2}f_2)\mathbf{kappa} C_\mathbf{kappa}(\mathbf{I}_p)k !$ and where $C(p, f_1, f_2) = \pi^{\frac{1}{2}p^2}\Gamma_p(\frac{1}{2}\nu)/\{\Gamma_p(\frac{1}{2} f_1)\Gamma_p(\frac{1}{2}f_2)\Gamma_p(\frac{1}{2}p)\}$ and $\alpha_p(\mathbf{L} = \mathbf{\prod}_{i>j}(l_i - l_j).$ In this paper the distribution of Pillai's $V^{(p)}$ criterion which is the trace of $\mathbf{L}$, [5], [6] and that of Roy's largest root criterion, $l_p$, [8], [10], have been obtained in series forms and certain constants involved in the series tabulated. In addition, the first four moments of $V^{(p)}$ are also obtained in the linear case, illustrating further use of some of the tabulations.

Of several possible versions of multiple correlation matrix between two sets of variables $\mathbf{x}$ and $\mathbf{y}$ (see, e.g., Khatri, 1964), we derive using the techniques of A. T. James (zonal polynomials), the non-null distribution of one version when (i) one of the two sets of variables is fixed, i.e., multivariate analysis of variance and covariance case (MANOVA), and when (ii) both sets of variables are random variables, i.e., canonical correlations case. These distributions are non-central multivariate $\beta$-distributions in much the same way as the two cases of multiple correlation commonly known as the multiple correlation of the second and the first kind respectively.

This investigation was originally motivated by the problem of determining the maximum number of variables which can be accommodated in a $2_v^{k-p}$ design in 256 runs and of constructing such a "saturated" design. This problem is solved through the application of an algorithm given by the authors in a previous paper (Draper and Mitchell (1967)) to the particular case $R = 5, q = k - p = 8$. To obtain the solution, the complete set of even 512-run designs of resolution $\geqq 6$ and the complete set of 256-run designs of resolution $\geqq 5$ are constructed. Tables are given which immediately provide generating relations for all of these designs, "optimally" blocked.

Let $\{X_i : i = 1, 2, \cdots\}$ be a sequence of random variables such that $X_1, \cdots, X_n$ are exchangeable and symmetric $(n = 1, 2, \cdots)$. Suppose that ties occur with probability zero among the partial sums $S_0 = 0, S_k = \sum^k_1X_i$. We study the laws of the variables $J_k$, the index of the $k$th positive sum in the sequence $S_1, S_2, \cdots (k = 1, 2, \cdots), N'_n$, the number of positive sums among $S_0, S_1, \cdots, S_{L_n}$, where $L_n$ is the index of $\max \{S_0, S_1, \cdots, S_n\}$. Brief attention is given to $J_k$ in [2], where the simple form of its law in the symmetric case is however not mentioned. The variable $N_n'$ does not seem to have been considered before. Setting $a_k = 2^{-2k}\binom{2k}{k},\quad k = 0, 1, 2, \cdots (a_0 = 1),$ we find the probabilities \begin{equation*}\tag{1.1}q_k(n) = P\lbrack J_k = n\rbrack = (k/n)a_ka_{n-k},\quad n = k, k + 1, \cdots,\end{equation*} \begin{equation*}\tag{1.2}p_n(i) = P\lbrack N_n' = i\rbrack = (2ia_i)^{-1}a_n,\quad i = 1, 2, \cdots, n (p_n(0) = a_n).\end{equation*} Let $\{X_t, 0 \leqq t \leqq T < \infty\}$ be a measurable, separable stochastic process which is continuous in probability and has exchangeable, symmetric increments. Relative to the bounded time interval $0 \leqq t \leqq T$, introduce the variables (1.3) \begin{align*}U = \text{"time spent in the positive half plane up to the moment when the process reaches its maximum,"} \\ V = \text {"time elapsed until the process reaches its maximum."}\end{align*} Asymptotic evaluations lead to THEOREM. $U/V$ is independent of $V/T$, and for $0 \leqq \alpha, \gamma \leqq 1$, $P\lbrack U < \alpha V\rbrack = 1 - (1 - \alpha)^{\frac{1}{2}},\quad P\lbrack U < \gamma T\rbrack = \gamma^{\frac{1}{2}}.$

Let $X_1, \cdots, X_n$ be independent random variables, identically distributed with continuous distribution function $F$. Let $X_{(1)}, \cdots, X_{(n)}$ denote the corresponding order statistics and $F_n$ the empiric distribution function, which we assume to be right continuous. Jung [4] found the asymptotic mean and variance of linear functions of the form $T_n = n^{-1} \sum^n_{i=1} J(i/n)X_{(i)} = \int^\infty_{-\infty} xJ(F_n(x)) dF_n(x)$ when the function $J$ has four bounded derivatives. More recently, it has been shown ([1], [2]) that under suitable restrictions \begin{equation*}\tag{1.1}\mathscr{L}\{n^{\frac{1}{2}}\lbrack T_n - \int^\infty_{-\infty} xJ(F(x)) dF(x)\rbrack\} \rightarrow N(0, \sigma^2)\end{equation*} where \begin{equation*}\tag{1.2}\sigma^2 = 2 \int\int_{s < t} J(F(s))J(F(t))F(s)\lbrack 1 - F(t)\rbrack ds dt.\end{equation*} Here $N(0, \sigma^2)$ denotes the normal distribution with mean zero and variance $\sigma^2$, and (1.1) uses a standard notation for convergence in distribution. The purpose of this note is to give a self-contained proof of (1.1) under the assumptions that the $X_i$ have finite mean and that $J'$ exists and is continuous and of bounded variation except at finitely many jumps of $J$. This theorem can be subsumed in a corrected version of Govindarajulu [2], where appeal is made to the results of [3]. The present proof is both more elementary and shorter. Chernoff, Gastwirth and Johns [1] do not require boundedness of $J$, but invoke compensating assumptions on $F$. Their methods of proof are quite different from those of [2] and of the present paper.

The robust-efficiency of the test and estimator based on Wilcoxon's [7] signed rank statistic when the sample observations are drawn from different populations is studied here. Let $X_1, \cdots, X_n$ be $n$ independent random variables distributed according to continuous cumulative distribution functions (cdf) $F_1(x), \cdots, F_n(x)$, respectively. Let $\mathscr{F}$ be the class of all continuous cdf's which are symmetric about their medians. If $F_1 = \cdots = F_n = F \varepsilon \mathscr{F}$, the Wilcoxon's [7] signed rank statistic provides a robust test for and estimator of the median of $F(x)$, (cf. [2], [4], [6], [7]). The asymptotic relative efficiency (ARE) of this test and estimator has been studied by Hodges and Lehmann [1]. The present investigation is concerned with the study of the robust-efficiency of the same when $F_1, \cdots, F_n$ are not necessarily identical.