A multivalued mapping from a space $X$ to a space $S$ carries a probability measure defined over subsets of $X$ into a system of upper and lower probabilities over subsets of $S$. Some basic properties of such systems are explored in Sections 1 and 2. Other approaches to upper and lower probabilities are possible and some of these are related to the present approach in Section 3. A distinctive feature of the present approach is a rule for conditioning, or more generally, a rule for combining sources of information, as discussed in Sections 4 and 5. Finally, the context in statistical inference from which the present theory arose is sketched briefly in Section 6.

In [14], Sanov proved that if $F_N$ is the empirical cumulative distribution function (cdf) of a sample drawn from a population whose true cdf is $F_0$ and $\Omega$ is a set of cdf's which satisfies certain regularity conditions and does not contain $F_0$, then $P\{F_N \epsilon \Omega\}$ is roughly $\exp \{-N \inf_{F\epsilon \Omega} \int \ln (dF/dF_0) dF\}$. This theory is extended to the $c$-sample case and to the case where the set of cdf's in question depends on $N$. These extensions are used to estimate the probability of a large deviation of those statistics which are, or can be approximated by, uniformly continuous functions of the empirical cdf's. As an example, the main result is applied to the Wilcoxon statistic, and the resulting formula is used to compute the exact Bahadur efficiency of the Wilcoxon test relative to the $t$-test.

If $X$ is a random variable with $EX = 0, \operatorname{var}(X) = \sigma^2$, then the inequalities \begin{equation*}\begin{align*}\tag{1.1} P\lbrack | X | \geqq \epsilon\rbrack \leqq \sigma^2/\epsilon^2, \\ \tag{1.2}P\lbrack X \geqq \epsilon\rbrack \leqq \sigma^2/(\epsilon^2 + \sigma^2)\end{align*},\end{equation*} where $\epsilon > 0$, are known as, respectively, the Tchebycheff and the one-sided Tchebycheff inequalities. It may be noted that under the stated conditions (1.2) can always be attained and (1.1) can be attained if $\sigma^2 \leqq \epsilon^2$. Let $\mathbf{X} = (X_1, X_2, \cdots, X_n)$ be a random vector with $E\mathbf{X} = \mathbf{0}, E\mathbf{X'X} = \mathbf{\Sigma, T}_+$ be a closed convex region in $R^n$, and $T = T_+ \bigcup \{\mathbf{X} \mid - \mathbf{X} \epsilon T_+\}$. In [3], Marshall and Olkin have obtained sharp upper bounds on $P\lbrack\mathbf{X} \epsilon T\rbrack$ and $P\lbrack\mathbf{X} \epsilon T_+\rbrack$ as multivariate generalizations of (1.1) and (1.2). They have used these bounds to obtain explicit sharp bounds on $P\lbrack \min X_i \geqq 1\rbrack, P\lbrack \min | X_i| \geqq 1\rbrack, P\lbrack |\prod X_i| \geqq 1\rbrack, P\lbrack \prod X_i \geqq 1, X_i > 0, i = 1, 2, \cdots, n\rbrack$ etc., when only the variances $\sigma_i^2$ of $X_i, i = 1, 2, \cdots, n$, are known. Let $\mathbf{Y} = (Y_1, Y_2, \cdots, Y_n)$ be a vector of $n$ nonnegative random variables with $E\mathbf{Y} = u$ and $\varphi \geqq 0$ be a homogeneous concave function on the nonnegative orthant $R_+^n$ of $R^n$. In Section 2 we have proved the main inequality of this paper, $P\lbrack\varphi(\mathbf{Y}) \geqq \epsilon\rbrack \leqq \varphi(\mathbf{u})/ \epsilon, \quad \epsilon > 0$, which is attainably sharp if $\varphi(\mathbf{u}) \leqq \epsilon$. In Section 3 this result has been used to obtain various generalizations of (1.1) and (1.2) to $n$ jointly distributed random variables or random vectors, which are also generalizations of the inequalities due to Marshall and Olkin for the case where only variances are known. In Section 4 we have obtained some sharp probability inequalities for some functions of symmetric psd (positive semidefinite) random matrices and have discussed their relation to inequalities due to Mudholkar [4].

The Stirling numbers $\{\sigma_n^j\}$ of the second kind are asymptotically normal. This result is similar to results achieved by Feller [1] and Goncarov [2] for other combinatorial distributions. Here the technique of proof is different; one of the most general forms of the central limit theorem is used. Interesting qualitative information about the Stirling numbers is also obtained from this result. Asymptotic estimates on the value of $\max_j \{\sigma^j_n\}$ are given.

In this paper the problem of testing the hypothesis $\Delta \leqq \Delta_0$ against $\Delta > \Delta_0$, where $\Delta$ is the ratio of variances in the one-way classification of the analysis of variance with variance components, is treated. The model is not restricted to equal class frequencies. It is found that the most powerful invariant test against an alternative $\Delta_1$ depends upon $\Delta_1$, but has the property of maximising the minimum power over the set of alternatives with $\Delta \geqq \Delta_1$. The test statistic is distributed like a ratio of linear combinations of independent chi-square distributed random variables. It is shown that a statistic used by Wald [6] to derive a confidence interval for $\Delta$ gives a test that is almost equal to the most powerful invariant tests against large alternatives $\Delta_1$. For the case $\Delta_0 = 0$ it is equal to the usual test in the fixed-effects model. In the balanced case the test reduces to the usual $F$-test which Herbach [2] has proved to be both uniformly most powerful invariant and uniformly most powerful unbiased.

The object of this article is to characterize the family of all distribution-free (DF) tests of independence, and those subfamilies which are optimal for specified alternative classes. It is found (Theorem 3.2) that each DF statistic is a function of a Pitman (or permutation) statistic; and (Theorem 3.4) that the rank statistics are those whose distributions depend appropriately on the maximal invariant [Lehmann (1959), p. 227]. For parametric alternatives the MP (most powerful) [Lehmann and Stein (1949)] and the locally MP tests are found to be Pitman tests based on the likelihood function (Lemma 4.1 and Theorem 4.1); while the corresponding optimal rank tests are analogous (Lemmas 4.2, 4.3) to those in the 2-sample case [e.g. Capon (1961)], and are closely related to those of Bhuchongkul (1964). In Section 5 randomized statistics similar to those of [4] are shown (Corollary 5.2) to be asymptotically equivalent to the optimal nonrandomized statistic for specified parametric alternatives. For one reasonable nonparametric class of alternatives one proves (Theorem 6.1) that the normal scores test is minimax; while for the other class an unexpected statistic (Theorem 6.2) is minimax. Finally, in Section 7 the ideas of monotone tests developed by Chapman (1958) and others are extended, and analogous results (Theorem 7.1) are obtained for minimum power.

Let $X_1, X_2, \cdots$ denote independent random variables having a common density $f$. The present paper considers estimates of $f$ of the following form ([8] and [10]): \begin{align}\tag{1.1}f_n(x) = \tau \int G(x, \tau(x - y)) dF_n(y) \\ = (\tau/n) \sum^n_{i = 1} G(x, \tau(x - X_i))\end{align} where $F_n$ denotes the sample distribution function of $X_1, \cdots, X_n, \tau = \tau(n) \rightarrow \infty$ and $\tau = 0(n)$ as $n \rightarrow \infty$, and $G$ is a non-negative function defined on $R^2$ satisfying regularity conditions to be listed in Section 2. More precisely, it considers the asymptotic behavior as $n \rightarrow \infty$ of the maximum deviation of $f_n(x)$ from $f(x)$ where $x$ varies in a compact interval which without loss of generality we take to be $\lbrack -1, 1\rbrack$. The main result, Theorem 3.1, states that under regularity conditions \begin{equation*}\tag{1.2}p\lim_{n \rightarrow \infty} \max_{| x| \leqq 1} (n/2\tau \log \tau)^{\frac{1}{2}} |(f_n(x) - f(x))/ |G_x |_2(f(x))^{\frac{1}{2}}| = 1\end{equation*} where $G_x$ denotes an $x$-section of $G$--i.e. $G_x(y) = G(x, y), y \epsilon R^1$--and $|\cdot |_p$ denotes the norm in $L_p = L_p(R^1$, Lebesgue measure). Also, a limiting distribution related to (1.2) is computed, and some sufficient conditions are given for the almost sure convergence of $\max_{|x| \leqq 1}| f_n(x) - f(x)|$ to zero as $n \rightarrow \infty$. Since under mild regularity conditions $f_n(x)$ and $f_n(y), x \neq y$, are asymptotically, independently, normally distributed when suitably normalized ([5]), one might expect (1.2) to follow from its analogue for normal random variables ([2]) by an argument involving the weak convergence of stochastic processes. However, having been unable to verify the necessary compactness conditions ([9]) with respect to any topology which makes the maximum functional in (1.2) continuous, we have adopted a more elementary approach. This approach uses a theorem on the large deviations of sums of independent, identically distributed random vectors to estimate the relevant probabilities directly. In order to develop the primary topic of the paper as quickly as possible, we have postponed the proof of the theorem on large deviations until Section 5, while using it in Sections 3 and 4 to prove our main theorems. Section 2 presents some preliminary material. We are aware of only two other papers which have considered the uniform convergence of $f_n$ to $f$, namely [8] and [7].

This paper is concerned with the problem of estimating a probability density which is known to be measurable with respect to a $\sigma$-lattice of subsets of the space on which it is defined. Our solution is represented as a conditional expectation. (Generally in the literature "conditional expectation" refers to "conditional expectation given a $\sigma$-field." However since we shall be concerned exclusively with "conditional expectation given a $\sigma$-lattice" we shall use this abbreviated terminology for the latter, more general concept.) Brunk [2] discusses conditional expectations and many of the extremum problems for which they provide solutions. We shall consider the case where the measure space, $(\Omega, \mathscr{A}, \mu)$ on which the density is defined is totally finite. Let $\mathscr{L}$ denote a $\sigma$-lattice of subsets of $\Omega(\mathscr{L} \subset \mathcal{A})$. A $\sigma$-lattice, by definition, is closed under countable unions and intersections and contains both $\Omega$ and the null set $\varnothing$. Let $\omega_1, \omega_2, \cdots, \omega_n$ be a sample of independent observations chosen in $\Omega$ according to the unknown, $\mathscr{L}$-measurable density $f$. We say that a point is chosen in $\Omega$ according to $f$ if the probability that it will lie in any set $A$ in $\mathscr{A}$ is given by $\int_Af d\mu$. The function $f$ is $\mathscr{L}$-measurable if the set $\lbrack f > a\rbrack$ is in $\mathscr{L}$ for each real number $a$. We shall use the maximum likelihood criterion for choosing an estimate. In other words we wish to find an $\mathscr{L}$-measurable density $\hat f$ such that the product of the values of $\hat f$ at the observed points is at least as large as the product of the values of any other $\mathscr{L}$-measurable density at those points. Such a function will be called a maximizing function. Clearly the $\sigma$-lattice $\mathscr{L}$ must satisfy some restrictions in order for the problem to be of any interest at all. For example if $(\Omega, \mathscr{A}, \mu)$ is a finite subinterval of the real line together with Borel subsets and Lebesgue measure and if $\mathscr{L} = \mathscr{A}$ then there are many obvious solutions if the density is bounded, and none at all if it is not bounded. The second section of this paper is devoted to the restrictions that we impose and to showing that these restrictions are satisfied in some problems which are of interest. The fourth section of this paper is devoted to some results on the asymptotic properties of our estimates in three special cases. The methods used are similar to those used by Marshall and Proschan [4]. The final section contains some observations on the problem of estimating a density on a non-totally finite measure space. Let $L_2$ denote the set of square integrable random variables and $L_2(\mathscr{L})$ the collection of all those members of $L_2$ which are $\mathscr{L}$-measurable. Let $R(\mathscr{L})$ denote the collection of all $\mathscr{L}$-measurable random variables. Let $\mathscr{B}$ denote the collection of Borel subsets of the real line. We shall adopt the following definition for the conditional expectation, $E_\mu(f \mid \mathscr{L})$, of a random variable given a $\sigma$-lattice. DEFINITION 1.1. If $f \epsilon L_2$ then $g \epsilon L_2(\mathscr{L})$ is equal to $E_\mu(f \mid \mathscr{L})$ if and only if $g$ has both of the following properties: \begin{equation*}\tag{1.1}\int (f - g)h d\mu \leqq 0 \text{for all} h \epsilon L_2(\mathscr{L})\end{equation*} and \begin{equation*}\tag{1.2}\int_B(f - g)d\mu = 0\quad\text{for all}\quad B \epsilon g^{-1}(\mathscr{B})\end{equation*}. (Brunk [1] shows that there is such a random variable $g$ associated with each $f \epsilon L_2$ and that $g$ is unique in the sense that if $g'$ is any other member of $L_2(\mathscr{L})$ having these properties then $g = g'\lbrack\mu\rbrack$.) In order to motivate the consideration of a problem such as the one we take up in this paper we introduce the following examples. The first example is discussed in [5]. EXAMPLE 1.1. Suppose $\Omega$ is a finite set and we denote its elements by $1, 2, \cdots, k$. Let $\mathscr{A}$ be the collection of all subsets of $\Omega$ and suppose $\mu$ assigns positive mass to each point in $\Omega$. Suppose $\mathscr{L}$ is an arbitrary $\sigma$-lattice of subsets of $\Omega$. Let $n_i$ denote the number of times the point $i$ is observed. We wish to find an $\mathscr{L}$-measurable density $\hat f$ on $\Omega$ such that $\prod^k_{i = 1} \hat f(i)^{n_i} \geqq \prod^k_{i = 1} h(i)^{n_i}$ for every other $\mathscr{L}$-measurable density $h$. It is shown in [5] that a solution is given by $\hat f = E_\mu(g \mid \mathscr{L})$ where $g(i) = n_i\lbrack n \cdot \mu(i)\rbrack^{-1}$. The problem posed in the next example was solved by Pyke (personal communication with Professor Brunk) and for a monotone density by Grenander [3]. EXAMPLE 1.2. Suppose $\Omega$ is a closed subinterval of the real line $(\Omega = \lbrack c, d\rbrack), \mathscr{a}$ is the collection of Borel subsets of $\Omega$ and $\mu$ is Lebesque measure. We wish to estimate the density $f$ which is known to be unimodal at some unknown point in $\Omega$. Suppose that our observations are ordered: $\omega_1 < \omega_2 < \cdots < \omega_n$. If $h$ is any unimodal density with mode at $a$ and $\omega_j < a < \omega_{j + 1}$ then define the function $g$ by: $g(x) = h(\omega_j)\quad (\omega_j \leqq x < a) \\ = h(\omega_{j + 1}) \quad (a \leqq x \leqq \omega_{j + 1}) \\ = h(x) \quad \text{otherwise}$. It is easily seen that the density $\hat f = \lbrack \int g d\mu\rbrack^{-1} \cdot g$ is unimodal with mode at $\omega_j$ or $\omega_{j+1}$ and that the product of the values of $\hat f$ at the observed points is at least as large as the product of the corresponding values of $h$. Hence our problem reduces to finding an estimate which has mode at one of the observed points. Similarly we can show that any maximizing estimate must be constant on every open interval joining two consecutive observed values. The next remark is easy to verify. REMARK 1.1. Let $\mathscr{L}$ be the $\sigma$-lattice of subsets of $\Omega$ consisting of all those intervals containing the point $a$. A function $f$ on $\Omega$ is unimodal at $a$ if and only if $f$ is $\mathscr{L}$-measurable. If we can estimate the density subject to the restriction that it is unimodal at a fixed point then we can select an estimate by comparing the ones we get by assuming the mode is at particular observations. We see thus that the problem of estimating a unimodal density reduces to estimating a density which is measurable with respect to a $\sigma$-lattice and constant on intervals joining consecutive observations. We shall see that these results for a unimodal density are typical of a larger class of problems. Now consider the problem of estimating a unimodal density on the reals together with Lebesgue measure. Clearly any estimate must be zero outside of the smallest closed interval containing all the observed points. Hence this problem reduces to the one discussed above.

Many optimization problems in mathematics and statistics are concerned with a quantity, or procedure, that yields some optimum value rather than the value itself. Frequently, they are referred to as searching problems. In this paper we consider a problem of this nature, which is described as follows. An object to be found is located in one of $R$ locations. A prior probability $p_i$ that the object is in $i$ is given $(\sum_ip_i = 1)$, along with an overlook probability $\alpha_i$ that if the object is in $i$, it is not found there on a given inspection of $i (0 < \alpha_i < 1, i = 1, \cdots, R)$. The $R$ locations--or boxes--can be searched one at a time, and it is assumed that all outcomes are independent, conditional on the location of the object and the inspection procedure used. Consequently, if the object is in $i$, the realization of the `first success'--the detection of the object--follows a geometric distribution with parameter $1 - \alpha_i$. A search procedure $\delta = (\delta_1, \delta_2, \cdots)$ is a sequence indicating the location to be inspected at each stage of the search, and one is interested in a $\delta$ that in some sense is optimal. In his notes on dynamic programming, Blackwell (see [3]) has shown that if, in addition, an inspection of $i$ costs $c_i$, the procedure minimizing the expected searching cost is a one-stage procedure, which instructs the searcher to inspect at each stage that box for which the ratio of the current detection probability, and the cost of an inspection there, is greatest. It is assumed here that $c_i = 1, i = 1, \cdots, R$. If $\delta^\ast = (\delta^\ast_1, \delta^\ast_2, \cdots)$ denotes an optimal one-stage procedure and $\delta^\ast_n = j$, then the detection probability for $j$ on the $n$th inspection is \begin{equation*}\tag(1.1) p_j\alpha_j^{m(j,n-1,\delta^\ast)} (1 - \alpha_j) = \max_i \{p_i\alpha^{m(i,n-1,\delta^\ast}_i (1 - \alpha_i)\}\end{equation*} where $m(i, n, \delta)$ is the number of inspections of $i$ among the first $n$ inspections made using $\delta$. In the next section $\delta^\ast$ is shown to be strongly optimal in the sense that \begin{equation*}\tag{1.2}P\lbrack N > n |\delta^\ast\rbrack \leqq P\lbrack N > n | \delta\rbrack,\end{equation*} for each $n = 0, 1, 2, \cdots$, and every search procedure $\delta$. (Here $N$ denotes the (random) number of inspections required to find the object.) The use of procedure $\delta^\ast$ thus ensures the greatest chance of finding the object within any fixed number of inspections, or, according to B. O. Koopman [2], it "optimally allocates the available search effort $\Phi$" for any number $\Phi$ of inspections. As a consequence of (1.2), $\delta^\ast$ also minimizes the expected `cost' of the search (since $E\lbrack N \mid \delta\rbrack = \sum^\infty_{n = 0} P\lbrack N > n \mid \delta\rbrack$), which is Blackwell's result for this case. Among the many interesting problems which arise in connection with this searching problem is the question of periodic features of the optimal procedure. Staroverov [5] and Matula [3] concerned themselves with this question, which was virtually answered by Matula who found necessary and sufficient conditions ensuring ultimate periodicity. We attempt to answer a different question in Section 3, where the same search problem is considered with the modification that $\sum_i p_i = 1 - q < 1$. That is, the object is in one of $R + 1$ locations, but searching is permitted only among the first $R$. (For instance, the $(R + 1)$st location could be the rest of the world.) In this problem every search procedure has positive probability of never terminating, making it necessary to couple a stopping rule $s$ (integer-valued) with any procedure $\delta$. A loss function is defined by imposing on the searcher a penalty cost $c(>0)$, payable when searching stops if the object has not been found. Thus, one either pays the cost of unsuccessful inspection plus the penalty cost, or simply the cost of inspection if the object is found prior to stopping. A procedure ($\delta, s$) is sought which minimizes the expected cost to the searcher, i.e. which yields the Bayes risk. Such a procedure exists and is shown to be $(\delta^\ast, s^\ast)$, where $\delta^\ast$ is a strongly optimal procedure satisfying (1.2). The determination of $S^\ast$ is the more difficult problem, however, and the main part of Section 3 is devoted to it. Instead of $s^\ast$ we consider the problem of finding the Bayes stopping region $S_B$, that set of posterior probabilities $p = (p_1, \cdots p_R)$ for which the Bayes risk equals the penalty cost $c$. Regions $S_L$ $S_U$ are given which bound $S_B$ in the sense that $S_L \subset S_B \subset S_U$. In Section 4, it is shown that for sufficiently large $C, S_L$ and $S_U$ `differ' by at most $R$ inspections taken according to $\delta^\ast$, and no more than one additional inspection is required in each of the $R$ boxes. If $(\delta^\ast, s_U)$ is a procedure which searches according to $\delta^\ast$, but stops once the posterior distribution $p \epsilon S_U$, a constant independent of $c$, is exhibited which bounds the difference between the Bayes risk and the expected risk using $(\delta^\ast, s_U)$.

A class of alternatives is here presented to Fisher's (1925) expansion of Student's $t$ density function. These expansions involve Appell's polynomials; and hence, recurrence schemes are available for the coefficients. Complete integrals of products of $t$ densities are of interest as Behrens-Fisher densities (viewed as Bayesian posterior distributions: Jeffreys, 1940; Patil, 1964) as moments of Bayesian posterior distributions (Anscombe, 1963; Tiao and Zellner, 1964). A symptotic expansion of complete integrals, obtained by term-by-term integration of these expansions, are favorably compared with those obtained from Fisher's expansion. Although expansions of complete integrals of products of multivariate $t$ densities can be developed from these expansions by the methods of Tiao and Zellner, the resulting coefficients are practically as complicated as the Tiao and Zellner coefficients; methods will be published soon (Dickey, 1965) for reducing the dimensionality of such integrals for quadrature. The paper concludes with a numerical study of the integral expansions.

For ordered samples of size $n$, with proportions $q_1$ and $q_2$ of the sample values censored from below and from above, respectively, from three-parameter Weibull and gamma populations, expressions are given for the elements of the maximum-likelihood information matrices, each element being the negative of the expected value of one of the second partial derivatives of the likelihood function with respect to the parameters. An important result is that, while one or more of the elements become infinite for values of the shape parameter less than or equal to 2 when $q_1 = 0$, this does not happen for $q_1 > 0$. If one lets $n \rightarrow \infty$ while holding $q_1$ and $q_2$ fixed and then inverts the information matrix and its submatrices, the results are the asymptotic variance-covariance matrices whose elements are the asymptotic variances and covariances of the joint maximum-likelihood estimators of all three parameters and of any one or two parameters, the other(s) being known. Tables are given of the coefficients of $(1/n)$ times powers of the scale parameter $\theta$ in the asymptotic variances and covariances for the cases $q_1 = 0.000(0.005)0.25, q_2 = 0.00(0.25)0.75$ for both Weibull and gamma populations with shape parameters 1, 2, and 3, omitting cases for which $q_1 = 0$ when the shape parameter is 1 or 2 and the location parameter is one of those being estimated. Results of a limited Monte Carlo study indicate that when at least one of the parameters is known, the variances and covariances of samples of size as small as 50 agree closely with the results given by the asymptotic formulas, but that when all three parameters are unknown, the variances and the absolute values of the covariances, even for samples of size as large as 100, are substantially in excess of the asymptotic values.

An extensive family of two-class PBIB designs is defined in Section 1 and given the title of NL designs. Several analogies of NL designs with Latin square type designs are discussed in Sections 1 and 2. Some of these designs have already appeared in scattered contexts. Section 3 gives a construction based on finite geometry for an infinite class of NL designs, and others can be constructed by methods to be taken up elsewhere. An extension to designs with more than two associate classes is indicated in Section 4.

The purpose of this note is to provide alternative proofs of results of Derman and Veinott [1]. The method of proof uses the potential theory for Markov chains developed by Kemeny and Snell in [2], [3], [4], [5]. In [2], Kemeny and Snell treat transient and recurrent chains separately, whereas Derman and Veinott consider chains having one positive recurrent class, $C, (0 \epsilon C)$, and a set of transient states, $T$, with $T \cup C = \{0, 1, 2, \cdots\}$. This difference does not cause any great difficulty. Kemeny and Snell also assume in [2] that their recurrent chains are noncyclic. The recurrent class, $C$, will therefore be assumed noncyclic in this note, the extension to the cyclic case being handled in the usual way. By relabelling the states, one may write the transition matrix, $P$, as \begin{equation*}\tag{1.1}P = \begin{pmatrix}P^1 R 0 \\ R Q\end{pmatrix}\end{equation*} where $P^1$ is the transition matrix of the recurrent class and $Q$ is the (substochastic) transition matrix of the set of transient states. In this note, the notation will be that of Derman and Veinott [1] and all references to the work of Kemeny and Snell will be to [2]. In matrix notation, the Derman-Veinott equation is \begin{equation*}\tag{1.2} (I - P)v = \omega^{\ast}\end{equation*} where $\omega{^\ast} = \omega - g\mathbf{1}, \omega = (\omega_0, \omega_1, \omega_2, \cdots)^T$ is a known vector, $\mathbf{1} = (1, 1, 1, \cdots)^T, v = (v_0, v_1, v_2, \cdots)^T$ is an unknown vector and $g$ is an unknown constant.

Bartlett [1] obtained an asymptotic relation connecting the periodogram of a linear process and the periodogram of the residuals with uniform spectrum in the case of one dimensional stationary time series, and using this relation he studied the asymptotic properties of the periodogram. In this paper we obtain some asymptotic relations between the co-periodogram and quadrature periodogram of the residuals in the case of a stationary Gaussian vector process. The covariances of the co-periodogram and quadrature periodogram have also been obtained. Two inequalities connecting the variance and the bandwidth of the cross spectral estimate have been derived.

It is shown that the distributions of certain multivariate analogues of $t$ and $F$ are dependent on the population covariance matrix. Suppose that $\mathbf{z} = (x_1, \cdots, x_p)'$ has the multivariate normal distribution with mean vector $\mathbf{\xi}$ and covariance matrix $\mathbf{\sigma}$. Let $\mathbf{S, S}^\ast$ be independent Wishart matrices also with covariance matrix $\mathbf{sigma}$, based on $n, n^\ast$ degrees of freedom respectively. Then if $\mathbf{T}$ is a $p \times p$ matrix such that \begin{equation*}\tag{1} \mathbf{TT}' = S\end{equation*} natural candidates for the multivariate analogues of $t$ and $F$ are \begin{equation*}\begin{align*}\tag{2}\mathbf{t} = \mathbf{t}^{-1} (\mathbf{z} - \mathbf{\xi}), \\ \tag{3}\mathbf{W} = \mathbf{T} ^{-1}\mathbf{S}^{\ast}\mathbf{T}^{'-1}.\end{align*}.\end{equation*} Olkin and Rubin (1964), Theorems 3.2 and 4.2, have shown that if $\mathbf{T}$ is taken to be upper or lower triangular, then $\mathbf{t}$ and $\mathbf{W}$ do in fact have distributions which independent of $\mathbf{sigma}$. However, if $\mathbf{T}$ is taken to by symmetrical and positive definite, $\mathbf{T} = \mathbf{S}^\frac{1}{2}$, then they remark (Section 3) that the distribution of $\mathbf{W}$ is unknown for general $\mathbf{\sigma}$. It seems worthwhile to present the following example, which shows that for $\mathbf{T = S}^\frac{1}{2}$ the distributions of $t$ and $\mathbf{W}$ depend on $\mathbf{\sigma}$; the contrary is occasionally asserted (see Bennett and Cornish (1964), p. 907. Let $p = 2$, and assume that\begin{equation*}\tag{4} \mathbf{\sigma}^{-1} = \begin{pmatrix}a^2@0\\0@b^2.\end{pmatrix}\end{equation*} Since $\mathbf{S}^\frac{1}{2}$ is positive definite, it may be written in the form\begin{equation*}\tag{5} \mathbf{S}^\frac{1}{2} = \begin{pmatrix}x,@(xy)^\frac{1}{2}q\\(xy)^\frac{1}{2}q,@y (\mathbf{0} < x, y < \infty, q^2 < 1).\end{pmatrix}\end{equation*} Noting the Jacobian\begin{equation*}\tag{6}\partial(\mathbf{S})/\partial(x,y,q) = 4(1 - q^2)(xy)^\frac{1}{2}(x + y),\end{equation*} it is readily found by transforming the Wishart distribution that $\mathbf{S}^\frac{1}{2}$ has the distribution. \begin{equation*}\begin{align*}\tag{7}f(\mathbf{S} ^\frac{1}{2}) d\mathbf{S}^\frac{1}{2} = \{(nab)^n/ \pi\Gamma(n - 1)\} (1 - q^2)^{n-2}(xy)^{n-\frac{3}{2}}(x + y) \\ \cdot\exp \{ -\frac{1}{2}n \lbrack a^2x^2 + (a^2 + b^2)xyq^2 + b^2y^2\rbrack\} dx dy dq, \\ (0 < x, y < \infty, q^2 < 1)\end{align*}.\end{equation*} (See Olkin and Rubin, p. 266.) Now \begin{equation*}\begin{align*}\tag{8} \epsilon(\mathbf{tt}' &= \epsilon\mathbf{W} \\ &= \epsilon (\mathbf{S}-^\frac{1}{2} \mathbf{\sigma S}^{-\frac{1}{2}}) &= \begin{bmatrix}\nu_1/a^2 + \nu_3/b^2, \quad - \nu_4/ a^2 - \nu_5/b^2 \\ - \nu_4/a^2 - \nu_5/b^2, \quad \nu_3/a^2 + \nu_2/b^2\end{bmatrix}\end{align*},\end{equation*} where\begin{equation*}\begin{align*}\tag{9}\nu_1 &= \mathscr{E}\lbrack x^{-2} (1 - q^2)^{-2}\rbrack, \quad \nu_2 = \mathscr{E}\lbrack y^{-2} (1 - q^2)^{-2}\rbrack, \quad \nu_3 = \mathscr{E} \lbrack q^2/xy (1 - q^2)^2\rbrack, \\ \nu_4 = \mathscr{E}\lbrack q/x^{\frac{3}{2}} y ^{\frac{1}{2}}(1 - q^2)^2\rbrack, \quad \nu_5 = \mathscr{E}\rbrack 1/x^{\frac{1}{2}} y^{\frac{3}{2}} (1 - q^2)^2\lbrack\end{align*}. \end{equation*} It will be sufficient to show that the matrices in (8) are not independent of $a$ and $b$. In order to simplify the calculations still further, we shall take the case $n = 4$, and consider only the elements $\epsilon(t^2_1), \epsilon(t^2_2)$ on the main diagonal in (8). The density (7) now takes the form: \begin{equation*}\tag{10}f(\mathbf{S}^{\frac{1}{2}} d\mathbf{S}^{\frac{1}{2}} = 2^{7_\pi^{-1}} (ab)^4(1 - q^2)^2(xy)^{\frac{5}{2}} (x + y) \cdot\exp \{-2\lbrack a^2x^2 + (a^2 + b^2)xyq^2 + b^2y^2\rbrack\} dx dy dq,\end{equation*} and we wish to evaluate $\nu_1, \nu_2, \nu_3$. For $\nu_1$, let us first consider \begin{equation*}\begin{align*}\tag{11}I(q) &= \int_0^\infty \int_0^\infty x^{-2}(1 - q^2)^{-2}f(\mathbf{S} ^{\frac{1}{2}}) dx dy \\ &= 2^{7_\pi^{-1}}(ab)^4 \int_0^\infty \int_0^\infty (x^{\frac{3}{2}}y^{\frac{5}{2}} + x^{\frac{1}{2}} y^{7/2}) e^{-2a2x2-2b2y2} \\ \cdot \sum^\infty_{k=0} K!^{-1}\lbrack -2(a^2 + b^2)xyq^2\rbrack^k dx dy\end{align*}.\end{equation*} Making use of the formulas \begin{equation*}\begin{align*}\tag{12}\Gamma(\omega)\Gamma(\omega + \frac{1}{2}) &= \pi^{\frac{1}{2}}\Gamma)(2\omega)/2^{2\omega-1}, \\ \int_0^\infty \omega^\lambda e^{-a^2\omega^{2}} d\omega &= \Gamma((\lambda + 1)/2)/2a^{\lambda+1},\\ \sum^\infty_{k=0} x^k\Gamma(k + \lambda)/k! = \Gamma(\lambda)(1 - x)^{-\lambda}, \quad (| x | < 1)\end{align*}\end{equation*}, it is found that \begin{equation*} \begin{align*}\tag{13}I(q) &= (2a^3/\pi b^{\frac{1}{2}} \sum^\infty_{k=0} k!^{-1}\lbrack -(a^2 + b^2)q^2/2ab\rbrack^k \cdot\{a\Gamma(k + \frac{3}{2}) + (a + b)\Gamma(k + \frac{5}{2})\} \\ \tag{14} &=(a^3/2b)^{\frac{1}{2}} \{a\phi^{-\frac{3}{2}}(q) + \frac{3}{2}(a + b)\phi^{-\frac{5}{2}}(q)\} \end{align*}\end{equation*} where \begin{equation*}\tag{15} \phi(q) = 1 + (a^2 + b^2)q^2/2ab\end{equation*} The series in (13) is convergent only for\begin{equation*}\tag{16} | q |^2 < 2ab/(a^2 + b^2)\end{equation*} However, since $I(q)$ and the expression (14) are both analytic functions of the complex variable $q = u + iv$ over the region $u^2 - v^2 > -2ab/(a^2 + b^2)$ in the $q$ plane, it follows by analytic continuation that $I(q)$ is certaintly equal to (14) for all real $q$. Noting that \begin{equation*}\begin{align*}\tag{17}\int_0^1\phi ^{-\frac{3}{2}}(q) dq &= (2ab)^{\frac{1}{2}}/(a + b), \\ \int_0^1\phi^{-\frac{5}{2}}(q) dq = \frac{2}{3}(2ab)^{\frac{1}{2}}(a^2 + 3ab + b^2)/(a + b)^3\end{align*},\end{equation*} we have \begin{equation*}\begin{align*}\tag{18}\nu_1 &= 2 \int_0^1 I(q) dq \\ &= 2a^2\lbrack 2 - (b/(a + b))^2\rbrack\end{align*},\end{equation*} $\nu_2$ being obtained by interchanging $a$ and $b$. Similarly, it is found that \begin{equation*}\tag{19}\nu_3 = 2(ab/(a + b))^2.\end{equation*} Hence \begin{equation*}\begin{align*}\tag{20}\mathscr{E}(t^2_1) 7= \nu_1/a^2 + \nu_3/b^2 = 4 + 2(a - b)/(a + b), \\ \mathscr{E}(t^2_2) &= \nu_3/a^2 + \nu_2/b^2 = 4 - 2(a - b)/(a + b)\end{align*},\end{equation*} and the matrices in (8) are not dependent of $a$ and $b$. As a check on the working, we have: \begin{equation*}\tag{21}\mathscr{E}(t^2_1 + t^2_2) = \mathscr{E} (\mathbf{z} -\xi)'\mathbf{S}^{-1}(\mathbf{z} - \xi) = 8,\end{equation*} which is independent of $\Sigma$ and in accordance with the known distribution of Hotelling's $T^2$.

Distribution problems in multivariate analysis are often related to the joint distribution of the characteristic roots of a matrix derived from sample observations. This well-known Fisher-Girshick-Hsu-Mood-Roy distribution (under certain null hypotheses) of $s$ non-null characteristic roots can be expressed in the form \begin{equation*}\tag{1.1}f(\theta_1, \cdots, \theta_s) = C(s, m, n) \prod^s_{i = 1} \theta^m_i (1 - \theta_i)^n \prod_{i > j} (\theta_i - \theta_j), 0 < \theta_i \leqq \cdots \leqq \theta_s < 1,\end{equation*} where \begin{equation*}\tag{1.2}C(s, m, n) = \pi^{\frac{1}{2}s^2}\Gamma_s(m + n + s + 1)/ \{\Gamma_s(\frac{1}{2}(2m + s + 1))\Gamma_s(\frac{1}{2}(2n + s + 1))\Gamma_s(\frac{1}{2}s)\}, \end{equation*} $\Gamma_s(\cdot)$ is the multivariate gamma function defined in [2], and $m$ and $n$ are defined differently for various situations described in [4], [6]. Pillai [3], [5] has given the density function of the larger of two roots, i.e. when $s = 2$, as a hypergeometric function. In this paper, the result is extended to the general case giving the density function of the largest of $s$ roots as a generalized hypergeometric function [1], [2]. The density function of the largest root of a sample covariance matrix derived by Sugiyama [7] can be obtained from the one derived here by considering the transformation $\frac{1}{2}\lambda_s = n\theta_s$, and making $n$ tend to infinity.