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Inference procedures for interelass and intraclass correlations are given in the multivariate context of familial data for which measurements are taken on more than one characteristic. Unified estimators are proposed based on a certain class of unbiased estimators of covariance matrices. Asymptotic distributions of the proposed estimators are derived under the assumption of multivariate normality. The results can be used to construct approximate confidence intervals and test procedures.

Let ( x , Xβ , V ) be a linear model and let A ′ = ( A ′ 1 , A ′ 2 ) be a p × p nonsingular matrix such that A 2 X = 0, Rank A 2 = p − Rank X . We represent the BLUE and its covariance matrix in alternative forms under the conditions that the number of unit canonical correlations between y 1 ( = A 1 x ) and y 2 ( = A 2 x ) is zero. For the second problem, let x ′ = ( x ′ 1 , x ′ 2 ) and let a g -inverse V − of V be written as ( V − )′ = ( A ′ 1 , A ′ 2 ). We investigate the reations (if any) between the nonzero canonical correlations { 1 ≧ ϱ 1 ≧ … ≧ ϱ 1 > 0 } due to y 1 ( = A 1 x ) and y 2 ( = A 2 x ), and the nonzero canonical correlations { 1 ≧ λ 1 ≧ … ≧ λ v+r > 0 } due to x 1 and x 2 . We answer some of the questions raised by Latour et al. (1987, in Proceedings, 2nd Int. Tampere Conf. Statist. (T. Pukkila and S. Puntanen, Eds.), Univ. of Tampere, Finland) in the case of the Moore-Penrose inverse V + = ( A ′ 1 , A ′ 2 ) of V .

It is well known that the James-Stein estimates of mean values of several populations can be derived as empirical Bayes estimates assuming a common prior distribution for all the mean values. But the superiority of such estimates over the usual unbiased estimates diminishes as the variability of the true mean values between populations increases. In such cases it is suggested that the populations may be split into two or more homogeneous groups and the James-Stein procedure applied to the mean values in each group separately. In this paper, we introduce a hierarchical prior distribution by considering the mean values within a group to have a common prior with some hyperparameters which are different from group to group. The hyperparameters in different groups are themselves considered to have a common prior distribution possibly with hyper-hyperparameters. Under some conditions on variabilities between and within groups, it is shown that the empirical Bayes estimates derived from a two stage prior distribution on the mean values are better than those obtained by applying the James-Stein procedure to all the mean values in one step or to the mean values in individual groups separately.

There is loss of efficiency when an estimated noise covariance matrix is used in the place of the unknown true noise covariance matrix in the construction of the optimum filter for signal detection. In the case of detecting a single signal specified by a real or a complex vector, we investigate the extent of this loss by obtaining an exact confidence bound for the realized signal-to-noise ratio. We also give an estimate of this ratio which is useful in optimum selection of features. Some of these results are extended to the case of discrimination between a number of given signals.

The usual t-statistic is not useful if the successive observations have some kind of linear trend. This generally arises in the drug testing experiment and it is clearly pointed out by Shah (1988). He suggests to use t'-statistic which is defined by for testing μ = 0 where , and y1.,…,yk are independent observations from . We generalize this statistic to multivariate situation and define T'-statistic as where .The exact null distribution of T' and an approximate null distribution of T' are obtained. For p = 1, this approximate values are compared.

This article presents methods for evaluating exact variance of combined inter- and intra-block estimates of treatment effects in incomplete block designs. The results are valid for all incomplete block designs and hold for a wide class of combined estimates including most in the literature. Tables of exact variance for some estimates are presented for some balanced designs and for some partially balanced designs.

SYNOPTIC ABSTRACTThis article presents explicitly the results on the asymptotic distributions of the likelihood ratio test statistic −2 log λ (= n ) when the sample is from the nonnormal populations possessing the first four moments similar to those of an elliptically contoured distribution. The statistics are obtained on the various structures of Σ for one or more populations. In all the situations, the asymptotic distributions of n are either noncentral chi–squares or those of a linear function of two noncentral chi–square variates, when the alternatives are close to the null hypotheses. For other alternatives, we get asymptotic normality of (–F0)/σ0 where E() = F0 + o (1) and V() = σ02/n + o(n−l).

Necessary and sufficient conditions for existence and expressions for general Hermitian and nonnegative definite solutions are obtained for the following three systems of linear equations: (I) $AX = C$, (II) $AX = C,XB = D$, (III) $AXB = C$.

A weaker generalized inverse (Rao's g-inverse; Graybill's c-inverse) can be used in place of the Moore-Penrose generalized inverse to obtain multiple and canonical correlations from singular covariance matrices.

Schatzoff [9] obtained the forms of the probability density function (pdf) and the cumulative distribution function (cdf) of the product of independent beta random variables when their parameters had some special values. The forms, however, did not indicate the constants explicitly. In this paper his approach is modified so as to allow presentation of explicit expressions for the pdf and cdf of the product of independent beta random variables (without restriction to the values of the parameters) in neat forms. Applications in multivariate analysis are given for the central and the non-central cases.

Fujikoshi (1982) obtained the necessary and sufficient conditions for the increased number of variables in the two sets of vectors not affecting the original nonzero canonical correlations and used these to obtain the likelihood ratio test procedure. He assumed a nonsingular covariance matrix due to random variables. Here, we study the same problem when the covariance matrix is singular and establish some further results. In this study, we note that the unit canonical correlations have to be separated in some of the situations. These results are valid for complex random vector variables and in some situations, the test for redundancy is given for complex random variables.

Let $\mathbf{A}_1$ and $\mathbf{A}_2$ be two symmetric matrices of order $p, \mathbf{A}_1$, positive definite and having a Wishart distribution ([2], [23]) with $f_1$ degrees of freedom and $\mathbf{A}_2$, at least positive semi-definite and having a (pseudo) non-central (linear) Wishart distribution ([1], [3], [23], [24]) with $f_2$ degrees of freedom. Now let $\mathbf{A}_2 = \mathbf{CYY}'\mathbf{C}'$ where $\mathbf{Y}$ is $p \times f_2$ and $\mathbf{C}$ is a lower triangular matrix such that $\mathbf{A}_1 + \mathbf{A}_2 = \mathbf{CC}'.$ Now consider the $s$( = minimum $(f_2, p))$ non-zero characteristic roots of the matrix $\mathbf{YY}'$. It can be shown that the density function of the characteristic roots of $\mathbf{Y}'\mathbf{Y}$ for $f_2 \leqq p$ can be obtained from that of the characteristic roots of $\mathbf{YY}'$ for $f_2 \geqq p$ if in the latter case the following changes are made: [23] \begin{equation*}\tag{1.1} (f_1, f_2, p) \rightarrow (f_1 + f_2 - p, p, f_2).\end{equation*} Now, in view of (1.1), we consider only the case $s = p$, based on the density function [12] of $L = \mathbf{YY}'$ for $f_2 \geqq p$. In this paper, some results are obtained first regarding the $i$th elementary symmetric function (esf) of the characteristic roots of a non-singular matrix $\mathbf{P} (\operatorname{tr}_i\mathbf{P})$ which are useful to compute the moments of $\operatorname{tr}_i\mathbf{L}$ and $\operatorname{tr}_i\{(\mathbf{I} - \mathbf{L})^{-1} - \mathbf{I}\}$. In particular, the first two moments of $\operatorname{tr}_2 \mathbf{L}$ are obtained in the non-central linear case. These two moments of the above criteria in the central case have been obtained earlier by Pillai ([18], [19]). Further, from a study of the first four moments of $U^{(p)} = \operatorname{tr}\{(\mathbf{I} - \mathbf{L})^{-1} - \mathbf{I}\}$, [11], [14], two approximations to the distribution of $U^{(p)}$ were obtained in the general non-central case. The approximations are generalizations of those given by Khatri and Pillai [10] for the linear case. The accuracy comparisons of the approximations are also made.

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.

Let $\mathbf{S}_i : p \times p (i = 1,2)$ be independently distributed as Wishart $(n_i, p, \mathbf{\sigma}_i)$. Let the characteristic (ch) roots of $\mathbf{S}_1\mathbf{S}^{-1}_2$ and $\mathbf{\sigma}_1\mathbf{\sigma}^{-1}_2$ be denoted by $f_i (i = 1, 2,\cdots, p)$ and $\lambda_i(i = 1, 2, \cdots, p)$ respectively such that $0 < f_1 < f_2 < \cdots < f_p < \infty$ and $0 < \lambda_1 \leqq \lambda_2 \leqq \cdots \leqq \lambda_p < \infty$. The distribution of $f_1, f_2, \cdots, f_P$ as stated by James [5] is not convenient for further development and is slowly convergent for higher values of $f_i's$. The distribution of $(f_1, \cdots, f_p)$ mentioned by James [5] can be written as \begin{equation*}\tag{1} c|\mathbf{\lambda}|^{-\frac{1}{2}n_1}|\mathbf{F}|^{\frac{1}{2}(n_1 - p - 1} \alpha_p(\mathbf{F} \int_{O(p)}|\mathbf{I}_p + \mathbf{\lambda}^{-1}\mathbf{HFH}'|^{-\frac{1}{2}(n_1 + n_2)} d\mathbf{H}\end{equation*} where \begin{equation*}\tag{2}c = \pi^{\frac{1}{2}p^2} \Gamma_p(\frac{1}{2} n_1 + \frac{1}{2}n_2)\{ \Gamma_p (\frac{1}{2}P) \Gamma_p (\frac{1}{2}n_1)\Gamma_p(\frac{1}{2}n_2)\}^{-1}, \Gamma_p(t) = \pi^{\frac{1}{4}p(p - 1)} \Pi^p_{j = 1} \Gamma (t - \frac{1}{2}j + \frac{1}{2},\end{equation*} \begin{equation*} {3}\alpha_p(\mathbf{F}) = \Pi ^{p - 1}_{i = 1} \Pi^p_{j = i + 1} (f_j - f_i), \quad \mathbf{F} = \operatorname{diag} (f_1, f_2, \cdots, f_p), \mathbf{\lambda} = \operatorname{diag} (\lambda_1, \lambda_2, \cdots, \lambda_p) \end{equation*} and the intergral is over an orthogonal group $O(p)$ with $\int_{O(p)} d\mathbf{H} = 1$. For testing the null hypothesis $H_0(\lambda\mathbf{\lambda} = \mathbf{I}_p), \lambda > 0$ being given, we have two statistics given by \begin{equation*}\tag{4}(i) l = |\lambda\mathbf{F}|^{n_1}/|\mathbf{I}_p + \lambda F|^{n_1 + n_2} \text{and} (ii) \lambda f_P \text{or} \lambda f_p/(1 + \lambda f_p)\end{equation*}. $l$ is considered by Anderson [1] and $\lambda f_p$ is obtained by Roy [7]. (1) is rewritten in such a way that the joint density function of $(\lambda f_1, \lambda f_2, \cdots, \lambda f_p)$ has noncentral parameters $\mathbf{I}_p - (\lambda\mathbf{\lambda})^{-1}$ and it is given by \begin{equation*}\tag{5}c|\lambda\mathbf{\lambda} |^{\frac{1}{2}n_1}|\lambda\mathbf{F}|^{\frac{1}{2}(n_1-p-1} \alpha_p(\lambda\mathbf{F})| \mathbf{I}_p + \lambda\mathbf{F})| ^{-\frac{1}{2}(n_1 + n_2)} _1F_0^{(p)} (\frac{1}{2}n_1 + \frac{1}{2}n_2; \mathbf{I}_p - (\lambda\mathbf{\lambda}^{-1}, \lambda\mathbf{F}(\mathbf{I}_p + \lambda\mathbf{F})^{-1})\end{equation*}. Hence, similar to testing of means, we propose the statistics $T = tr (\lambda\mathbf{F})$ and $V = tr (\lambda\mathbf{F})(\mathbf{I}_p + \lambda\mathbf{F})^{-1}$ for testing the hypothesis $H_0$ and obtain their distribution of $T$ only while the moment generating function of $V$ is given. Moreover, if in the null hypothesis $H'_0(\lambda\mathbf{\lambda} = \mathbf{I}_p, \lambda > 0$ is unknown, then the test procedure will depend on the ratios of roots, and hence, we consider the joint distribution of $(x_1, x_2, \cdots, x_{p - 1})$, where $x_i = f_i/f_p$ for $i = 1, 2, \cdots, p - 1$, and $f_p$. Under null hypothesis $H'_0$, we obtain the density functions of $x_1$ (or $x_{p - 1}$) and of $(y_2, y_3, \cdots, y_{p - 1})$ for $y_i = (f_i - f_1)/f_p - f_1), i = 2, 3, \cdots, p - 1$.

By applying Roy's union—intersection principle to Rider's statistic (the ratio of two ranges) and Murty's statistic (the ratio of two maximum values), two statistics are obtained for testing the equality of ranges in k rectangular populations and their 5% points are tabulated.

Cohen [1] has obtained the likelihood estimates of the parameters in case of normal and exponential distributions when the samples are progressively censored. He assumed that the parameters remain the same at each stage of censoring. But there are situations in which it is reasonable to assume that the parameters of the distribution under consideration might change at each stage of censoring, e.g. the remaining items at each stage are checked and the defects are eliminated whenever possible. Srivastava [2] has treated this problem when the items under consideration come from an exponential distribution with different failure rates at different stages. If two stage progressively censored samples are considered, we have the following four types of censoring. Let n be the total number of the items under consideration. Let T1 and T2 be the times of censoring (T2 > TJ). Let ni be the number of items failed on or before T, and ri be the number of items eliminated after T,(i = 1, 2). Then we have the following schemes of experiment when n and r1 are given: (i) Let ni(i = 1, 2) and r, be fixed. Then Ti (i = 1, 2) are random variables which is termed as Type II censoring by Cohen [1]. (ii) Let n, and T2 be fixed. Then T1 and n2 are random variables. The experiment is stopped at the first stage if T1 2 T2, and otherwise the experiment is carried to the second stage. (iii) Let T1 and n2 be fixed. Then, T2 and n1 are random variables. If n >2 n-r, n2 , the experiment is stopped at the first stage, and T2 = T1 and otherwise the experiment is carried to the second stage. (iv) Let T1 and T2 be fixed. Then n1 and n2 are random variables. If n, > n r, , the experiment is stopped at the first stage, and otherwise the experiment is carried to the second stage. This type of censoring is termed as Type I by Cohen [1]. When there are k stages of censoring, there are in the above way 2k types of censoring and it is difficult to consider all of them at a time. The maximum likelihood estimates for all these types will remain the same, except for variances. Hence, we are considering here the last type of censoring for log-normal and logistic distributions and obtain the maximum likelihood estimates of the parameters involved in each stage of censoring. It is noted that due to different

We note that (1) is a weighted beta distribution with weights given by (3). I t is well-known that for large (n-t-k)and small pZ with (n-t-k)p2= (constant), (3) tends to a Poisson one. Hence, the distribution of R ~ under these conditions can be approximated to a non-central beta one. For the product-moment correlation coefficient r , assuming gO, n)r/ (1--r~) 1/2 to be distributed as non-central t with 2n degrees of freedom (d.f.), Harley [1] obtained gO, n) and the non-central parameter by using the first three moments, and she pointed out that this approximation is valid for all practical values of n. Hence, we study the transformed multiple correlation coefficient given by

In the curve E(y) = α + δx + βρ x , the estimation of the parameter ρ is of vital importance. Shah and Pate1 [2] have described a least-squares solution by using tables. A simple approach of finding a quick estimate of ρ with high efficiency is discussed here in detail. A table is given in order to simplify the calculation of the estimate. Two Chemical Engineering examples are discussed to illustrate the method.

Estimates of parameters of a singly truncated bivariate normal distribution have been directly obtained using eight (or nine) moments. Asymptotic variances and covariances of these estimates are given and they are compared with those of maximum likelihood estimates.

Let $\mathbf{A}_1$ and $\mathbf{A}_2$ be two symmetric matrices of order of $p, \mathbf{A}_1$, positive definite and having a Wishart distribution [2], [12] with $f_1$ degrees of freedom, and $\mathbf{A}_2$, at least positive semi-definite and having a (pseudo) non-central (linear) Wishart distribution ([1], [3], [4], [12], [13]) with $f_2$ degrees of freedom. Now let $\mathbf{A}_2 = \mathbf{CYY'C'}$ where $\mathbf{C}$ is a lower triangular matrix such that $\mathbf{A}_1 + \mathbf{A}_2 = \mathbf{CC}'$ and the density function of $\mathbf{Y} : p \times f_2$ is given by \begin{equation*}\tag{1.1}k_1e^{-\lambda^2} \sum^\infty_{j = 0} (2\lambda y_{11})^j\Gamma\lbrack\frac{1}{2}(f_1 + f_2 + j)\rbrack |\mathbf{I}_p - \mathbf{YY}'|^{\frac{1}{2}}(f_{1 - p - 1)}/j!\end{equation*} where $\mathbf{I}_p$ is an identity matrix of order $p$, $k_1 = \prod^p_{i = 2} \Gamma\lbrack\frac{1}{2}(f_1 + f_2 - i + 1)\rbrack/ \pi^{\frac{1}{2}pf_2} \prod^p_{i = 1} \Gamma\lbrack f_1 - i = 1)/2\rbrack,$ $\lambda$ is the only non-centrality parameter in the linear case and $y_{11}$ is the element in the top left corner of the $\mathbf{Y}$ matrix. Now $V^{(s)}$ criterion suggested by Pillai and $U^{(s)}$ (a constant times Hotelling's $T_0^2$), [7], [8], [9], [10] are the sums of the non-zero characteristic roots of the matrix $\mathbf{YY}'$ and $(\mathbf{I}_p - \mathbf{YY}')^{-1} - \mathbf{I}_p$ respectively. Here $s$ is minimum $(f_2, p)$. Also we may note that $V^{(s)} = \text{trace} \mathbf{YY}' = \text{trace} \mathbf{Y'Y}$ and $U^{(s)} = \mathrm{tr} (\mathbf{I}_p - \mathbf{YY}')^{-1} - p = \mathrm{tr} (\mathbf{I}_{f_2} - \mathbf{Y'Y})^{-1} - f_2$. It can be shown that the density function of the characteristic roots of the matrix $\mathbf{Y'Y}$ for $f_2 \leqq p$ can be obtained from that of the characteristic roots of $\mathbf{YY}'$ for $f_2 \geqq p$ if in the latter case the following changes are made: ([12], [5]) \begin{equation*}\tag{1.2}(f_1, f_2, p) \rightarrow (f_1 + f_2 - p, p, f_2).\end{equation*} Hence, for the criterion $V^{(s)}$, (and similarly for $U^{(s)}$), we shall only consider the density function of $\mathbf{L} = \mathbf{YY}'$ for $f_2 \geqq p$ which is given by [6] \begin{equation*}\tag{1.3}f(\mathbf{L}) = ke^{-\lambda^2}_1F_1\{\frac{1}{2} (f_1 + f_2), \frac{1}{2}f_2, \lambda^2l_{11}\}|\mathbf{L}|^{ (f_2 - p - 1)/2} |\mathbf{I}_p - \mathbf{L}|^{(f_1 - p - 1)/2}, \end{equation*} where $k = \pi^{-p(p - 1)/4} \prod^p_{i = 1} \Gamma\lbrack\frac{1}{2}(f_1 + f_2 + 1 - i)\rbrack/\{\Gamma\lbrack \frac{1}{2}(f_1 + 1 - i)\rbrack\Gamma\lbrack\frac{1}{2}(f_2 + 1 - i)\rbrack\}.$ $l_{11}$ is the element in the top left corner of the matrix $\mathbf{L}$ and $_1F_1$ denotes the confluent hypergeometric function. We shall call the distribution of $\mathbf{L}: p \times p$ the non-central (linear) multivariate beta distribution with $f_2$ and $f_1$ degrees of freedom. Pillai [11] had noted that the elements of the matrix $\mathbf{L}$ can be transformed into independent beta variables which he showed for $p = 2, 3, 4$ and $5$. In this paper we give a theorem which proves the general case. In addition, when $\lambda = 0$ the first and second order moments of $l_{ij}$ are obtained and used to derive the first two moments of $V^{(s)}$ in the non-central case when $f_2 \geqq p$. The moments of $V^{(s)}$ for $f_2 \leqq p$ can be written down with the help of (1.2). Similar results are obtained for $U^{(s)}$.

In this paper a decision procedure based on the Tukey Studentized range ([5], [6], [8]) has been shown to be an optimum procedure for a particular type of slippage of means of univariate normal populations based on a common but unknown variance. The method given here is similar to that used by Paulson [2] and Truax [7].

The paper extends the results of Khatri to complex elliptical variates. Asymptotic confidence bounds on location parameters for the linear growth curve for the complex variates, the asymptotic distribution of the canonical correlations for the two sets of complex variates, and the asymptotic confidence bounds for the discriminatory values for the linear Fisher's discriminator for the future complex observation z are developed in this paper on the lines given by Khatri.

Let \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x} ' = (x_1 ,x_2 , \ldots ,x_n ) \) be a random vector and let its joint density be $$ f(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x} '\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x} ) = (2\Pi )^{ - \frac{1} {2}n} \exp ( - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x} '\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x} /2)for\;all\;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x} \in R^n , $$ (1.1) where R n is a set of n-vectors defined on the real field R.

This paper provides proof of conjectures about the expected values of the $j$th elementary symmetric function of the roots (i.e. $E(\operatorname{tr}_j A)$) of a non-central Wishart matrix. Utilizing this result we have obtained computationally convenient formula for the expected value of the $j$th elementary symmetric function of the roots of central positive definite quadratic form in matrix argument.

In a two-factor design, a design which is optimal for one factor is shown to be optimal jointly for both the factors with respect to each of $A$-, $D$-, and $E$-optimalities. As an interesting consequence we have that a linked block design is optimal for the estimation of treatment differences. Similar results are also obtained for a class of three factor designs.

Certain generalizations of characterizing the normal law by constancy of regression are discussed in this paper.

Kantorovich gave an upper bound for the product (x'Vx)(x'V -1 x) where x is an n-vector of unit length and V is an nXn positive definite matrix. Bloomfield, Watson and Knott found the bound to |X'VXX'V -1 X|, and we found bounds for the trace and determinant of X'VYY'V -1 -1X where X and Y are nXk matrices such that X'X=Y'Y=I. In the present paper we establish bounds for traces and determinants of X'VYY'V -1 -1X and X'BYY'CX when X and Y are matrices of different orders. A review of previous results on generalizations of the Kantorovich inequality and a number of new results of independent interest are also given.

Let the column vectors of \(\mathop {\text{X}}\limits_ \sim :\) pxn be distributed as independent normals with common covariance matrix \(\mathop \sum \limits_ \sim \). Then, the quadratic form in normal vectors is denoted by \(\mathop {\text{X}}\limits_ \sim \mathop {\text{A}}\limits_ \sim \mathop {\text{X}}\limits_ \sim '\, = \,\mathop {\text{S}}\limits_ \sim \) where \(\mathop {\text{A}}\limits_ \sim :\,{\text{nxn}}\) is a symmetric matrix which is assumed to be positive definite. This paper deals with a series representation of the density function of \(\mathop {\text{S}}\limits_ \sim \) when \({\text{E(}}\mathop {\text{X}}\limits_ \sim {\text{)}}\, \ne \,\mathop 0\limits_ \sim \), extending the idea of the author (1971) and Kotz et al. (1967b) to the multivariate non- central case. It is pointed out that this method gives a result which is not easy to obtain directly by integrating over an orthogonal space in the sense of James (1964).