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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 $\mathbf{X}$ 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.

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.

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

In practical situations, one is generally faced with multivariate problems in the form of testing the hypotheses or obtaining a set of simultaneous confidence bounds on certain parameters of interest. We shall consider here the variates under study to be normally distributed. A lot of work on the univariate and multivariate normal populations for the simultaneous confidence bounds on the location and scale parameters has been done, (see references, not necessarily exhaustive). Establishing certain inequalities for normal variates, we try to give shorter confidence bounds on variances and on a given set of linear functions of location parameters when this set is previously chosen for study. For the univariate case, Dunn [6], [8] using the Bonferroni inequality, obtained shorter confidence bounds when the number of linear functions is not too large. We may note that Nair [12], David [5], Dunn [6], [7], [8] and Siotani [22], [24] have studied the closeness of the Bonferroni inequality while deriving the percentage points of certain statistics in univariate and multivariate normal cases. In this paper, we improve the Bonferroni inequality in all the situations considered by Siotani [22], [23], [24] and Dunn [6], [7], [8], and point out various uses of these results in obtaining simultaneous confidence bounds on variances and on linear functions of means (or location parameters) with confidence greater than or equal to $1 - \alpha$ where $\alpha$ is the size of the test. We mention our main results in Section 2 for those who are interested in results and not in proofs. Since our results are extensions of Dunn [6], [8], Siotani [22], [24] and Banerjee [2], [3], their comments on the shortness of the confidence bounds apply to our cases too.