Consider the positive roots of the determinental equation $\det|YJY^\ast - x^2YY^\ast| = 0$ for a $p(n)$ by $n$ sample matrix of independent unit Gaussians $Y$ with transpose $Y^\ast$ and a projection matrix $J$ of rank $m(n).$ We prove that the empirical measure of these roots converges in probability to a nonrandom limit $F$ as $p(n), m(n),$ and $n$ go to infinity with $p(n)/n \rightarrow \beta$ and $m(n)/n \rightarrow \mu$ in $(0, 1).$ Along with possible atoms at zero and one, $F$ has a density proportional to $((x - A)(x + A)(B - x)(B + x))^\frac{1}{2}/\lbrack x(1 - x)(1 + x) \rbrack$ between $A = |(\mu - \mu \beta)^\frac{1}{2} - (\beta - \mu \beta)^\frac{1}{2}|$ and $B = |(\mu - \mu\beta)^\frac{1}{2} + (\beta - \mu \beta)^\frac{1}{2}|.$ On the basis of this result, tables of quantiles are given for probability plotting of multiple discriminant ratios, canonical correlations, and eigenvalues arising in MANOVA under the usual null hypotheses when the dimension and degree of freedom parameters are large.

Asymptotic representations of the joint densities of the canonical correlation coefficients, calculated from a sample from a multivariate normal population, and of the latent roots of $B(B + W)^{-1},$ where $B$ is $W_p(n_1, \Sigma, \Omega)$ and $W$ is $W_p(n_2, \Sigma),$ are obtained by deriving asymptotic representations of the hypergeometric functions in the joint densities. The results hold in the first case for large sample size and arbitrary values of the population canonical correlations and in the second case for large $n_2$ and $\Omega = n_2\Theta,$ where the latent roots of the noncentrality matrix $\Omega$ are arbitrary.

Let $(X_{1,m}, X_{2,m}, \cdots X_{k,m}), 1 \leqslant m \leqslant n,$ be a sample of size $n$ from the $k$ dimensional normal distribution with mean vector $\mu$ and covariance matrix $\Sigma.$ Let $V = (\nu_{ij}), 1 \leqslant i, j \leqslant k,$ denote the symmetric scatter matrix where $v_{ij} = \sum_m(X_{i,m} - \mu_i)(X_{j,m} - \mu_j).$ The problem posed is to characterize the eigenfunctions of the expectation operators of the Wishart distribution, i.e., those scalar valued functions, $f(V),$ such that $E(f(V)) = \lambda_{n,k}f(\Sigma).$ If $f$ is an eigenfunction then (a) for nonsingular $T,f(T'VT)$ is an eigenfunction and (b) for integral $p, |V|^{p/2}f(V)$ is an eigenfunction. For $k \leqslant 2,$ a complete solution of the problem is given. For $k = 1$ the functions $f(v) = cv^\alpha$ are the only eigenfunctions. For $k = 2,$ a function $f$ is an eigenfunction if and only if (i) $f$ is homogeneous and (ii) $4 \frac{\partial^2 f}{\partial v_{22}\partial v_{11}} - \frac{\partial^2 f}{\partial^2v_{21}} = C|V|^{-1}f.$ A representation of eigenfunctions is given in terms of sums of associated Legendre functions. Relationships between eigenfunctions and harmonic functions are indicated. Any homogeneous polynomial is proved to be a linear combination of polynomial eigenfunctions.

A goodness-of-fit test is proposed for the simple hypothesis specifying a continuous $p$-variate distribution. For a suitably defined empirical probability measure, the proposed test is based on the supremum of the absolute differences between hypothesized and empirical probabilities, the supremum being taken over all possible events. This test statistic is shown to be distribution free in the general $p$-variate case, its exact null distribution is indicated, and its asymptotic null distribution is obtained.

This paper introduces and studies the large sample properties of an estimator for the mean survival time from censored samples. Let $X_1, \cdots, X_n$ be independent identically distributed random variables with $F(x) = P\lbrack X_1 > x \rbrack.$ Let $Y_1, \cdots, Y_n$ be independent identically distributed (and independent of $X_1, \cdots, X_n$) censoring times with $G(y) = P\lbrack Y_1 > y\rbrack.$ Based on observing only $Z_i = \min(X_i, Y_i)$ and which observations are censored (i.e., $X_i > Y_i$), we give a class of estimators of the mean survival time $\mu = \int^\infty_0F(x) dx.$ The estimators are of the form $\hat{\mu} = \int^{M_n}_0 \hat{F}(x)dx,$ where $M_n \uparrow \infty$ as $n \uparrow \infty$ and $\hat{F}$ is an estimator of $F$ depending on the $Z_i$'s and the censoring pattern. Conditions of $F, G$ and $\{M_n\}$ for the asymptotic normality of $\hat{\mu}$ are stated and proved in Section 2 based on approximations detailed in Section 3. Section 4 gives conditions for strong consistency of $\hat{\mu}$ with rates, while Section 5 examines the meaning of the conditions for the case of the negative exponential distributions for $F$ and $G.$

In a previous work, the author showed how linear combinations of the orthogonal components of the Cramer-von Mises statistic could be used to test fit to a fully specified distribution function. In this paper, the results are extended to the case where $r$ parameters are estimated from the data. It is shown that if the coefficient vector of the linear combination is orthogonal to a specified $r$ dimensional subspace, then the asymptotic distribution of that combination is the same whether the parameters are estimated or known exactly.

In this paper, we consider the estimation of isotonic, convex or related functions by means of splines. It is shown that certain classes of isotone or convex functions can be represented as a positive cone embedded in a Hilbert space. Using this representation, we give an existence and characterization theorem for isotonic or convex splines. Two special cases are examined showing the existence of a globally monotone cubic smoothing spline and a globally convex quintic smoothing spline. Finally, we examine a regression problem and show that the isotonic-type of spline provides a strongly consistent solution. We also point out several other statistical applications.

The technique of ridge regression, first proposed by Hoerl and Kennard, has become a popular tool for data analysts faced with a high degree of multicollinearity in their data. By using a ridge estimator, one hopes to both stabilize one's estimates (lower the condition number of the design matrix) and improve upon the squared error loss of the least squares estimator. Recently, much attention has been focused on the latter objective. Building on the work of Stein and others, Strawderman and Thisted have developed classes of ridge regression estimators which dominate the usual estimator in risk, and hence are minimax. The unwieldy form of the risk function, however, has led these authors to minimax conditions which are stronger than needed. In this paper, using an entirely new method of proof, we derive conditions that are necessary and sufficient for minimaxity of a large class of ridge regression estimators. The conditions derived here are very similar to those derived for minimaxity of some Stein-type estimators. We also show, however, that if one forces a ridge regression estimator to satisfy the minimax conditions, it is quite likely that the other goal of Hoerl and Kennard (stability of the estimates) cannot be realized.

Conditions for the pointwise consistency of weighted least squares estimators from multivariate regression models with martingale difference errors are given in terms of the relative rates at which certain quadratic forms diverge to infinity.

Kariya and Eaton and Kariya studied a robustness property of the usual tests for serial correlation against departure from normality. When the results were applied to a regression model $y = X \beta + u(X: nxk),$ it was assumed that the column space of $X$ is spanned by some $k$ latent vectors of the covariance matrix of error term $u.$ In this paper we delete this assumption and in a much broader class of distributions derive a locally best invariant test for a one-sided problem and a locally best unbiased and invariant test for a two-sided problem. The null distributions of these tests are the same as those under normality.

Under general conditions strong consistency of certain estimates of the maximum lags of an autoregressive moving average process is established. A theorem on weak consistency is also proved and in certain cases where consistency does not hold the probability of over-estimation of a maximum lag is evaluated.

Whittle has proved that the least-squares estimator of a scalar parameter of the spectrum of a purely nondeterministic time series possesses a certain optimality property independently of the distribution of the residuals. In this paper we furnish a proof in full detail of the corresponding result for a vector parameter and also provide some examples which illustrate the application of the result.

It is shown that the estimation procedure of Walker leads to estimates of the parameters of a Gaussian moving average process which are asymptotically equivalent to the maximum likelihood estimates proposed by Whittle and represented by Godolphin.

Let $\{X(t), - \infty < t < \infty\}$ be a stationary time series with spectral density function $\phi(\lambda).$ Let $\{t_n\}$ be a stationary Poisson point process on the real line. The existence of consistent estimates of $\phi(\lambda)$ based on the discrete-time observations $\{X(t_n)\}^N_{n = 1},$ when the actual sampling times are not known, has been an open question (Beutler). Using an orthogonal series method, a class of spectral estimates is considered and its uniform and integrated uniform consistency in quadratic mean is established. Rates of convergence are established and are compared with the optimal rates of the available (Brillinger, Masry) kernel-type estimates based on the observations $\{X(t_n), t_n\}^N_{n = 1}$.

The problem considered is that of testing sequentially between two separated composite hypotheses concerning the mean of a normal distribution with known variance. The parameter space is the real line, on which is assumed an a priori distribution, $W,$ with full support. A family $\{\delta(c)\}$ of sequential tests is defined and shown to be asymptotically Bayes, as the cost, $c$, per observation tends to zero, relative to a large class of fully supported a priori distributions. The ratio of the integrated risk of the Bayes procedure to that of $\delta(c)$ is shown to be $1 - 0(\log\log c^{-1}/\log c^{-1})$, as $c$ tends to zero, for every $W.$

The final decision of a fixed-sample likelihood ratio test is often determined before the entire sample is taken. Such a test can be curtailed as soon as the final decision becomes obvious. The construction and properties of these curtailed tests are described. A particular class of such tests contains sequential tests which are uniformly most powerful. The asymptotic efficiency of this class is investigated.

Consider a polynomial regression of degree $n$ on an interval. Explicit optimal designs are given for minimizing the determinant of the covariance matrix of the least squares estimators of the highest $s$ coefficients. The designs are calculated using continued fractions.

Let $X_1, \cdots, X_n$ be independent and normally distributed variables, such that $0 < \operatorname{Var} X_i = \sigma^2, i = 1, \cdots, n$ and $E(X_1, \cdots, X_n)' = A'\beta$ where $A$ is a $k \times n$ matrix with known coefficients and $\beta = (\beta_1, \cdots, \beta_k)'$ is an unknown vector. $\sigma$ may be known or unknown. Denote the experiment obtained by observing $X_1, \cdots, X_n$ by $\mathscr{E}_A.$ Let $A$ and $B$ be matrices of dimension $n_A \times k$ and $n_B \times k.$ The deficiency $\delta(\mathscr{E}_A, \mathscr{E}_B)$ is computed when $\sigma$ is known and for some cases, including the case $BB' - AA'$ positive semidefinite and $AA'$ nonsingular, also when $\sigma$ is unknown. The technique used consists of reducing to testing a composite hypotheses and finding a least favorable distribution.

The estimation of the generalized residual variance is considered when an observable Wishart random matrix is available. It is shown that when the loss function is normalized squared error, the natural fully equivariant estimator is dominated by an alternative. The latter uses either one of two estimators depending on the result of a preliminary test of significance. This alternate estimator has an everywhere smaller mean normalized squared error than the natural estimator.

To improve the rate of decrease of integrated mean square error for nonparametric kernel density estimators beyond $0(n^{-\frac{4}{5}}),$ we must relax the constraint that the density estimate be a bonafide density function, that is, be nonnegative and integrate to one. All current methods for kernel (and orthogonal series) estimators relax the nonnegativity constraint. In this paper we show how to achieve similar improvement by relaxing the integral constraint only. This is important in applications involving hazard function and likelihood ratios where negative density estimates are awkward to handle.

A result for constructing an "optimal" selection rule for selecting a subset of $k(\geqslant 2)$ populations is given. Attention is restricted to the class of rules for which the infimum of the probability of a correct selection, over a subset of the parameter space, is guaranteed to be a specified number. In this class a rule is derived which minimizes the supremum of the expected size of the selected subset.

Correcting a result of Krauth we show that there does not exist a UMP sign test for certain nonparametric hypotheses and treat the asymptotic properties of a one-sided sign test.

For a given family $\mathscr{F}$ of continuous cdf's $n$ i.i.d. random variables with cdf $F \in \mathscr{F}$ are observed sequentially with the object of choosing the largest. An upper bound for the greatest asymptotic probability of choosing the largest is $\alpha \doteq .58,$ the optimal asymptotic value when $F$ is known, and a lower bound is $e^{-1},$ the optimal value when the choice is based on ranks. It is known that if $\mathscr{F}$ is the family of all normal distributions a minimax stopping rule gives asymptotic probability $\alpha$ of choosing the largest while if $\mathscr{F}$ is the family of all uniform distributions a minimax rule gives asymptotic value $e^{-1}.$ This note considers a case intermediate to these extremes.