Let $(X, Y)$ be an $\mathbb{R}^d \times \mathbb{R}$-valued random vector and let $(X_1, Y_1), \cdots, (X_n, Y_n)$ be a random sample drawn from its distribution. We study the consistency properties of the kernel estimate $m_n(x)$ of the regression function $m(x) = E\{Y\mid X = x\}$ that is defined by $m_n(x) = \Sigma^n_{i=1} Y_ik((X_i - x)/h_n)/\Sigma^n_{i=1}k((X_i - x)/h_n)$ where $k$ is a bounded nonnegative function on $\mathbb{R}^d$ with compact support and $\{h_n\}$ is a sequence of positive numbers satisfying $h_n \rightarrow_n0, nh^d_n \rightarrow_n\infty$. It is shown that $E\{\int|m_n(x) - m(x)|^p\mu(dx)\} \rightarrow_n 0$ whenever $E\{|Y|^p\} < \infty(p \geqslant 1)$. No other restrictions are placed on the distribution of $(X, Y)$. The result is applied to verify the Bayes risk consistency of the corresponding discrimination rules.

Stone proved that nearest neighbor estimates of a nonparametric regression function are "universally" consistent. We show that the same holds for window estimates, and we obtain a rate of convergence under some restrictions.

Random samples of size $N_\alpha$ are taken from the $p$-variate normal populations $N_p(\mu^{(\alpha)}, \Sigma^{(\alpha)}), 1 \leqslant \alpha \leqslant k$, with $\mu^{(\alpha)}$ and $\Sigma^{(\alpha)}$ unknown. Bartlett's modification of the likelihood ratio test (LRT) for the hypothesis $H_1:\Sigma^{(1)} = \cdots = \Sigma^{(k)}$ rejects $H_1$ for large values of $|S|^n/\Pi|S^{(\alpha)}|^{n_\alpha}$, where $S = \Sigma S^{(\alpha)}, n_\alpha = N_\alpha - 1, n = \Sigma n_i$, and $S^{(\alpha)}$ is the sample covariance matrix from the $\alpha$th population. The (unmodified) LRT for the hypothesis $H_1: \mu^{(1)} = \cdots = \mu^{(k)}, \Sigma^{(1)} = \cdots = \Sigma^{(k)}$ rejects $H_2$ for large values of $|S + T|^N/\Pi|S^{(\alpha)}|^{N_\alpha}$, where $N = \Sigma N_\alpha, T = \Sigma N_\alpha(\bar{X}^{(\alpha)} - \bar{X}^{(+)})(\bar{X}^{(\alpha)} - \bar{X}^{(+)})', \bar{X}^{(\alpha)}$ is the $\alpha$th sample mean, and $\bar{X}^{(+)}$ is the grand mean. It is proved that each of these tests is unbiased against all alternatives.

Inference about the correlation coefficient $\rho$ in a bivariate normal distribution is considered when observations from the distribution are available only in the form of a broken random sample. In other words, a random sample of $n$ pairs is drawn from the distribution but the observed data are only the first components of the $n$ pairs and, separately, some unknown permutation of the second components of the $n$ pairs. Under these conditions, the estimation of $\rho$ is, as Samuel Johnson put it, "like a dog's walking on his hinder legs. It is not done well; but you are surprised to find it done at all." We study the maximum likelihood estimation of $\rho$ and present some effective procedures for estimating the sign of $\rho$.

For $p \geqslant 4$ and one observation $X$ on a $p$-dimensional spherically symmetric distribution, minimax estimators of $\theta$ whose risks are smaller than the risk of $X$ (the best invariant estimator) are found when the loss is a nondecreasing concave function of quadratic loss. For $n$ observations $X_1, X_2, \cdots, X_n$, we have classes of minimax estimators which are better than the usual procedures, such as the best invariant estimator, $\bar{X}$, or a maximum likelihood estimator.

Generalised residuals, as defined by Cox and Snell, may be thought of as residuals from generalised regression. Under regularity conditions on the regression and on the estimator of the unknown parameters, the asymptotic behaviour of the empirical distribution of these residuals is determined. The addition of a random adjustment to the maximum likelihood estimator leads to the familiar Brownian bridge as the limit.

It is shown that, under some regularity conditions on the kernel, a one-sample $U$-statistic with kernel of degree two admits an asymptotic expansion with remainder term $o(N^{-1})$.

Almost sure asymptotic linearity of signed rank order statistic in shift parameter is studied under suitable regularity conditions and the results are extended to stationary $\phi$-mixing processes as well.

Let $(\phi_\nu)^\infty_{\nu=0}$ be a complete orthonormal system of functions on a interval $\lbrack a, b\rbrack$ and let $w$ be a function defined on $\mathbb{R}$ with support $\lbrack a, b\rbrack$ and strictly positive on $(a, b)$. Let $(x_j)^\infty_{j=1}$ be i.i.d.rv's absolutely continuous with density function $f$ with respect to Lebesgue measure $\mu$ on $\mathbb{R}$. Let $f_n$ be the estimate of $f$ defined for $x\in (a, b)$ by $f_n(x) = \Sigma^m_{\nu=0}a_\nu(m)\hat{d}_\nu\phi_\nu(x)/w(x)$, where $a_\nu(m), (m = 0, 1, \cdots, \nu = 0, 1, \cdots, m)$ is a sequence in $\mathbb{R}$ and $\hat{d}_\nu = n^{-1}\Sigma^n_{j=1}\phi_\nu(X_j)w(X_j)$. In this work the asymptotic distributions of the functionals $T_n = n(m + 1)^{-1} \int^b_a(f_n - Ef_n)^2w^2 dx$ and $T^\ast_n = n(m + 1)^{-1} \int^b_a(f_n - f)^2w^2 dx$ are found. These results are used to construct tests of goodness-of-fit analogous to those proposed by Bickel and Rosenblatt. The basic idea in obtaining the results consists in finding the asymptotic distribution of $T_n(T^\ast_n)$ with $f_n$ replaced by a conveniently chosen Gaussian process and showing that the two functionals converge to the same law. For this the normalized and centered sample distribution function is approximated by an appropriate Brownian motion process by using a Skorohod-like imbedding due to Brillinger and Breiman.

Given a sample set $X_1, \cdots, X_N$ of independent identically distributed real-valued random variables, each with the unknown probability density function $f(\cdot)$, the problem considered is to estimate $f$ from the sample set. The function $f$ is assumed to be in $L_2(a, b); f$ is not assumed to be in any parametric family. This paper constructs an adaptive "two-pass" solution to the problem: in a preprocessing step (the first pass), a preliminary rough estimate of $f$ is obtained by means of a standard orthogonal-series estimator. In the second pass, the preliminary estimate is used to transform the orthogonal series. The new, transformed orthogonal series is then used to obtain the final estimate. The paper establishes consistency of the estimator and derives asymptotic (large sample set) estimates of the bias and variance. It is shown that the adaptive estimator offers reduced bias (better resolution) in comparison to the conventional orthogonal series estimator. Computer simulations are presented which demonstrate the small sample set behavior. A case study of a bimodal density confirms the theoretical conclusions.

We consider problems of sequential testing when the loss function is the sum of a component due to an error in the terminal decision and a cost of observation component. In all cases we establish a characterization of a complete class or an essentially complete class. In order to obtain such results for testing a null hypothesis against an alternative hypothesis we establish complete class results for testing the closure of the null hypothesis against the closure of the alternative hypothesis. A complete class for testing closure of null against closure of alternative is an essentially complete class for testing null against alternative. Furthermore, a complete class for testing closure of null against closure of alternative is a complete class for testing null against alternative when the risks have certain continuity properties. Such continuity properties do hold in many cases. Three models are treated. The first is when the closure of the null space is compact and the cost of the first observation is positive. Under very unrestrictive conditions it is shown that the Bayes tests form a complete class. This result differs considerably from most fixed sample analogues that have been studied. The second model is when the closure of the null space is compact, the distributions are exponential family, and the cost of the first observation is zero. The third model is for the one dimensional exponential family case when the hypotheses are one sided.

Some estimability facts for partitioned linear models with constraints are presented. For a model $E(Y) = X_1\pi_1 + X_2\pi_2$ with constraints on $\pi_1$ and $\pi_2$ a reduced model is derived that contains all information regarding the estimability (and also regarding the blues) of parametric functions $b'\pi_2$. For a model $E(Y) = X_0\pi_0 + X_1\pi_1 + X_2\pi_2$ with constraints on $\pi_0, \pi_1$ and $\pi_2$, several necessary and sufficient conditions are given for when estimability of $b'\pi_2$ in the original model is equivalent to estimability in the simpler model $E(Y) = X_0\pi_0 + X_2\pi_2$.

The limiting behavior of estimators for several errors-in-variables models is investigated. It is assumed that an estimator of the covariance matrix of the measurement error is available. Models are delineated on the basis of the prior knowledge of the error structure. In all cases the limiting distribution of the estimators, standardized by $n^{\frac{1}{2}}$, is normal. Modifications of the estimators that guarantee finite moments and improve the small sample behavior of the estimators are presented.

This paper deals with the asymptotic properties of so-called autoregressive integrated moving average processes. Moreover, it is shown that least squares estimates of the parameters of a Gaussian autoregressive integrated process are consistent and also best asymptotically normal.

Some asymmetrical weighing and $2^n$ fractional factorial designs are proved to be optimal over all possible designs with respect to a very general class of criteria. This strengthens and unifies many previously published results in this area. An easy method to prove E-optimality is also presented.

Orthogonal arrays with variable numbers of symbols are shown to be universally optimal as fractional factorial designs. The orthogonality of completely regular Youden hyperrectangles ($F$-hyperrectangles) is defined as a generalization of the orthogonality of Latin squares, Latin hypercubes, and $F$-squares. A set of mutually orthogonal $F$-hyperrectangles is seen to be a special kind of orthogonal array with variable numbers of symbols. Theorems on the existence of complete sets of mutually orthogonal $F$-hyperrectangles are established which unify and generalize earlier results on Latin squares, Latin hypercubes, and $F$-squares.