This paper contains a review of certain multivariate probability inequalities. The inequalities discussed include the FKG inequalities and the related association inequalities, inequalities resulting from Schur convexity and its extension to reflection groups, and inequalities for probabilities of certain convex symmetric sets.

Sufficient conditions for asymptotic lognormality of exact and approximate, unconditional and conditional $P$-values are established. It is pointed out that the mean, which is half the Bahadur slope, and the standard deviation of the asymptotic distribution of the log transformed $P$-value together, but not the mean alone, permit approximation of both the level and power of the test. This provides a method of discriminating between tests that have Bahadur efficiency one. The asymptotic distributions of the log transformed $P$-values of the common one- and two-sample tests for location are derived and compared.

The normal, Poisson, gamma, binomial, and negative binomial distributions are univariate natural exponential families with quadratic variance functions (the variance is at most a quadratic function of the mean). Only one other such family exists. Much theory is unified for these six natural exponential families by appeal to their quadratic variance property, including infinite divisibility, cumulants, orthogonal polynomials, large deviations, and limits in distribution.

The problem of estimating a $p$-variate normal mean under arbitrary quadratic loss when $p \geq 3$ is considered. Any estimator having uniformly smaller risk than the maximum likelihood estimator $\delta^0$ will have significantly smaller risk only in a fairly small region of the parameter space. A relatively simple minimax estimator is developed which allows the user to select the region in which significant improvement over $\delta^0$ is to be achieved. Since the desired region of improvement should probably be chosen to coincide with prior beliefs concerning the whereabouts of the normal mean, the estimator is also analyzed from a Bayesian viewpoint.

In this study, we consider the simultaneous estimation of the parameters of the distributions of $p$ independent Poisson random variables using the loss function $L_k(\lambda, \hat{\lambda}) = \sum (\lambda_i - \hat{\lambda}_i)^2/\lambda^k_i$ for a given positive integer $k$. New estimators are derived, which include the minimax estimators proposed by Clevenson and Zidek (1975), as special cases. The case when more than one observation is taken from some of the variables is considered.

A general class of models for analysis of censored survival data with covariates is considered. If $n$ individuals are observed over a time period divided into $I(n)$ intervals, it is assumed that $\lambda_j(t)$, the hazard rate function of the time to failure of the individual $j$, is constant and equal to $\lambda_{ij} > 0$ on the $i$th interval, and that the vector $\ell = \{\log \lambda_{ij}: j = 1, \ldots, n; i = 1, \ldots, I(n)\}$ lies in a linear subspace. The maximum likelihood estimate $\hat{\ell}$ of $\ell$ provides a simultaneous estimate of the underlying hazard rate function, and of the effects of the covariates. Maximum likelihood equations and conditions for existence of $\hat{\ell}$ are given. The asymptotic properties of linear functionals of $\hat{\ell}$ are studied in the general case where the true hazard rate function $\lambda_0(t)$ is not a step function, and $I(n)$ increases without bound as the maximum interval length decreases. In comparison with recent work on regression analysis of survival data, the asymptotic results are obtained under more relaxed conditions on the regression variables.

This paper is concerned with the development and application of diagnostic checks for vector linear time series models. A hypothesis testing procedure based upon the score, or Lagrangean multiplier, principle is advocated and the distributions of the test statistic both under the null hypothesis and under a Pitman sequence of alternatives are discussed. Consideration of alternative models with singular sensitivity matrices when the null hypothesis is true leads to an interpretation of the score test as a pure significance test and to a notion of an equivalence class of local alternatives. Portmanteau tests of model adequacy are also investigated and are seen to be equivalent to score tests.

Let $\mathbf{R}$ be an infinite dimensional stationary covariance matrix, let $\mathbf{R}(k)$ and $\mathbf{W}(k)$ denote the top $k \times k$ left hand corners of $\mathbf{R}$ and $\mathbf{R}^{-1}$ respectively and let $\mathbf{\Sigma}(k)$ and $\mathbf{\Gamma}(k)$ denote the approximations for $\mathbf{R}(k)^{-1}$ suggested by Whittle (1951) and Shaman (1976) respectively. We consider quadratic forms of the type $Q(k) = \beta(k)' \mathbf{R}(k)^{-1}\alpha (k)$, when the vectors $\beta(k)$ and $\alpha(k)$ constitute the first $k$ elements of the infinite absolutely summable sequences $\{\beta_j\}$ and $\{\alpha_j\}$. If $\chi_1(k) = \beta (k)' \mathbf{W}(k) \mathbf{\alpha}(k)$ and $\chi_2(k) = \beta (k)' \mathbf{\Sigma(k)}\mathbf{\alpha}(k)$, then, as $k \rightarrow \infty, Q(k)$ and $\chi_1(k)$ converge to the same limiting value for all such $\alpha (k)$ and $\beta(k)$, but $\chi_2(k)$ does not necessarily do so. Further, if $\tilde\mathbf{\alpha}(k) = (\alpha_k, \cdots, \alpha_1)'$ and $\tilde\mathbf{\beta}(k) = (\beta_k, \cdots, \beta_1)'$ then $\chi_1(k) \equiv \tilde\mathbf{\beta}(k)'\mathbf{\Gamma}(k)\tilde\mathbf{\alpha}(k)$. We discuss the use of $\mathbf{W}(k)$ for evaluating the asymptotic covariance structure of the autoregressive estimates of the inverse covariance function and the moving average parameters.

A central limit theorem is proved for the sample covariances of a linear process. The sufficient conditions for the theorem are described by more natural ones than usual. We apply this theorem to the parameter estimation of a fitted spectral model, which does not necessarily include the true spectral density of the linear process. We also deal with estimation problems for an autoregressive signal plus white noise. A general result is given for efficiency of Newton-Raphson iterations of the likelihood equation.

Strong consistency and asymptotic normality of least squares estimates in stochastic regression models are established under certain weak assumptions on the stochastic regressors and errors. We discuss applications of these results to interval estimation of the regression parameters and to recursive on-line identification and control schemes for linear dynamic systems.

Let $(X_1, X_2, \cdots)$ be a sequence of random variables and let the p.d.f. of $\mathbf{X}_n = (X_1, \cdots, X_n)$ be $p(\mathbf{x}_n, \theta)$, where $\theta = (\theta_1, \theta_2)$. An estimating equation rule for $\theta_1$ is a sequence of functions $g(x_1, \theta_1), g(x_1, x_2, \theta_1), \cdots$. If the random sample size $N = n$, we estimate $\theta_1$ through the estimating equation $g(\mathbf{X}_n, \theta_1) = 0$. In this paper, optimum estimation rules are obtained and, in particular, sufficient conditions for the optimality of the maximum conditional likelihood estimation rule are given. In addition, Bhapkar's concept of information in an estimating equation is used to discuss stopping criteria.

A simple class of models for possibly contaminated data is considered, for which the effect of large observations on our beliefs and on our procedures is small. Various properties are derived, and the effects of differing prior opinions are considered.

Necessary and sufficient conditions are obtained for an exchangeable sequence of random variables to be a mixture of symmetric location families.

The coverage of a random sample from a multinomial population is defined to be the sum of the probabilities of the observed classes. The problem is to estimate the coverage of a random sample given only the number of classes observed exactly once, twice, etc. This problem is related to the problem of estimating the number of classes in the population. Non-parametric confidence intervals are given when the coverage is low such that a Poisson approximation holds. These intervals are related to a coverage estimator of Good (1953).

For a fixed probability $0 < \gamma < 1$, the "most outlying" $100(1 - \gamma){\tt\%}$ subset of the data from a location model may be located with a Grubbs outlier subset test statistic. This subset is essentially located in terms of its complement, which is the connected $100\gamma{\tt\%}$ span of the data which supports the smallest sample variance. We show that this range of the data may be characterized approximately as the $100\gamma{\tt\%}$ span such that its midpoint is equal to the trimmed mean averaged over the span. Such a range forms a tolerance interval for predicting a future observation from the location model, and the asymptotic laws for its location, coverage, and center are presented.

An asymptotic notion of robust tests is studied which is based on the requirement of equicontinuous error probabilities. If the test statistics are consistent, their robustness in Hampel's sense and robustness of the associated tests turn out to be equivalent. Uniform extensions are considered. Moreover, test breakdown points are defined. The main applications are on rank statistics: they are generally robust, under a slight condition even uniformly so; their points of final breakdown coincide with the breakdown points of the corresponding $R$ - estimators.

Let $X_1, X_2, \cdots, X_n$ be i.i.d random variables with d.f. $F$. Suppose the $\{\hat{T}_n = \hat{T}_n(X_1, X_2, \cdots, X_n); n \geq 1\}$ are real-valued statistics and the $\{T_n(F); n \geq 1\}$ are centering functionals such that the asymptotic distribution of $n^{1/2}\{\hat{T}_n - T_n(F)\}$ is normal with mean zero. Let $H_n(x, F)$ be the exact d.f. of $n^{1/2}\{\hat{T}_n - T_n(F)\}$. The problem is to estimate $H_n(x, F)$ or functionals of $H_n(x, F)$. Under regularity assumptions, it is shown that the bootstrap estimate $H_n(x, \hat{F}_n)$, where $\hat{F}_n$ is the sample d.f., is asymptotically minimax; the loss function is any bounded monotone increasing function of a certain norm on the scaled difference $n^{1/2}\{H_n(x, \hat{F}_n) - H_n(x, F)\}$. The estimated first-order Edgeworth expansion of $H_n(x, F)$ is also asymptotically minimax and is equivalent to $H_n(x, \hat{F}_n)$ up to terms of order $n^{- 1/2}$. On the other hand, the straightforward normal approximation with estimated variance is usually not asymptotically minimax, because of bias. The results for estimating functionals of $H_n(x, F)$ are similar, with one notable difference: the analysis for functionals with skew-symmetric influence curve, such as the mean of $H_n(x, F)$, involves second-order Edgeworth expansions and rate of convergence $n^{-1}$.

The concept of Choquet's 2-alternating capacity is explored from the viewpoint of Le Cam's experiment theory. It is shown that there always exists a least informative binary experiment for two sets of probability measures generated by 2-alternating capacities. This result easily implies the Neyman-Pearson lemma for capacities. Moreover, its proof gives a new method of construction of minimax tests for problems in which hypotheses are generated by 2-alternating capacities. It is also proved that the existence of least informative binary experiments is sufficient for a set of probability measures to be generated by a 2-alternating capacity. This gives a new characterization of 2-alternating capacities, closely related to that of Huber and Strassen.

Let $Y_1, Y_2, \cdots, Y_n (n = 1, 2, \cdots)$ be independent random variables (r.v.'s) uniformly distributed over the $d$-dimensional unit cube, and let $\alpha_n(\cdot)$ be the empirical process based on this sequence of random samples. Let $V_{n, d}(\cdot)$ be the distribution function of the Cramer-von Mises functional of $\alpha_n(\cdot)$, and define $V_d(\cdot) = \lim_{n \rightarrow \infty} V_{n, d}(\cdot), \Delta_{n, d} = \sup_{0 < x < \infty}|V_{n, d}(x) - V_d(x)|$. We deduce that $\Delta_{n,d} = O(n^{-1}), d \geq 1$, and calculate also the "usual" levels of significance of the distribution function $V_d(\cdot)$ for $d = 2$ to 50, using expansion methods. Previously these were known only for $d = 1, 2, 3$.

Necessary and sufficient conditions for a linear estimator to be admissible among linear estimators are described. The model assumed is general, allowing for relations between elements of the mean vector and covariance matrix, and allowing the covariance matrix to vary in an arbitrary subset of nonnegative definite symmetric matrices.

In a linear (or affine) functional model the principal parameter is a subspace (respectively an affine subspace) in a finite dimensional inner product space, which contains the means of $n$ multivariate normal populations, all having the same covariance matrix. A relatively simple, essentially algebraic derivation of the maximum likelihood estimates is given, when these estimates are based on single observed vectors from each of the $n$ populations and an independent estimate of the common covariance matrix. A new derivation of least squares estimates is also given.

The problem of combining several independent Chi squared or $F$ tests is considered. The data consist of $n$ independent Chi squared or $F$ variables on which tests of the null hypothesis that all noncentrality parameters are zero are based. In each case, necessary conditions and sufficient conditions for a test to be admissible are given in terms of the monotonicity and convexity of the acceptance region. The admissibility or inadmissibility of several tests based upon the observed significance levels of the individual test statistics is determined. In the Chi squared case, Fisher's and Tippett's procedures are admissible, the inverse normal and inverse logistic procedures are inadmissible, and the test based upon the sum of the significance levels is inadmissible when the level is less than a half. The results are similar, but not identical, in the $F$ case. Several generalized Bayes tests are derived for each problem.

The maximum likelihood estimator of a nondecreasing regression function with normally distributed errors has been considered in the literature. Its asymptotic distribution at a point is related to a solution of the heat equation, and its rate of convergence to the underlying regression function is of order $n^{-1/3}$. This estimator can be modified by grouping adjacent observations and then "isotonizing" the corresponding means. It is shown that the resulting estimator has an asymptotic normal distribution for certain group sizes and its rate of convergence is of order $n^{-2/5}$. The results of a simulation study for small sample sizes are presented and grouping procedures are discussed.

Isotonic estimation involves the estimation of a function which is known to be increasing with respect to a specified partial order. For the case of a linear order, a general theorem is given which simplifies and extends the techniques of Prakasa Rao and Brunk. Sufficient conditions for a specified limit distribution to obtain are expressed in terms of a local condition and a global condition. It is shown that the rate of convergence depends on the order of the first non-zero derivative and that this result can obtain even if the function is not monotone over its entire domain. The theorem is applied to give the asymptotic distributions of several estimators.

A sharp inequality comparing the probability content of the $\ell_1$ ball and that of $\ell_\infty$ ball of the same volume is proved. The result is generalized to bound the probability content of the $\ell_p$ ball for arbitrary $p \geq 1$. Examples of the type of bound include $P\{(|X_1|^p + |X_2|^p)^{1/p} \leq c\} \geq F^2(c/2^{1/2p}),\quad p \geq 1,$ where $X_1, X_2$ are independent each with distribution function $F$. Applications to multiple comparisons in Behrens-Fisher setting are discussed. Multivariate generalizations and generalizations to non-independent and non-exchangeable distributions are also discussed. In the process a majorization result giving the stochastic ordering between $\Sigma a_i X_i$ and $\Sigma b_i X_i$, when $(a^2_1, a^2_2, \cdots, a^2_n)$ majorizes $(b^2_1, b^2_2, \cdots, b^2_n)$, is also proved.

In testing hypotheses involving order restrictions on a collection of parameters, distributions arise which are mixtures of standard distributions. Since tractable expressions for the mixing proportions generally do not exist even for parameter collections of moderate size, the implementation of these tests may be difficult. Stochastic upper and lower bounds are obtained for such test statistics in a variety of these kinds of problems. These bounds are also shown to be tight. The tightness results point out some situations in which the bounds could be used to obtain approximate methods. These results can also be applied to obtain the least favorable configuration when testing the equality of two multinomial populations versus a stochastic ordering alternative.

A general class of recursive residuals is defined by means of lower-triangular, orthonormal transformations. For these residuals, some weak invariance principles are established under appropriate regularity conditions. The theory is then incorporated in the study of robustness of some tests for change of parameters occurring at unknown time points.

This paper gives a proof of a conjecture of P. W. Holland concerning the non-existence of certain covariance stabilizing transformations.

We consider conditions in a nonlinear regression model for the consistency of the estimator obtained by minimizing the $L_1$-norm, i.e. the sum of absolute deviations.