The classical chi-square test for independence in a two-way contingency table often rejects the independence hypothesis at an extremely small significance level, particularly when the sample size is large. This paper proposes some alternative distributions to independence, to help interpret the $\chi^2$ statistic in such situations. The uniform alternative, in which every possible contingency table of the given dimension and sample size receives equal probability, leads to the volume test, as originally suggested in a regression context by H. Hotelling. Exponential family theory is used to generate a class of intermediate alternatives between independence and uniformity, leading to a random effects model for contingency tables.

Empirical partially Bayes methods are considered as a means of improving efficiency in a class of problems in which the number of nuisance parameters increases to infinity. In the method used, the parameter of interest is estimated in an asymptotically unbiased way while James-Stein shrinkage is applied to the nuisance parameter estimates. When the shrinkage estimators are carefully chosen, this yields estimators generally more efficient than maximum likelihood. In the models considered, the conditional structure imposed allows construction of a simple estimator which is broadly consistent and efficient.

Wald's general analytic conditions that imply strong consistency of the approximate maximum likelihood estimators (AMLEs) have been extended by Le Cam, Kiefer and Wolfowitz, Huber, Bahadur, and Perlman. All these conditions use the log likelihood ratio of the type $\log\lbrack f(x, \theta)/f(x, \theta_0)\rbrack$, where $\theta_0$ is the true value of the parameter. However these methods usually fail in the nonparametric case. Thus, in this paper, for each $\theta \neq \theta_0$, we look at the log likelihood ratio of the type $\log\lbrack f(x, \theta)/f(x, \theta_r(\theta))\rbrack$, where $\theta_r(\theta)$ is a certain parameter selected in a neighborhood $V_r$ of $\theta_0$. Some general analytic conditions that imply strong consistency of the AMLE are given. The results are shown to be applicable to several nonparametric families having densities, e.g., concave distributions functions, and increasing failure rate distributions. In particular, they can be applied to several censored data cases.

When the number of design points goes to infinity, we show that the minimum logit chi-squared estimator of the parameter in a linear logistic regression model for binomial response data is asymptotically normal. We also give conditions under which it is consistent.

If a discrete-time random walk, consisting of the sum of independent increments from an unknown underlying distribution, is observed at every time instant then it is clear that the underlying distribution can be consistently estimated. If, however, we are restricted to observing only a subset of the times, and if this subset is too sparse, then a central limit effect takes over and only two moments can be consistently estimated. We show that divergence of the sum of the reciprocals of the observation time-intervals is a necessary and sufficient condition to permit consistent estimation of the third moment. Corresponding conditions permit consistent estimation of moments of higher order. An explicit consistent estimator for the distribution itself is presented when all moments can be consistently estimated and the distribution is determined by its moments.

Linear estimation is considered in nonparametric regression models of the form $Y_i = f(x_i) + \varepsilon_i, x_i \in (a, b)$, where the zero mean errors are uncorrelated with common variance $\sigma^2$ and the response function $f$ is assumed only to have a bounded square integrable $q$th derivative. The linear estimator which minimizes the maximum mean squared error summed over the observation points is derived, and the exact minimax rate of convergence is obtained. For practical problems where bounds on $\|f^{(q)}\|^2$ and $\sigma^2$ may be unknown, generalized cross-validation is shown to give an adaptive estimator which achieves the minimax optimal rate under the additional assumption of normality.

For nonparametric regression estimation on a bounded interval, optimal rates of decrease for integrated mean square error are known but not the best possible constants. A sharp result on such a constant, i.e., an analog of Fisher's bound for asymptotic variances is obtained for minimax risk over a Sobolev smoothness class. Normality of errors is assumed. The method is based on applying a recent result on minimax filtering in Hilbert space. A variant of spline smoothing is developed to deal with noncircular models.

We suggest a sieve for the estimation of the spectral density of a Gaussian stationary stochastic process. In contrast to the standard periodogram-based estimates this one aims at exploiting the full Gaussian nature of the process.

A data-driven method of choosing the bandwidth, $h$, of a kernel density estimator is heuristically motivated by considering modifications of the Kullback-Leibler or pseudo-likelihood cross-validation function. It is seen that this means of choosing $h$ is asymptotically equivalent to taking the $h$ that minimizes some compelling error criteria such as the average squared error and the integrated squared error. Thus, for a given kernel function, the bandwidth can be chosen optimally without making precise smoothness assumptions on the underlying density.

We introduce two nonparametric multivariate density estimators that are particularly suitable for application in interactive computing environments. These estimators are statistically comparable to kernel methods and computationally comparable to histogram methods. Asymptotic theory of the estimators is presented and examples with univariate and simulated trivariate Gaussian data are illustrated.

A sample $X_1, \cdots, X_n$ of i.i.d. $R^d$-valued random vectors with common density $f$ is used to construct the density estimate $f_n(x) = (1/n) \sum^n_{i = 1} H^{-d}_{ni}K((x - X_i)/H_{ni}),$ where $K$ is a given density on $R^d$, and the $H_{ni}$'s are positive functions of $n, i$ and $X_1, \cdots, X_n$ (but not of $x$). The $H_{ni}$'s can be thought of as locally adapted smoothing parameters. We give sufficient conditions for the weak convergence to 0 of $\int |f_n - f|$ for all $f$. This is illustrated for the estimate of Breiman, Meisel and Purcell (1977).

We introduce a new estimate of the exponent of a distribution whose tail varies regularly at infinity. This estimate is expressed as the convolution of a kernel with the logarithm of the quantile function, and includes as particular cases the estimates introduced by Hill and by De Haan. Under very weak conditions, we prove asymptotic normality, consistency and discuss the optimal choices of the kernel and of the bandwidth parameter.

In the framework of Huber's theory of robust estimation of a location parameter, minimax variance M-estimators are studied for error distributions with densities of the form $f(x) = (1 - \varepsilon)h(x) + \varepsilon g(x)$, where $g$ is unknown. A well-known result of Huber (1964) is that when $h$ is strongly unimodal, the least informative density $f_0 = (1 - \varepsilon)h + \varepsilon g_0$ has exponential tails. We study the minimax variance solutions when the known density $h$ is not necessarily strongly unimodal, and definitive results are obtained under mild regularity conditions on $h$. Examples are given where the support of the least informative contaminating density $g_0$ is a set of form: (i) $(-b, -a) \cup (a, b)$ for some $0 < a < b < \infty$; (ii) $(- < a < \infty;$ and (iii) a countable collection of disjoint sets. Minimax variance problems for multivariate location and scale parameters are also studied, with examples given of least informative distributions that are substochastic.

Suppose that $F_0$ is a population which is symmetric about zero, so that $F(\cdot) = F_0(\cdot - \theta)$ is symmetric about $\theta$. We consider the problem of estimating $F_0$ (shape parameter), both $\theta$ and $F_0$, and $F$ based on a random sample from $F$. First, some asymptotically minimax bounds are obtained. Then, some estimates are constructed which are asymptotically minimax-efficient (the risks of which achieve the minimax bounds uniformly). Furthermore, it is pointed out that one can estimate $F_0$, the shape of $F$, as well without knowing the location parameter $\theta$ as with knowing it. After a slight modification, Stone's (1975) estimator is proved to be asymptotically minimax-efficient in the Hellinger neighborhood.

The problem is to estimate a quantile of a symmetric distribution. The cases of known and unknown center are studied for small and large samples. The estimators for known center are the sample quantile, the symmetrized sample quantile, the sample quantile from flipped over data, the Rao-Blackwellized sample quantile, and a Bayes estimator using a Dirichlet prior. For center unknown, we study the analogues of the first four estimators listed above. For small samples and center known, the Rao-Blackwellized sample quantile performs very well for normal and double exponential distributions while for the Cauchy distribution the flipped over estimator did well. In the center known case, the latter four estimators are asymptotically equivalent, asymptotically optimal in the sense of Hajek's convolution, and asymptotically minimax in the Hajek-LeCam sense. For center unknown, those properties remain true if one uses an adaptive estimator of the center for the symmetrized sample quantile, the flipped over estimator, and the Rao-Blackwell estimator.

We may take observations sequentially from a population with unknown mean $\theta$. After this sampling stage, we are to decide whether $\theta$ is greater or less than a known constant $\nu$. The net worth upon stopping is either $\theta$ or $\nu$, respectively, minus sampling costs. The objective is to maximize the expected net worth when the probability measure of the observations is a Dirichlet process with parameter $\alpha$. The stopping problem is shown to be truncated when $\alpha$ has bounded support. The main theorem of the paper leads to bounds on the exact stage of truncation and shows that sampling continues longest on a generalized form of neutral boundary.

Two families of invariant tests for independence of random variables on compact Riemannian manifolds are proposed and studied. The tests are based on Gine's Sobolev norms which are obtained by mapping the manifolds into Hilbert spaces. For general compact manifolds, randomization tests are suggested. For the bivariate circular case, distribution-free tests based on uniform scores are considered.

In this paper we introduce a class of linear serial rank statistics for the problem of testing white noise against alternatives of ARMA serial dependence. The asymptotic normality of the proposed statistics is established, both under the null as well as alternative hypotheses, using LeCam's notion of contiguity. The efficiency properties of the proposed statistics are investigated, and an explicit formulation of the asymptotically most efficient score-generating functions is provided. Finally, we study the asymptotic relative efficiency of the proposed procedures with respect to their normal theory counterparts based on sample autocorrelations.

This paper first characterizes the invariant structure of a model for which nonnull robustness holds. Applications of this result yield the nonnull robustness and optimality robustness of some tests for covariance structure including a test for sphericity. Second, we show the optimality robustness of the LBI tests in the GMANOVA(MANOVA) problem, and the problem of testing independence. In the GMANOVA problem, a robustness property of an essentially complete class of invariant tests is also shown.

This paper examines how a Bayesian decision maker would update his/her probability $p$ for the occurrence of an event $A$ in the light of a number of expert opinions expressed as probabilities $q_1, \cdots, q_n$ of $A$. It is seen, among other things, that the linear opinion pool, $\lambda_0p + \sum^n_{i = 1} \lambda_iq_i$, corresponds to an application of Bayes' Theorem when the decision maker has specified only the mean of the marginal distribution for $(q_1, \cdots, q_n)$ and requires his/her formula for the posterior probability of $A$ to satisfy a certain consistency condition. A product formula similar to that of Bordley (1982) is also derived in the case where the experts are deemed to be conditionally independent given $A$ (and given its complement).

To the classical model of searching for one object out of $n$, we add uncertainty about the parameters $\mathbf{\pi}$ of the distribution of the $n$ objects among the $m$ boxes. Adopting a Bayesian approach, we study the optimal sequential search strategy. For the case $n = 1$, we obtain a generalization of the fundamental result of Blackwell: the strategy which searches at each stage in the "most inviting" box is optimal. This strategy is also optimal for $m = 2$ and arbitrary $n$. However, for $n > 1$ the optimal strategy may be very different from that of the classical model, even when the uncertainty about $\mathbf{\pi}$ is very small.

A one-step version $M^{(1)}_n$ and a two-step version $M^{(2)}_n$ of a general $M$-estimator $M_n$ are suggested such that $M_n - M^{(1)}_n = O_p(n^{-1})$ and $M_n - M^{(2)}_n = O_p(n^{-3/2})$ for every $n^{1/2}$-consistent initial estimator and under some regularity conditions. In the special case of maximum likelihood estimation, this among other yields that the second-order efficiency properties of $M^{(2)}_n$ coincide with those of $M_n$. An application to the Pitman estimator of location is considered.

For fixed censoring models that contain at most one intermediate censoring point, we obtain exact algebraic expressions for the asymptotic variances of (i) the quantiles of the Kaplan-Meier (KM, 1958) survival estimator and (ii) the KM estimator itself at fixed time points. The relationship between (i) and (ii) is found to be the same as the one derived by Sander (1975) and Reid (1981b) for the random censorship model. Confidence intervals for the quantiles based on (i) are briefly discussed and compared to previously known procedures. Although Greenwood's Formula is recommended over (ii) in practice because of its (desirable) conditioning on the observed censoring pattern, (ii) is of theoretical interest as an asymptotic limit for Greenwood's Formula in closed form.

Let $X$ be a positive random variable and assume that both $a = EX^{-1}$ and $\mu = EX$ are finite. Define $c^2 = 1 - (a\mu)^{-1}$. This quantity serves as a measure of variability for $X$ which is reflected in the behavior of completely monotone functions of $X$. For $g$ completely monotone with $g(0) < \infty$: $0 \leq Eg(X) - g(EX) \leq c^2g(0) \text{and}\operatorname{Var} g(X) \leq c^2g^2(0).$

Two random quantities $x$ and $y$, taking values in sets $X$ and $Y$, are to be observed sequentially. A predicter (bookie) posts odds on $(x, y)$ and on $y$ given $x$ according to functions $P$ and $q(x)$, respectively. The predicter is coherent (the bookie can avoid a sure loss) if and only if $P$ is a finitely additive probability distribution on $X \times Y$ and $q$ satisfies a general law of total probability: $P(A) = \int q(x)(Ax)P_0(dx)$ for $A \subset X \times Y, Ax = \{y: (x, y) \in A\}, P_0 = \text{marginal of} P \text{on} X.$