For point processes comprising i.i.d. copies of a multiplicative intensity process, it is shown that even though log-likelihood functions are unbounded, consistent maximum likelihood estimators of the unknown function in the stochastic intensity can be constructed using the method of sieves. Conditions are given for existence and strong and weak consistency, in the $L^1$-norm, of suitably defined maximum likelihood estimators. A theorem on local asymptotic normality of log-likelihood functions is established, and applied to show that sieve estimators satisfy the same central limit theorem as do associated martingale estimators. Examples are presented. Martingale limit theorems are a principal tool throughout.

Modified versions of linear rank statistics $S_N(\varphi)$ with score function $\varphi$ are studied for the two-sample testing problem of randomness. Depending on the unknown underlying alternatives, some score function $\varphi = b$ is known to be approximately optimal. Behnen, Neuhaus and Ruymgaart (1983) proposed estimating $b$ by some estimator $\hat b_N$ obtained by the kernel method based on ranks and used the quadratic rank statistic $S_N(\hat{b}_N)$ for testing the hypothesis of randomness $H_0$ versus the omnibus alternative. In the present paper the behavior of the corresponding test as well as that of a variant adapted to stochastically larger alternatives is studied by means of local asymptotic results with bandwidth of the kernel fixed. It turns out that the present asymptotics fit finite sample Monte Carlo results much better than previous results do and is able to explain to a large extent the power behavior of the proposed tests. Critical values as well as recommendations for the use of the tests in practice are included.

We consider efficient estimation of the slope in the errors in variables model with normal error when either the ratio of error variances is known and the distribution of the independent is arbitrary and unknown or the distribution of the independent variable is not Gaussian or degenerate. We calculate information bounds and exhibit estimates achieving these bounds using an initial minimum distance estimate and suitable estimates of the efficient score function.

This paper reports Monte Carlo evidence on the fixed sample size properties of adaptive maximum likelihood estimates of a linear regression. The focus is on the problem of selecting the smoothing and trimming parameters used in estimating the score function. We examine the performance of adaptive maximum likelihood estimators when these parameters are preselected or, alternatively, are determined by a data-based bootstrap method.

Let $f$ be a continuously differentiable function from [0, 1] to the complex plane. Suppose that "at time $n$" we are given the random set $\{f(k/n) + e_{n, k}: 1 \leq k \leq n\}$, where the random errors $e_{n, k}$ are i.i.d. and $(\mathrm{Re} e_{n, 1}, \mathrm{Im} e_{n, 1})$ is $N \big( (0, 0), \sigma^2 \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix} \big)$ with $\sigma^2$ known. We do not know which datum belongs to which position $\theta = k/n, k = 1, 2, \cdots, n$. In general, $f$ cannot be determined. In this paper it is shown that a random set $T_n$ can be constructed such that with probability one, $T_n$ converges in the Hausdorff sense to the trajectory $f(\lbrack 0, 1 \rbrack)$.

In this paper it is shown that in the monotone regression model the unknown regression function can be consistently estimated by the least-squares method.

The regression model $\mathbf{y} = g(\mathbf{x}) + \mathbf{\varepsilon}$ and least-squares estimation are studied in a general context. By making use of empirical process theory, it is shown that entropy conditions on the class $\mathscr{G}$ of possible regression functions imply $L^2$-consistency of the least-squares estimator $\hat{\mathbf{g}}_n$ of $g$. This result is applied in parametric and nonparametric regression.

Given a $(p + 1)$-dimensional random vector $(X, Y)$ where $f$ is the unknown density of $X$, the parameters of the multiple linear regression function $\alpha(x) = E(Y/X = x) = x\beta$ may be estimated from a sample $\{(X_1, Y_1), \cdots, (X_n, Y_n)\}$ by minimizing the functional $\hat{\psi}(\beta) = \int(\hat{\alpha}_n(x) - x\beta)^2\hat{f}_n(x) dx$, where $\hat{\alpha}_n$ and $\hat{f}_n$ may be any of a large class of nonparametric estimators of $\alpha$ and $f$. The strong consistency and asymptotic normality of the estimators so obtained are proved in this article under conditions on $(X, Y)$ that are less restrictive than those assumed by Faraldo Roca and Gonzalez Manteiga for $p = 1$. This class of estimators includes ordinary and generalized ridge regression estimators as special cases.

Consider the regression model $Y_i = g(t_i) + \varepsilon_i, 1 \leq i \leq n$, with nonrandom design variables $(t_i)$ and measurements $(Y_i)$ for the unknown regression function $g(\cdot)$. We assume that the data are heteroscedastic, i.e., $E(\varepsilon^2_i) = \sigma^2_i \not\equiv \operatorname{const.}$ and investigate how to estimate $\sigma^2_i$. If $\sigma^2_i = \sigma^2(t_i)$ with a smooth function $\sigma^2(\cdot)$, initial estimators $\tilde{\sigma}^2_i$ can be improved by kernel smoothers and the resulting class of estimators is shown to be uniformly consistent. These estimates can be used to improve the estimation of the regression function $g$ itself in parametric and nonparametric models. Further applications are suggested.

We propose and investigate the asymptotic properties of a class of estimators of the regression parameter in the general linear model. The estimators depend on a preliminary estimate of the regression parameter and the residuals based on it. For the location model, the estimators are linear combinations of the order statistics and the robustness and efficiency properties of this class of estimators carry over to the general linear model. The estimators settle the doubts raised by Bickel (1973) about the feasibility of the construction of a general class of reparametrization invariant estimators of a regression parameter which are linear combinations of order statistics in the location problem.

A class of robust estimates for the linear model is introduced. These estimates, called MM-estimates, have simultaneously the following properties: (i) they are highly efficient when the errors have a normal distribution and (ii) their breakdown-point is 0.5. The MM-estimates are defined by a three-stage procedure. In the first stage an initial regression estimate is computed which is consistent robust and with high breakdown-point but not necessarily efficient. In the second stage an M-estimate of the errors scale is computed using residuals based on the initial estimate. Finally, in the third stage an M-estimate of the regression parameters based on a proper redescending psi-function is computed. Consistency and asymptotical normality of the MM-estimates assuming random carriers are proved. A convergent iterative numerical algorithm is given. Finally, the asymptotic biases under contamination of optimal bounded influence estimates and MM-estimates are compared.

The asymptotic distribution of an $M$-estimator is studied when the underlying distribution is discrete. Asymptotic normality is shown to hold quite generally within the assumed parametric family. When the specification of the model is inexact, however, it is demonstrated that an $M$-estimator whose score function is not everywhere differentiable, e.g., a Huber estimator, has a nonnormal limiting distribution at certain distributions, resulting in unstable inference in the neighborhood of such distributions. Consequently, smooth score functions are proposed for discrete data.

A mixed-Dirichlet prior was previously used to model the hypotheses of "independence" and "dependence" in contingency tables, thus leading to a Bayesian test for independence. Each Dirichlet has a main hyperparameter $\kappa$ and the mixing is attained by assuming a hyperprior for $\kappa$. This hyperparameter can be regarded as a flattening or shrinking constant. We here review the method, generalize it and check the robustness and sensitivity with respect to variations in the hyperpriors and in their hyperhyperparameters. The hyperpriors examined included generalized log-Students with various numbers of degrees of freedom $\nu$. When $\nu$ is as large as 15 this hyperprior approximates a log-normal distribution and when $\nu = 1$ it is a log-Cauchy. Our experiments caused us to recommend the log-Cauchy hyperprior (or of course any distribution that closely approximates it). The user needs to judge values for the upper and lower quartiles, or any two quantiles, of $\kappa$, but we find that the outcome is robust with respect to fairly wide variations in these judgments.

If a pair of two-by-two contingency tables are amalgamated by addition it can happen that a measure of association for the amalgamated table lies outside the interval between the association measures of the individual tables. We call this the amalgamation paradox and we show how it can be avoided by suitable designs of the sampling experiments. We also study the conditions for the "homogeneity" of two subpopulations with respect to various measures of association. Some of the proofs have interesting geometrical interpretations.

The problem of minimizing $\sum^n_{i = 1}f(x_i)$ subject to the constraints $\sum^n_{i = 1}x_i = A, \sum^n_{i = 1}g(x_i) = B$ and $x_i \geq 0$ is solved. The solutions are different depending upon whether $(\operatorname{sgn} g")f"/g"$ is an increasing or decreasing function. The result is used to show that for certain designs, if they are optimal with respect to two criteria, then they are also optimal with respect to many other criteria.

Let $X_1, X_2, \ldots, X_n$ be independent identically distributed random variables with $EX^2_1 = \infty$ but $X_1$ belonging to the domain of attraction of a stable law. It is known that the sample mean $\bar{X}_n$ appropriately normalized converges to a stable law. It is shown here that the bootstrap version of the normalized mean has a random distribution (given the sample) whose limit is also a random distribution implying that the naive bootstrap could fail in the heavy tailed case.

The present paper considers inference for the statistical model specifying proportionality between independent Wishart distributions. It is shown that the maximum likelihood estimate of the common covariance matrix and the constants of proportionality is the unique solution to the likelihood equation and an iterative procedure determining the estimator is given. The model is an exponential transformation model, which implies the existence of an exact ancillary (a maximal invariant), and an asymptotic expansion for the distribution of the estimator conditionally on the ancillary is given. In addition, it is shown that the estimator is unbiased to order $O(n^{-3/2})$. Finally, we derive the Bartlett adjustment to the likelihood ratio statistic for the hypothesis of proportionality and subsequently for the hypothesis specifying that all of the constants of proportionality are equal to one. A small simulation study shows that the Bartlett adjustment can be very effective in improving the accuracy of chi-squared approximations.

Suppose one is able to observe sequentially a series of independent observations $X_1, X_2, \ldots$, such that $X_1, X_2, \ldots, X_{\nu - 1}$ are i.i.d. with known density $f_0$ and $X_\nu, X_{\nu + 1}, \ldots$ are i.i.d. with density $f_\theta$ where $\nu$ is unknown. Define $R(n, \theta) = \sum^n_{k = 1} \prod^n_{i = k} \frac{f_\theta(X_i)}{f_0(X_i)}.$ It is known that rules, which call for stopping and raising an alarm the first time $n$ that $R(n, \theta)$ or a mixture thereof exceeds a prespecified level $A$, are optimal methods of detecting that the density of the observations is not $f_0$ any more. Practical applications of such stopping rules require knowledge of their operating characteristics, whose exact evaluation is difficult. Here are presented asymptotic $(A \rightarrow \infty)$ expressions for the expected stopping times of such stopping rules (a) when $\nu = \infty$ and (b) when $\nu = 1$. We assume that the densities $f_\theta$ form an exponential family and that the distribution of $\log(f_\nu(X_1)/f_0(X_1))$ is (strongly) nonlattice. Monte Carlo studies indicate that the asymptotic expressions are very good approximations, even when the expected sample sizes are small.

This paper introduces a new stochastic process, a collection of $U$-statistics indexed by a family of symmetric kernels. Conditions are found for the uniform almost-sure convergence of a sequence of such processes. Rates of convergence are obtained. An application to cross-validation in density estimation is given. The proofs adapt methods from the theory of empirical processes.

In statistical decision theory, an important question is to characterize the admissible rules. In this paper, we establish complete class theorems for estimating the noncentrality parameter of noncentral chi-square and noncentral $F$ distributions under squared error loss. Under a minor assumption, any admissible estimator must be a generalized Bayes rule. Using this result, we prove that the positive part of the UMVUE is inadmissible.

Let $X_i, i = 1, 2, \ldots, k$, be independent random variables distributed according to a one-parameter exponential family with parameter $\theta_i$. Assume also that the probability density function of $X_i$ is a Polya frequency function of order two $(PF_2)$. Consider the null hypothesis $H_0: \theta_1 = \theta_2 = \cdots = \theta_k$ against the alternative $K$: not $H_0$. We show that any permutation invariant test of size $\alpha$, whose conditional (on $T = \sum^k_{i = 1}X_i)$ acceptance sections are convex, is unbiased. A stronger result is that any size $\alpha$ test function $\varphi$, which is Schur-convex for fixed $t$, is unbiased. Previously, such a result was known only for the normal and Poisson cases.

The paper develops a class of James-Stein estimators that dominates the sample mean under sequential sampling schemes of Ghosh, Sinha and Mukhopadhyay (1976). Asymptotic risk expansions of the sample mean and James-Stein estimators are provided up to the second-order term. Also, a Monte Carlo study is undertaken to compare the risks of these estimators.

Let $\succ$ denote a preference relation on a set $F$ of lottery acts. Each $f$ in $F$ maps a state space $S$ into a set $P$ of lotteries on decision outcomes. The paper discusses axioms for $\succ$ on $F$ which imply the existence of an SSB (skew-symmetric bilinear) functional $\phi$ on $P \times P$ and a finitely additive probability measure $\pi$ on $2^S$ such that, for all $f$ and $g$ in $F$, $f \succ g \Leftrightarrow \int_S \phi(f(s), g(s)) d\pi(s) > 0.$ This $S^3B$ (states SSB) model generalizes the traditional Ramsey-Savage model in which $\phi$ decomposes as $\phi(p, q) = u(p) - u(q)$, where $u$ is a linear functional on $P$. The $S^3B$ model preserves the probability structure of the Ramsey-Savage model while weakening their assumptions of transitivity and independence.

The main concerns of this paper are the definition of coherent inference (dF-coherent inference) in conformity with a theory of conditional probabilities derived from de Finetti's coherence principle, and a critical comparison of such a definition with one proposed by Heath, Lane and Sudderth.

The problem addressed is that of finding the "closest" symmetric distribution (density) to a given theoretical or empirical distribution (density) function. Measures of "closeness" considered include: weighted sup norm, weighted $L_p$ norm and Hellinger distance. Explicit formulas are given for the closest symmetric distribution function to the empirical distribution function in both sup norm and integrated square error.

An empirical Bayes test for testing $\vartheta \leq \vartheta_0$ against $\vartheta > \vartheta_0$ for the uniform distribution on $\lbrack 0, \vartheta)$ is discussed. The relation is shown with the estimation of a decreasing density on $\lbrack 0, \infty)$ and a monotone empirical Bayes test is derived based on the least-concave majorant of the empirical distribution function. The asymptotic distribution of the Bayes risk is obtained and some Monte Carlo results are given.

We generalize the projection pursuit procedure of Friedman and Stuetzle (abstract version) and prove strong convergence. This answers a question of Huber.