Without parametric assumptions, high-dimensional regression analysis is already complex. This is made even harder when data are subject to censoring. In this article, we seek ways of reducing the dimensionality of the regressor before applying nonparametric smoothing techniques. If the censoring time is independent of the lifetime, then the method of sliced inverse regression can be applied directly. Otherwise, modification is needed to adjust for the censoring bias. A key identity leading to the bias correction is derived and the root-$n$ consistency of the modified estimate is established. Patterns of censoring can also be studied under a similar dimension reduction framework. Some simulation results and an applica-tion to a real data set are reported.

This paper presents a solution to an open problem in astrophysics, namely that of estimating nonparametrically the strength of the pulsed signal in a series of high-energy photon arrival times. The newly proposed estimator, based on a modified maximal symmetric $2s$-spacing, is shown to be strongly consistent and asymptotically normally distributed, and a Monte Carlo study shows that its small and moderate sample behavior is very satisfactory. Additionally, new results regarding the weak and strong limiting behavior of modified maximal $2s$-spacings are derived.

The quasi-likelihood function proposed by Wedderburn broadened the scope of generalized linear models by specifying the variance function in-stead of the entire distribution. However, complete specification of variance functions in the quasi-likelihood approach may not be realistic. We define a nonparametric quasi-likelihood by replacing the specified variance function in the conventional quasi-likelihood with a nonparametric variance function estimate. This nonparametric variance function estimate is based on squared residuals from an initial model fit. The rate of convergence of the nonparametric variance function estimator is derived. It is shown that the asymptotic limiting distribution of the vector of regression parameter estimates is the same as for the quasi-likelihood estimates obtained under correct specification of the variance function, thus establishing the asymptotic efficiency of the nonparametric quasi-likelihood estimates. We propose bandwidth selection strategies based on deviance and Pearson’s chi-square statistic. It is demonstrated in simulations that for finite samples the proposed nonparametric quasi-likelihood method can improve upon extended quasi-likelihood or pseudo-likelihood methods where the variance function is assumed to fall into a parametric class with unknown parameters. We illustrate the proposed methods with applications to dental data and cherry tree data.

We characterize those coherent design criteria which depend only on the dispersion matrix (assumed proper and nonsingular) of the “state of nature,” which may be a parameter-vector or a set of future observables, and describe the associated decision problems. Connections are established with the classical approach to optimal design theory for the normal linear model, based on concave functions of the information matrix. Implications of the theory for more general models are also considered.

Much contrary to popular belief and even a published result, it is seen that orthogonal array plus one run plans are not necessarily optimal, within the relevant class, for general $s_1 \times \dots \times s_m$ factorials. A broad sufficient condition on $s_1,\dots, s_m$ ensuring the optimality of such plans has been worked out. This condition covers, in particular, all symmetric factorials and thus strengthens some previous results.

The expectation-maximization (EM) algorithm is a powerful computational technique for locating maxima of functions. It is widely used in statistics for maximum likelihood or maximum a posteriori estimation in incomplete data models. In certain situations, however, this method is not applicable because the expectation step cannot be performed in closed form. To deal with these problems, a novel method is introduced, the stochastic approximation EM (SAEM), which replaces the expectation step of the EM algorithm by one iteration of a stochastic approximation procedure. The convergence of the SAEM algorithm is established under conditions that are applicable to many practical situations. Moreover, it is proved that, under mild additional conditions, the attractive stationary points of the SAEM algorithm correspond to the local maxima of the function presented to support our findings.

This paper presents convergence conditions for a Markov chain constructed using Gibbs sampling, when the equilibrium distribution is the conditional sampling distribution of sufficient statistics from a generalized linear model. For cases when this unidimensional sampling is done approximately rather than exactly, the difference between the target equilibrium distribution and the resulting equilibrium distribution is expressed in terms of the difference between the true and approximating univariate conditional distributions. These methods are applied to an algorithm facilitating approximate conditional inference in canonical exponential families.

A Dirichlet mixture of normal densities is a useful choice for a prior distribution on densities in the problem of Bayesian density estimation. In recent years, efficient Markov chain Monte Carlo method for the computation of the posterior distribution has been developed. The method has been applied to data arising from different fields of interest. The important issue of consistency was however left open. In this paper, we settle this issue in affirmative.

Z In a class of semiparametric mixture models, the score function (and consequently the effective information) for a finite-dimensional parameter can be made arbitrarily small depending upon the direction taken in the parameter space. This result holds for a broad range of semiparametric mixtures over exponential families and includes examples such as the gamma semiparametric mixture, the normal mean mixture, the Weibull semiparametric mixture and the negative binomial mixture. The near-zero information rules out the usual parametric $\sqrt{n}$ rate for the finite-dimensional parameter, but even more surprising is that the rate continues to be unattainable even when the mixing distribution is constrained to be countably discrete. Two key conditions which lead to a loss of information are the smoothness of the underlying density and whether a sufficient statistic is invertible.

We consider time series $Y_t = G(X_t)$ where $X_t$ is Gaussian with long memory and $G$ is a polynomial. The series $Y_t$ may or may not have long memory. The spectral density $g_\theta(x)$ of $Y_t$ is parameterized by a vector $\theta$ and we want to estimate its true value $\theta_0$ . We use a least-squares Whittle-type estimator $\hat{\theta}_N$ for $\theta_0$, based on observations $Y_1,\dots,Y_N$. If $Y_t$ is Gaussian, then $\sqrt{N}(\hat{\theta}_N-\theta_0)$ converges to a Gaussian distribution. We show that for non-Gaussian time series $Y_t$ , this $\sqrt{N}$ consistency of the Whittle estimator does not always hold and that the limit is not necessarily Gaussian. This can happen even if $Y_t$ has short memory.

This paper studies a class of tests useful for testing the goodness-of-fit of an autoregressive model. These tests are based on a class of empirical processes marked by certain residuals. The paper first gives their large sample behavior under null hypotheses. Then a martingale transformation of the underlying process is given that makes tests based on it asymptotically distribution free. Consistency of these tests is also discussed briefly.

Motivated by Gaussian tests for a time series, we are led to investigate the asymptotic behavior of the residual empirical processes of stochastic regression models. These models cover the fixed design regression models as well as general AR$(q)$ models. Since the number of the regression coeffi-cients is allowed to grow as the sample size increases, the obtained results are also applicable to nonlinear regression and stationary AR$(\infty)$ models. In this paper, we first derive an oscillation-like result for the residual em-pirical process. Then, we apply this result to autoregressive time series. In particular, for a stationary AR$(\infty)$ process, we are able to determine the order of the number of coefficients of a fitted AR$(q_n)$ model and obtain the limiting Gaussian processes. For an unstable AR$(q)$ process, we show that if the characteristic polynomial has a unit root 1, then the limiting process is no longer Gaussian. For the explosive case, one of our side results also provides a short proof for the Brownian bridge results given by Koul and Levental.

We answer two open questions concerning the existence of universal schemes for classification and regression estimation from stationary ergodic processes. It is shown that no measurable procedure can produce weakly consistent regression estimates from every bivariate stationary ergodic process, even if the covariate and response variables are restricted to take values in the unit interval. It is further shown that no measurable procedure can produce weakly consistent classification rules from every bivariate stationary ergodic process for which the response variable is binary valued. The results of the paper are derived via reduction arguments and are based in part on recent work concerning density estimaton from ergodic processes.

One of the most difficult problems occurring with stepwise multiple test procedures for a set of two-sided hypotheses is the control of direc-tional errors if rejection of a hypothesis is accomplished with a directional decision. In this paper we generalize a result for so-called step-down procedures derived by Shaffer to a large class of stepwise or closed multiple test procedures. In a unifying way we obtain results for a large class of order statistics procedures including step-down as well as step-up procedures (Hochberg, Rom), but also a procedure of Hommel based on critical values derived by Simes. Our method of proof is also applicable in situations where directional decisions are mainly based on conditionally independent $t$-statistics. A closed $F$-test procedure applicable in regression models with orthogonal design, the modified $S$-method of Scheffé applicable in the Analysis of Variance and Fisher’s LSD-test for the comparison of three means will be considered in more detail.

For independent $d$-variate random variables $X_1,\dots,X_m$ with common density $f$ and $Y_1,\dots,Y_n$ with common density $g$, let $R_{m,n}$ be the number of edges in the minimal spanning tree with vertices $X_1,\dots,X_m$, $Y_1,\dots,Y_n$ that connect points from different samples. Friedman and Rafsky conjectured that a test of $H_0: f = g$ that rejects $H_0$ for small values of $R_{m,n}$ should have power against general alternatives. We prove that $R_{m,n}$ is asymptotically distribution-free under $H_0$ , and that the multivariate two-sample test based on $R_{m,n}$ is universally consistent.

Z.Given measurements $(x_i,y_i) = 1,\dots,n$, we discuss methods to assess whether an underlying regression function is smooth (continuous or differentiable) or whether it has discontinuities. The variance of the measurements is assumed to be unknown, and is estimated simultane-ously. By regressing squared differences of the data formed with various span sizes on the span size itself, we obtain an asymptotic linear model with dependent errors. The parameters of this asymptotic linear model include the sum of the squared jump sizes as well as the variance of the measurements. Both parameters can be consistently estimated, with mean squared error rates of convergence of $n^{-2/3}$ for the sum of squared jump sizes and $n^{-1}$ for the error variance. We derive the asymptotic constants of the mean squared error (MSE) and discuss the dependence of MSE on the maximum span size $L$. The test for the existence of jumps is formulated for the null hypothesis that the sum of squared jump sizes is 0. The asymptotic distribution of the test statistic is obtained essentially via a central limit theorem for $U$-statistics. We motivate and illustrate the methods with data surrounded by a scientific controversy concerning the question whether the growth of children occurs smoothly or rather in jumps.

The paper is concerned with testing uniformity versus a monotone den-sity. This problem arises in two important contexts, after transformations, testing whether a sample is a simple random sample or a biased sample, and testing whether the intensity function of a nonhomogeneous Poisson process is constant against monotone alternatives. A penalized likelihood ratio test ($P$-test) and a Dip likelihood test ($D$-test) are developed. The $D$-test is analogous to Hartigan and Hartigan’s (1985) Dip test for bump hunting problems. While nonparametric, both the $P$- and $D$-tests are quite efficient in comparison to the most powerful (MP) tests for some simple alternatives and also the Laplace test developed for nonhomogeneous Poisson process. The $P$- and $D$-tests have higher power than the above MP tests under different sets of monotone alternatives and so have greater applicability. Moderate sample size performance and applications of our tests are illustrated via simulations and examination of an air-conditioning equipment data set.

Suppose that X is a random variable with density $f(x|\theta)$ and that $\pi(\theta|x)$ is a proper posterior corresponding to an improper prior $\nu(\theta)$. The prior is called $\mathscr{P}$-admissible if the generalized Bayes estimator of every bounded function of $\theta$ is almost-$\nu$-admissible under squared error loss. Eaton showed that recurrence of the Markov chain with transition density $R(\eta|\theta) = \int \pi(\eta|x)f(x|\theta) dx$ is a sufficient condition for $\mathscr{P}$-admissibility of $\nu(\theta)$. We show that Eaton’s Markov chain is recurrent if and only if its conjugate partner, with transition density $\tilde{R}(y|x) = \int f(y|\theta) \pi(\theta|x) d\theta$, is recurrent. This provides a new method of establishing $\mathscr{P}$-admissibility. Often, one of these two Markov chains corresponds to a standard stochastic process for which there are known results on recurrence and transience. For example, when $X$ is Poisson $(\theta)$ and an improper gamma prior is placed on $\theta$, the Markov chain defined by $\tilde{R}(y|x)$ is equivalent to a branching process with immigration. We use this type of argument to establish $\mathscr{P}$-admissibility of some priors when $f$ is a negative binomial mass function and when $f$ is a gamma density with known shape.

Let $\mu$ be a positive measure on $\mathbb{R}^d$ and let $F(\mu) = \{P(\theta,\mu); \theta \in \Theta\}$ be the natural exponential family generated by $\mu$. The aim of this paper is to show that if $\mu$ is infinitely divisible then the generalized variance of $\mu$, .i.e., the determinant of the covariance operator of $P(\theta,\mu)$, is the Laplace transform of some positive measure $\rho(\mu)$ on $mathbb{R}^d$. We then investigate the effect of the transformation $\mu \to \rho(\mu)$ and its implications for the skewness vector and the conjugate prior distribution families of $F(\mu)$. .

In this paper, we compare the asymptotic behavior of some common block bootstrap methods based on nonrandom as well as random block lengths. It is shown that, asymptotically, bootstrap estimators derived using any of the methods considered in the paper have the same amount of bias to the first order. However, the variances of these bootstrap estimators may be different even in the first order. Expansions for the bias, the variance and the mean-squared error of different block bootstrap variance estimators are obtained. It follows from these expansions that using overlapping blocks is to be preferred over nonoverlapping blocks and that using random block lengths typically leads to mean-squared errors larger than those for nonrandom block lengths.

Based on a revised Lancaster-type representation of the additive interactions associated with a probability measure, a new approach for the analysis of high-dimensional contingency tables is proposed. The approach is essentially model-free because the additive interaction tensor is merely a convenient reparameterization of the given table. Single interaction terms are investigated using the bootstrap method whose first-order asymptotic validity is immediate. The global structure can be investigated by using the multiple $p$-values given by Holm’s sequentially rejecting multiple testing procedure. The procedure is based on a characterization of the Moebius function as a solution of the simultaneous eigenproblem for all intersection operators in a finite lattice.