Let $X_1, X_2, \dots, X_n$ be an i.i.d. sample from the Poisson mixture distribution $p(x) = (1/x!) \int_0^{\infty} s^x e^{-s}f(s) ds$. Rates of convergence in mean integrated squared error (MISE) of orthogonal series estimators for the mixing density f supported on $[a, b]$ are studied. For the Hölder class of densities whose rth derivative is Lipschitz $\alpha$, the MISE converges at the rate $(\log n/ \log \log n)^{-2(r +\alpha)}$. For Sobolev classes of densities whose rth derivative is square integrable, the MISE converges at the rate $(\log n/ \log \log n)^{-2r}$. The estimator is adaptive over both these classes.

For the Sobolev class, a lower bound on the minimax rate of convergence is $(\log n/ \log \log n)^{-2r}$, and so the orthogonal polynomial estimator is rate optimal.

A new variable bandwidth selector for kernel estimation is proposed. The application of this bandwidth selector leads to kernel estimates that achieve optimal rates of convergence over Besov classes. This implies that the procedure adapts to spatially inhomogeneous smoothness. In particular, the estimates share optimality properties with wavelet estimates based on thresholding of empirical wavelet coefficients.

Let $X_1, \dots, X_n$ be independent identically distributed observations from an unknown probability density $f(\cdot)$. Consider the problem of estimating the level set $G = G_f(\lambda) = {x \epsilon\mathbb{R}^2: f(x) \geq \lambda}$ from the sample $X_1, \dots, X_n$, under the assumption that the boundary of G has a certain smoothness. We propose piecewise-polynomial estimators of G based on the maximization of local empirical excess masses. We show that the estimators have optimal rates of convergence in the asymptotically minimax sense within the studied classes of densities. We find also the optimal convergence rates for estimation of convex level sets. A generalization to the N-dimensional case, where $N > 2$, is given.

The Grenander estimator of a decreasing density, which is defined as the derivative of the concave envelope of the empirical c.d.f., is known to be a very good estimator of an unknown decreasing density on the half-line $\mathbb{R}^+$ when this density is not assumed to be smooth. It is indeed the maximum likelihood estimator and one can get precise upper bounds for its risk when the loss is measured by the $\mathbb{L}^1$-distance between densities. Moreover, if one restricts oneself to the compact subsets of decreasing densities bounded by H with support on $[0, L]$ the risk of this estimator is within a fixed factor of the minimax risk. The same is true if one deals with the maximum likelihood estimator for unimodal densities with known mode. When the mode is unknown, the maximum likelihood estimator does not exist any more. We shall provide a general purpose estimator (together with a computational algorithm) for estimating nonsmooth unimodal densities. Its risk is the same as the risk of the Grenander estimator based on the knowledge of the true mode plus some lower order term. It can also cope with small departures from unimodality.

We show that local irregularity of observed sample paths provides additional information which allows nonparametric estimators and predictors for continuous time processes to reach parametric rates in mean square as well as in a.s. uniform convergence. For example, we prove that under suitable conditions the kernel density estimator $f_T$ associated with the observed sample path $(X_t, 0 \leq t \leq T)$ satisfies $$\sup_{x \epsilon \mathbb{R}}|f_T(x) - f(x)| = o(\ln_k T(\frac{\ln T}{T})^{1/2}) \quad \text{a.s.}, k \geq 1$$ where f denotes the unknown marginal density of the stationary process and where $\ln_k$ denotes the kth iterated logarithm.

The proof uses a special Borel-Cantelli lemma for continuous time processes together with a sharp large deviation inequality. Furthermore the parametric rate obtained in (1) is preserved by using a suitable sampling scheme.

Local linear density estimators achieve automatic boundary corrections and enjoy some typical optimal properties. Proper choice of the smoothing parameters is crucial for their performance. A data-based bandwidth selector is developed in the spirit of plug-in rules. Consistency and asymptotic normality of the selected bandwidth are demonstrated. The bandwidth is very efficient regardless of whether there are non-smooth boundaries in the support of the density or not.

Consider a partial linear model, where the expectation of a random variable Y depends on covariates $(x, z)$ through $F(\theta_0 x + m_0(z))$, with $\theta_0$ an unknown parameter, and $m_0$ an unknown function. We apply the theory of empirical processes to derive the asymptotic properties of the penalized quasi-likelihood estimator.

We show that the nonparametric maximum likelihood estimator (NPMLE) of a distribution function based on balanced ranked set samples is consistent, converges weakly to a Gaussian process and is asymptotically efficient. The covariance function of the limiting process is described in terms of the solution to a Fredholm integral equation of the second kind.

By approximating the classical product-limit estimator of a distribution function with an average of iid random variables, we derive sufficient and necessary conditions for the rate of (both strong and weak) uniform convergence of the product-limit estimator over the whole line. These findings somehow fill a longstanding gap in the asymptotic theory of survival analysis. The result suggests a natural way of estimating the rate of convergence. We also prove a related conjecture raised by Gill and discuss its application to the construction of a confidence interval for a survival function near the endpoint.

The product-limit estimator (PLE) and weighted empirical processes are two important ingredients of almost any censored regression analysis. A link between them is provided by the generalized PLEs introduced in this paper. These generalized PLEs are the product-limit integrals of the empirical cumulative hazard function estimators in which weighted empirical processes are used to replace the standard empirical processes. The weak convergence and some large sample approximations of the generalized PLEs are established. As an application these generalized PLEs are used to define some minimum distance estimators which are shown to be asymptotically normal. These estimators are qualitatively robust. In some submodels an optimal choice of the weight matrix is the covariate matrix and some of these estimators are quite efficient at a few common survival distributions. To implement these estimators some computational aspects are discussed and an algorithm is given. From a real data example and some preliminary simulation results, these estimators seem to be very competitive to and more robust than some more traditional estimators such as the Buckley-James estimator.

We consider a coherent system S consisting of m independent components for which we do not know the distributions of the components' lifelengths. If we know the structure function of the system, then we can estimate the distribution of the system lifelength by estimating the distributions of the lifelengths of the individual components. Suppose that we can collect data under the "autopsy model," wherein a system is run until a failure occurs and then the status (functioning or dead) of each component is obtained. This test is repeated n times. The autopsy statistics consist of the age of the system at the time of breakdown and the set of parts that are dead by the time of breakdown. We develop a nonparametric Bayesian estimate of the distributions of the component lifelengths and then use this to obtain an estimate of the distribution of the lifelength of the system. The procedure is applicable to machine-test settings wherein the machines have redundant designs. A parametric procedure is also given.

The construction and analysis of repair models is an important area in reliability. A commonly used model is the minimal repair model. Under this model, repair restores the state of the system to its level prior to failure. Kijima introduced repair models that could be classified as "better-than-minimal." Under Kijima's models, the system, upon repair, is functionally the same as a working system of lesser age which has never experienced failure. In this paper, we present a new approach to the modeling of better-than-minimal repair models. Using this approach, we construct a general repair model that contains Kijima's models as special cases. We also study the problem of estimating the distribution of the time to first failure of a system maintained by general repair. We make use of counting processes to show strong consistency of the estimator and prove results on weak convergence. Finally, we derive a Hall-Wellner type asymptotic confidence band for the distribution of the time to first failure of the system.

In this paper efficient designs are determined when Anderson's procedure is applied in order to identify the degree of a multivariate polynomial regression model. It is shown that the optimal designs are very closely related to model robust designs which maximize a weighted p-mean of D-efficiencies. As a consequence we obtain designs with high efficiency for model discrimination and for the statistical analysis in the identified model.

Chen and Hedayat and Tang and Wu studied and characterized minimum aberration $2^{n-m}$ designs in terms of their complemetary designs. Based on a new and more powerful approach, we extend the study to identify minimum aberration $q^{n-m}$ designs through their complementary designs. By using MacWilliams identities and Krawtchouk polynomials in coding theory, we obtain some general and explicit relationships between the wordlength pattern of a $q^{n-m}$ design and that of its complementary design. These identities provide a powerful tool for characterizing minimum aberration $q^{n-m}$ designs. The case of $q = 3$ is studied in more details.

This paper addresses the problems of determination and construction of universally optimal designs in two-dimensional blocks of size $p \times 2$, assuming within-block observations are correlated. Generalized least-squares estimation of treatment contrasts is considered in four fixed block-effects models: (I) with fixed row and column effects, (II) with the row effects only, (III) with the column effects only, and (IV) with neither row nor column effects. For a general dependence structure and $p = 2$, optimal designs for Model I are found to coincide with the least-squares optimal designs. For general p, Models I-IV, and the within-block correlation pattern described by the doubly geometric process, interesting nonbinary block patterns are found for the universally optimal designs. Only for Model IV for small, positive correlations do binary blocks turn out to be best, though binarity with respect to rows or columns is often required. Regardless of the model, the conditions frequently coincide with those for optimal nested row-column designs with uncorrelated errors: one class of these designs is found to be optimal for at least some values of the correlation parameters under all four models, and others are found to be optimal for particular models. The exact form of the blocks for a universally optimum design is found to be quite sensitive to the blocksize and to the magnitude of the correlations under both Models III and IV.

For the canonical problem of estimating the mean of a multivariate normal distribution with a known covariance matrix using a squared error loss, we give a general method for finding estimates that have risk functions identical to that of a given inadmissible estimate. In the case of more than one dimension, the estimates considered are spherically symmetric, but in one dimension no such assumption is made. Generally speaking, we characterize all estimates which have the risk duplication property. It is proven that every James-Stein estimator except $(1 - (p - 2)/ \parallel X \parallel^2) \rm X$ can be duplicated in risk by infinitely many estimators. A general theorem is presented from which a principal inadmissibility result for spherically symmetric estimates in Brown follows. This and the other results all basically depend on solving Riccati differential equations of an appropriate kind. A curious result is that two smooth estimates whose graphs intersect cannot have identical risk. Several results for the entire discrete exponential family and the binomial case demonstrate that the phenomena are fundamentally different in continuous and discrete cases. Our results indicate a new method for constructing explicit dominating estimates that may work in many problems.

The classical problem of testing goodness-of-fit of a parametric family is reconsidered. A new test for this problem is proposed and investigated. The new test statistic is a combination of the smooth test statistic and Schwarz's selection rule. More precisely, as the sample size increases, an increasing family of exponential models describing departures from the null model is introduced and Schwarz's selection rule is presented to select among them. Schwarz's rule provides the "right" dimension given by the data, while the smooth test in the "right" dimension finishes the job. Theoretical properties of the selection rules are derived under null and alternative hypotheses. They imply consistency of data driven smooth tests for composite hypotheses at essentially any alternative.

When studying some process or development in different subjects or units--be it biological, chemical or physical--we usually see a typical pattern, common to all curves. Yet there is variation both in amplitude and dynamics between curves. Following some ideas of structural analysis introduced by Kneip and Gasser, we study a method--dynamic time warping with a proper cost function--for estimating the shift or warping function from one curve to another to align the two functions. For some models this method can identify the true shift functions if the data are noise free. Noisy data are smoothed by a nonparametric function estimate such as a kernel estimate. It is shown that the proposed estimator is asymptotically normal and converges to the true shift function as the sample size per subject goes to infinity. Some simulation results are presented to illustrate the performance of this method.

Schmid extended the classical EWMA control chart to autocorrelated processes. Here, we consider the tail probability of the run length in the in-control state. The in-control process is assumed to be a stationary Gaussian process. It is proved that the tails for the autocorrelated process are larger than in the case of independent variables if all autocovariances are greater than or equal to zero. The inequality is strict. Moreover, this result is still valid for stationary processes having elliptically contoured marginal distributions.

Observations are taken independently and sequentially. Detection of a change in distribution is studied when the problem has an invariance structure. The prechange distribution is assumed to be a member of a specified family and is assumed known up to a nuisance parameter. We provide a general method of constructing surveillance schemes in the presence of a nuisance parameter and give sufficient conditions for approximating their average run lengths to false alarm. Applications include detecting a change in scale of i.i.d. gamma variates with unknown initial scale, detecting a change in location of i.i.d. normal variates with unknown initial mean, and a non-parametric scheme based on ranks for detecting a change to a stochastically larger distribution.

Applications of extreme value theory to problems of statistical inference typically involve estimating tail probabilities well beyond the range of the data, without the benefit of a concise mathematical model for the sampling distribution. The available model is generally only an asymptotic one. That is, an approximation to probabilities of extreme deviation is supposed, which is assumed to become increasingly accurate as one moves further from the range of the data, but whose concise accuracy is unknown. Quantification of the level of accuracy is essential for optimal estimation of tail probabilities. In the present paper we suggest a practical device, based on a nonstandard application of the bootstrap, for determining empirically the accuracy of the approximation and thereby constructing appropriate estimators.

Recently, the problem of characterizing monotone unidimensional latent variable models for binary repeated measures was studied by Ellis and van den Wollenberg and by Junker. We generalize their work with a de Finetti-like characterization of the distribution of repeated measures $\rm X = (X_1, X_2, \dots)$ that can be represented with mixtures of likelihoods of independent but not identically distributed random variables, where the data satisfy a stochastic ordering property with respect to the mixing variable. The random variables $X_j$ may be arbitrary real-valued random variables. We show that the distribution of X can be given a monotone unidimensional latent variable representation that is useful in the sense of Junker if and only if this distribution satisfies conditional association (CA) and a vanishing conditional dependence (VCD) condition, which asserts that finite subsets of the variables in X become independent as we condition on a larger and larger segment of the remaining variables in X. It is also interesting that the mixture representation is in a certain ordinal sense unique, when CA and VCD hold. The characterization theorem extends and simplifies the main result of Junker and generalizes methods of Ellis and van den Wollenberg to a much broader class of models.

Exchangeable sequences of binary random variables also satisfy both CA and VCD, as do exchangeable sequences arising as location mixtures. In the same way that de Finetti's theorem provides a path toward justifying standard i.i.d.-mixture components in hierarchical models on the basis of our intuitions about the exchangeability of observations, this theorem justifies one-dimensional latent variable components in hierarchical models, in terms of our intuitions about positive association and redundancy between observations. Because these conditions are on the joint distribution of the observable data X, they may also be used to construct asymptotically power-1 tests for unidimensional latent variable models.

We provide a new characterization of the Dirichlet distribution. Let $\theta_{ij}, 1 \leq i \leq k, 1 \leq j \leq n$, be positive random variables that sum to unity. Define $\theta_{i \cdot} = \Sigma_{j=1}^n \theta_{ij}, \theta_{I \cdot} = {\theta_{i \cdot}_{i=1}^{k-1}, \theta_{j|i} = \theta_{ij}/ \Sigma_j \theta_{ij}$ and \theta_{J|i} = {\theta_{j|i}}_{j=1}^{n-1}$. We prove that if ${\theta_{I \cdot}, \theta_{J|1}, \dots, \theta_{J|k}}$ are mutually independent and ${\theta_{\cdot J}, \theta_{I|1}, \dots, \theta_{I|n}}$ are mutually independent (where $\theta_{\cdot J}$ and $\theta_{I|j}$ are defined analogously, and each parameter set has a strictly positive pdf, then the pdf of $\theta_{ij}$ is Dirichlet. This characterization implies that under assumptions made by several previous authors for selecting a Bayesian network structure out of a set of candidate structures, a Dirichlet prior on the parameters is inevitable.