The simple hypothesis is tested that the distribution of independent random variables $X_1, X_2, \cdots, X_n$ is a given probability measure $P_0$. Let $\pi_n$ be any sequence of partitions. The alternative hypothesis is the set of probability measures $P$ with $\sum_{A\in \pi_n}|P(A) - P_0(A)| \geq \delta$, where $\delta > 0$. Note the dependence of this set of alternatives on the sample size. It is shown that if the effective cardinality of the partitions is of the same…
Projection pursuit regression and kernel regression are methods for estimating a smooth function of several variables from noisy data obtained at scattered sites. Methods based on local averaging can perform poorly in high dimensions (curse of dimensionality). Intuition and examples have suggested that projection based approaches can provide better fits. For what sorts of regression functions is this true? When and by how much do projection methods reduce the curse of…
Pattern theory offers concepts for modelling images and methods for making inferences from observed images. This will be described briefly and illustrated by examples. We shall present limit theorems for the Markov processes on graphs (that are basic to pattern theory) motivated by computational considerations. They will yield approximations that have been exploited to make the inference algorithms computationally feasible. We shall also consider the problem of estimating…
Spherical regression studies models which postulate that the unit vector $v$ is equal to an unknown rotation $P$ of the unit vector $u$ "plus" an experimental error. The case where the experimental errors follow a Fisher-von Mises distribution with a large concentration parameter $\kappa$ is considered in this work. Asymptotic $(\kappa \rightarrow \infty)$ inferential procedures for $P$ are proposed when $n$, the sample size, is fixed. Diagnostic methods for spherical…
The rate of convergence in the central limit theorem for functions of independent random variables is studied in a unifying approach. The basic result sharpens and extends a theorem of van Zwet. Applications to $U$-, $L$- and $R$-statistics are also given, improving or extending the results of Helmers and van Zwet, Helmers and Huskova, Does and van Es and Helmers.
We define and investigate classes of statistical models for the analysis of associations between variables, some of which are qualitative and some quantitative. In the cases where only one kind of variables is present, the models are well-known models for either contingency tables or covariance structures. We characterize the subclass of decomposable models where the statistical theory is especially simple. All models can be represented by a graph with one vertex for each…
It is an open problem in the literature to derive optimum tests for the equality of treatment effects in an unbalanced two-way classification model. For such models without interaction, optimum tests are derived in the following cases: (i) the locally best invariant unbiased test for the random effects model corresponding to an equiblock and equireplicate design, (ii) the locally best invariant unbiased test for the mixed effects model with mixed treatment effects…
We prove that the distribution of least-squares estimates in autoregressions can be bootstrapped with accuracy $o(n^{-1/2})$ a.s., thereby improving the normal approximation error of $O(n^{-1/2})$.
It is shown in this article that the classical i.i.d. bootstrap remains a valid procedure for estimating the sampling distributions of certain symmetric estimators of location, as long as the random observations are independently drawn from distributions with (essentially) a common location. This may be viewed as a robust property of the classical i.i.d. bootstrap. Also included is a study of the second order properties of a different bootstrap procedure proposed by Wu in…
A Bayesian bootstrap for a finite population is introduced; its small-sample distributional properties are discussed and compared with those of the frequentist bootstrap for a finite population. It is also shown that the two are first-order asymptotically equivalent.
A class of procedures is proposed for testing the stationarity of a Gaussian process or the homogeneity of independent processes. Requiring very limited prior knowledge of model structure, the methods can detect changes or differences in mean, in variance, in covariances and even in law. Although the theory of the stationarity test is worked out only for processes whose realizations are stationary over "epochs" separated by known change points, Monte Carlo evidence indicates…
The purpose of this paper is to investigate the convergence rates of a sequence of empirical Bayes decision rules for the two-action decision problems where the distributions of the observations belong to a discrete exponential family. It is found that the sequence of the empirical Bayes decision rules under study is asymptotically optimal, and the order of associated convergence rates is $O(\exp(-cn))$, for some positive constant $c$, where $n$ is the number of accumulated…
Identifiability problems have previously precluded a general approach to testing the hypothesis of a "pure" distribution against the alternative of a mixture of distributions. Three types of identifiability are defined, and it is shown that $B$-identifiability allows a Bayesian solution to the testing problem. First, an equivalence relation is defined over parametrizations of probability functions. Then the projection onto the quotient space is shown to give a $B$-identifiabl…
A procedure for estimating the mixing distribution for a normal mean is presented. The estimator is shown to be consistent regardless of the mixing distribution, and it is suggested that the estimation techniques may be applied in a similar manner to estimate mixing distributions for a wide class of location parameter families. Implied marginal density and derivative estimates for the normal case are shown to be consistent, converging at near-optimal rates. In addition,…
Controlling Bayes and/or minimax risks under possibly different loss functions is formulated as a problem faced by two or more statisticians who must compromise and agree on the use of a single decision procedure. The theory characterizing solutions to Bayes compromise problems and minimax-Bayes compromise problems is presented. In a Bayes compromise problem, Bayes risks under different prior distributions and/or loss functions are minimized simultaneously. In a…
Discrete and multinomial analogs are defined for classical (continuous) invariant nonparametric problems of estimating the sample cumulative distribution function (sample c.d.f.) and the sample median. Admissibility of classical estimators and their analogs is investigated. In discrete (including multinomial) settings the sample c.d.f. is shown to be an admissible estimator of the population c.d.f. under the invariant weighted Cramer-von Mises loss function $L_1(F, \hat{F}) = \int \big\lbrack (F(t) - \hat{F}(t))^2/(F(t)(1 - F(t))) \big\rbrack dF(t).$…
Imputation and multiple-imputation procedures have been used in practice to handle the problem of ignorable nonresponse in sample surveys. We examine the large-sample properties of these procedures where covariates are available for the case when the complete-data analysis is based on least squares. The results provide a formal justification for the inference procedures discussed by Rubin and Schenker for the location problem and suggest new procedures for the regression…
We study the asymptotic behavior of the statistical estimators that maximize a not necessarily differentiable criterion function, possibly subject to side constraints (equalities and inequalities). The consistency results generalize those of Wald and Huber. Conditions are also given under which one is still able to obtain asymptotic normality. The analysis brings to the fore the relationship between the problem of finding statistical estimators and that of finding the…
We present a solution to the goodness-of-fit problem for multivariate observations, using the innovation process for the (sequential) empirical distribution function with respect to a conveniently chosen linear ordering or scanning system in $R^m$.
An interval censoring model is carefully defined and the range of its applicability is illustrated. A class of rank tests for the two- and $k$-sample problems is proposed. The statistic is based on the exact ranks of the observed responses and resembles the rank statistic in the usual uncensored case. This statistic is shown to be asymptotically equivalent to the efficient scores test statistic under an assumed parametric model. An alternative rank statistic, based on the…
We study semiparametric models where for a fixed value of the finite-dimensional parameter there exists a sufficient statistic for the nuisance parameter. An asymptotically normal sequence of estimators for the parametric component is constructed, which is efficient under the assumption that projecting on the set of nuisance scores is equivalent to taking conditional expectations given the sufficient statistic. The latter property is checked for a number of examples, in…
Strong uniform consistency rates are established for kernel type estimators of functionals of the conditional distribution function, under general conditions. The present treatment unifies a number of specific problems previously studied separately in the literature. Some of these applications we treat in detail, including regression curve estimation, density estimation, estimation of conditional df's, $L$-smoothing and $M$-smoothing. Various previous results in the…
A kernel density estimator is defined to be admissible if no other kernel estimator has (among all densities and sample sizes) uniformly smaller mean integrated squared error. Admissible kernel density estimators are precisely those using kernels with nonnegative Fourier transforms bounded by 1. Several examples are given.
This paper discusses the possibility of truly nonparametric inference about functionals of an unknown density. Examples considered include: discrete functionals, such as the number of modes of a density and the number of terms in the true model; and continuous functionals, such as the optimal bandwidth for kernel density estimates or the widths of confidence intervals for adaptive location estimators. For such functionals it is not generally possible to make two-sided…
The simple hypothesis is tested that the distribution of independent random variables X1,X2,⋯,Xn is a given probability measure P0. Let πn be any sequence of partitions. The alternative hypothesis is the set of probability measures P with ∑A∈πn∣P(A)−P0(A)∣≥δ, where δ>0. Note the dependence of this set of alternatives on the sample size. It is shown that if the effective cardinality of the partitions is of the same…
Spherical regression studies models which postulate that the unit vector v is equal to an unknown rotation P of the unit vector u "plus" an experimental error. The case where the experimental errors follow a Fisher-von Mises distribution with a large concentration parameter κ is considered in this work. Asymptotic (κ→∞) inferential procedures for P are proposed when n, the sample size, is fixed. Diagnostic methods for spherical…
The rate of convergence in the central limit theorem for functions of independent random variables is studied in a unifying approach. The basic result sharpens and extends a theorem of van Zwet. Applications to U-, L- and R-statistics are also given, improving or extending the results of Helmers and van Zwet, Helmers and Huskova, Does and van Es and Helmers.
We prove that the distribution of least-squares estimates in autoregressions can be bootstrapped with accuracy o(n−1/2) a.s., thereby improving the normal approximation error of O(n−1/2).
The purpose of this paper is to investigate the convergence rates of a sequence of empirical Bayes decision rules for the two-action decision problems where the distributions of the observations belong to a discrete exponential family. It is found that the sequence of the empirical Bayes decision rules under study is asymptotically optimal, and the order of associated convergence rates is O(exp(−cn)), for some positive constant c, where n is the number of accumulated…
Identifiability problems have previously precluded a general approach to testing the hypothesis of a "pure" distribution against the alternative of a mixture of distributions. Three types of identifiability are defined, and it is shown that B-identifiability allows a Bayesian solution to the testing problem. First, an equivalence relation is defined over parametrizations of probability functions. Then the projection onto the quotient space is shown to give a B-identifiabl…
Discrete and multinomial analogs are defined for classical (continuous) invariant nonparametric problems of estimating the sample cumulative distribution function (sample c.d.f.) and the sample median. Admissibility of classical estimators and their analogs is investigated. In discrete (including multinomial) settings the sample c.d.f. is shown to be an admissible estimator of the population c.d.f. under the invariant weighted Cramer-von Mises loss function L1(F,F^)=∫[(F(t)−F^(t))2/(F(t)(1−F(t)))]dF(t).…
An Innovation Approach to Goodness-of-Fit Tests in Rm
We present a solution to the goodness-of-fit problem for multivariate observations, using the innovation process for the (sequential) empirical distribution function with respect to a conveniently chosen linear ordering or scanning system in Rm.
An interval censoring model is carefully defined and the range of its applicability is illustrated. A class of rank tests for the two- and k-sample problems is proposed. The statistic is based on the exact ranks of the observed responses and resembles the rank statistic in the usual uncensored case. This statistic is shown to be asymptotically equivalent to the efficient scores test statistic under an assumed parametric model. An alternative rank statistic, based on the…
Strong uniform consistency rates are established for kernel type estimators of functionals of the conditional distribution function, under general conditions. The present treatment unifies a number of specific problems previously studied separately in the literature. Some of these applications we treat in detail, including regression curve estimation, density estimation, estimation of conditional df's, L-smoothing and M-smoothing. Various previous results in the…