Showing papers on "Asymptotic distribution published in 1989"

Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses

Abstract: In this paper, we propose a classical approach to model selection. Using the Kullback-Leibler Information measure, we propose simple and directional likelihood-ratio tests for discriminating and choosing between two competing models whether the models are nonnested, overlapping or nested and whether both, one, or neither is misspecified. As a prerequisite, we fully characterize the asymptotic distribution of the likelihood ratio statistic under the most general conditions.

The Robust Inference for the Cox Proportional Hazards Model

Abstract: We derive the asymptotic distribution of the maximum partial likelihood estimator β for the vector of regression coefficients β under a possibly misspecified Cox proportional hazards model. As in the parametric setting, this estimator β converges to a well-defined constant vector β*. In addition, the random vector n 1/2(β – β*) is asymptotically normal with mean 0 and with a covariance matrix that can be consistently estimated. The newly proposed robust covariance matrix estimator is similar to the so-called “sandwich” variance estimators that have been extensively studied for parametric cases. For many misspecified Cox models, the asymptotic limit β* or part of it can be interpreted meaningfully. In those circumstances, valid statistical inferences about the corresponding covariate effects can be drawn based on the aforementioned asymptotic theory of β and the related results for the score statistics. Extensive studies demonstrate that the proposed robust tests and interval estimation procedures...

Regression analysis of multivariate incomplete failure time data by modeling marginal distributions

Abstract: Many survival studies record the times to two or more distinct failures on each subject. The failures may be events of different natures or may be repetitions of the same kind of event. In this article, we consider the regression analysis of such multivariate failure time observations. Each marginal distribution of the failure times is formulated by a Cox proportional hazards model. No specific structure of dependence among the distinct failure times on each subject is imposed. The regression parameters in the Cox models are estimated by maximizing the failure-specific partial likelihoods. The resulting estimators are shown to be asymptotically jointly normal with a covariance matrix that can be consistently estimated. Simultaneous inferential procedures are then proposed. Extensive Monte Carlo studies indicate that the normal approximation is adequate for practical use. The new methods allow time-dependent covariates, missing observations, and arbitrary patterns of censorship. They are illustrat...

A method of simulated moments for estimation of discrete response models without numerical integration

Abstract: This paper proposes a simple modification of a conventional method of moments estimator for a discrete response model, replacing response probabilities that require numerical integration with estimators obtained by Monte Carlo simulation. This method of simulated moments (MSM) does not require precise estimates of these probabilities for consistency and asymptotic normality, relying instead on the law of large numbers operating across observations to control simulation error, and hence can use simulations of practical size. The method is useful for models such as high-dimensional multinomial probit (MNP), where computation has restricted applications.

Semiparametric Estimation of Index Coefficients

Abstract: This paper gives a solution to the problem of estimating coefficients of index models, through the estimation of the density-weighted average derivative of a general regression function. The estimators, based on sample analogues of the product moment representation of the average derivative, are constructed using nonparametric kernel estimators of the density of the regressors. Asymptotic normality is established using extensions of classical U-statistic theorems, and asymptotic bias is reduced through use of a higher-order kernel

Efficient parameter estimation for self-similar processes

Abstract: Asymptotic normality of the maximum likelihood estimator for the parameters of a long range dependent Gaussian process is proved. Furthermore, the limit of the Fisher information matrix is derived for such processes which implies efficiency of the estimator and of an approximate maximum likelihood estimator studied by Fox and Taqqu. The results are derived by using asymptotic properties of Toeplitz matrices and an equicontinuity property of quadratic forms.

On the statistical analysis of capture experiments

Abstract: SUMMARY A procedure is given for estimating the size of a closed population in the presence of heterogeneous capture probabilities using capture-recapture data when it is possible to model the capture probabilities of individuals in the population using covariates. The results include the estimation of the parameters associated with the model of the capture probabilities and the use of these estimated capture probabilities to estimate the population size. Confidence intervals for the population size using both the asymptotic normality of the estimator and a bootstrap procedure for small samples are given.

On the relation between S-estimators and M-estimators of multivariate location and covariance

Abstract: We discuss the relation between S-estimators and M-estimators of multivariate location and covariance. As in the case of the estimation of a multiple regression parameter, S-estimators are shown to satisfy first-order conditions of M-estimators. We show that the influence function IF (x;S F) of S-functionals exists and is the same as that of corresponding M-functionals. Also, we show that S-estimators have a limiting normal distribution which is similar to the limiting normal distribution which is similar to the limiting normal distribution of M-estimators. Finally, we compare asymptotic variances and breakdown point of both types of estimators.

Approximate Distributions of Order Statistics: With Applications to Nonparametric Statistics

Abstract: 0 Introduction.- 0.1. Weak and Strong Convergence.- 0.2. Approximations.- 0.3. The Role of Order Statistics in Nonparametric Statistics.- 0.4. Central and Extreme Order Statistics.- 0.5. The Restriction to Independent and Identically Distributed Random Variables.- 0.6. Graphical Methods.- 0.7. A Guide to the Contents.- 0.8. Notation and Conventions.- I Exact Distributions and Basic Tools.- 1 Distribution Functions, Densities, and Representations.- 1.1. Introduction to Basic Concepts.- 1.2. The Quantile Transformation.- 1.3. Single Order Statistics, Extremes.- 1.4. Joint Distribution of Several Order Statistics.- 1.5. Extensions to Continuous and Discontinuous Distribution Functions.- 1.6. Spacings, Representations, Generalized Pareto Distribution Functions.- 1.7. Moments, Modes, and Medians.- 1.8. Conditional Distributions of Order Statistics.- P.1. Problems and Supplements.- Bibliographical Notes.- 2 Multivariate Order Statistics.- 2.1. Introduction.- 2.2. Distribution Functions and Densities.- P.2. Problems and Supplements.- Bibliographical Notes.- 3 Inequalities and the Concept of Expansions.- 3.1. Inequalities for Distributions of Order Statistics.- 3.2. Expansions of Finite Length.- 3.3. Distances of Measures: Convergence and Inequalities.- P.3. Problems and Supplements.- Bibliographical Notes.- II Asymptotic Theory.- 4 Approximations to Distributions of Central Order Statistics.- 4.1. Asymptotic Normality of Central Sequences.- 4.2. Expansions: A Single Central Order Statistic.- 4.3. Asymptotic Independence from the Underlying Distribution Function.- 4.4. The Approximate Multivariate Normal Distribution.- 4.5. Asymptotic Normality and Expansions of Joint Distributions.- 4.6. Expansions of Distribution Functions of Order Statistics.- 4.7. Local Limit Theorems and Moderate Deviations.- P.4. Problems and Supplements.- Bibliographical Notes.- 5 Approximations to Distributions of Extremes.- 5.1. Asymptotic Distributions of Extreme Sequences.- 5.2. Hellinger Distance between Exact and Approximate Distributions of Sample Maxima.- 5.3. The Structure of Asymptotic Joint Distributions of Extremes.- 5.4. Expansions of Distributions of Extremes of Generalized Pareto Random Variables.- 5.5. Variational Distance between Exact and Approximate Joint Distributions of Extremes.- 5.6. Variational Distance between Empirical and Poisson Processes.- P.5. Problems and Supplements.- Bibliographical Notes.- 6 Other Important Approximations.- 6.1. Approximations of Moments and Quantiles.- 6.2. Functions of Order Statistics.- 6.3. Bahadur Approximation.- 6.4. Bootstrap Distribution Function of a Quantile.- P.6. Problems and Supplements.- Bibliographical Notes.- 7 Approximations in the Multivariate Case.- 7.1. Asymptotic Normality of Central Order Statistics.- 7.2. Multivariate Extremes.- P.7. Problems and Supplements.- Bibliographical Notes.- III Statistical Models and Procedures.- 8 Evaluating the Quantile and Density Quantile Function.- 8.1. Sample Quantiles.- 8.2. Kernel Type Estimators of Quantiles.- 8.3. Asymptotic Performance of Quantile Estimators.- 8.4. Bootstrap via Smooth Sample Quantile Function.- P.8. Problems and Supplements.- Bibliographical Notes.- 9 Extreme Value Models.- 9.1. Some Basic Concepts of Statistical Theory.- 9.2. Efficient Estimation in Extreme Value Models.- 9.3. Semiparametric Models for Sample Maxima.- 9.4. Parametric Models Belonging to Upper Extremes.- 9.5. Inference Based on Upper Extremes.- 9.6. Comparison of Different Approaches.- 9.7. Estimating the Quantile Function Near the Endpoints.- P.9. Problems and Supplements.- Bibliographical Notes.- 10 Approximate Sufficiency of Sparse Order Statistics.- 10.1. Comparison of Statistical Models via Markov Kernels.- 10.2. Approximate Sufficiency over a Neighborhood of a Fixed Distribution.- 10.3. Approximate Sufficiency over a Neighborhood of a Family of Distributions.- 10.4. Local Comparison of a Nonparametric Model and a Normal Model.- P. 10. Problems and Supplements.- Bibliographical Notes.- Appendix 1. The Generalized Inverse.- Appendix 2. Two Technical Lemmas on Expansions.- Appendix 3. Further Results on Distances of Measures.- Author Index.

Statistical Analysis of Random Fields

Abstract: 1. Elements of the Theory of Random Fields.- 1.1 Basic concepts and notation.- 1.2 Homogeneous and isotropic random fields.- 1.3 Spectral properties of higher order moments of random fields.- 1.4 Some properties of the uniform distribution.- 1.5 Variances of integrals of random fields.- 1.6 Weak dependence conditions for random fields.- 1.7 A central limit theorem.- 1.8 Moment inequalities.- 1.9 Invariance principle.- 2. Limit Theorems for Functionals of Gaussian Fields.- 2.1 Variances of integrals of local Gaussian functionals.- 2.2 Reduction conditions for strongly dependent random fields.- 2.3 Central limit theorem for non-linear transformations of Gaussian fields.- 2.4 Approximation for distribution of geometric functional of Gaussian fields.- 2.5 Reduction conditions for weighted functionals.- 2.6 Reduction conditions for functionals depending on a parameter.- 2.7 Reduction conditions for measures of excess over a moving level.- 2.8 Reduction conditions for characteristics of the excess over a radial surface.- 2.9 Multiple stochastic integrals.- 2.10 Conditions for attraction of functionals of homogeneous isotropic Gaussian fields to semi-stable processes.- 3. Estimation of Mathematical Expectation.- 3.1 Asymptotic properties of the least squares estimators for linear regression coefficients.- 3.2 Consistency of the least squares estimate under non-linear parametrization.- 3.3 Asymptotic expansion of least squares estimators.- 3.4 Asymptotic normality and convergence of moments for least squares estimators.- 3.5 Consistency of the least moduli estimators.- 3.6 Asymptotic normality of the least moduli estimators.- 4. Estimation of the Correlation Function.- 4.1 Definition of estimators.- 4.2 Consistency.- 4.3 Asymptotic normality.- 4.4 Asymptotic normality. The case of a homogeneous isotropic field.- 4.5 Estimation by means of several independent sample functions.- 4.6 Confidence intervals.- References.- Comments.

Nonparametric Estimation of Time-Varying Parameters

Abstract: A sequence of observations yt, t = 1, 2,, N, is generated by the time-varying multiple regression model $${y_t} = {\beta '_t}{x_t} + {\sigma _t}{u_t}, t = 1,2, \ldots ,N,$$ where, for t = 1, 2,, N, u t is an unobservable random variable with zero mean and unit variance, x t is an observable p-vector-valued variable, and σ t and β t are, respectively, unobservable scalar and p-vector-valued parameters No model (stochastic or nonstochastic) is assumed for the σ t or β t ; instead they are assumed to be smoothly varying over t, in a certain sense A class of estimators of the β t , σ t is proposed, for each value of t; the estimators optimize a criterion prompted by Gaussian maximum likelihood considerations, and may be viewed as analogous to certain nonparametric function fitting estimators, employing a kernel function and band-width parameter, both selected by the practitioner Consistency and asymptotic normality are established in case of independent u t , and a consistent estimator of the asymptotic covariance matrix of the β t estimators is given Such results are also possible for serially correlated u t We discuss questions of implementation, in particular the choice of kernel function and band-width Generalization of the class of estimators to include certain robust estimators is possible, as is generalization of the methods to more general models involving time-varying parameters

Nonparametric Estimation of Probability Densities and Regression Curves

Abstract: 1. Asymptotic Properties of Certain Measures of Deviation for Kernel-Type Non Parametric Estimators of Probability Densities.- 1. Integrated Mean Square Error of Nonparametric Kernel-Type Probability Density Estimators.- 2. The Mean Square Error of Nonparametric Kernel-Type Density Estimators.- 2. Strongly Consistent in Functional Metrics Estimators of Probability Density.- 1. Strong Consistency of Kernel-Type Density Estimators in the Norm of the Space C.- 2. Convergence in the L2 Norm of Kernel-Type Density Estimators.- 3. Convergence in Variation of Kernel-Type Density Estimators and its Application to a Nonparametric Estimator of Bayesian Risk in a Classification Problem.- 3. Limiting Distributions of Deviations of Kernel-Type Density Estimators.- 1. Limiting Distribution of Maximal Deviation of Kernel-Type Estimators.- 2. Limiting Distribution of Quadratic Deviation of Two Nonparametric Kernel-Type Density Estimators.- 3. The Asymptotic Power of the Un1n2-Test in the Case of' singular' Close Alternatives.- 4. Testing for Symmetry of a Distribution.- 5. Independence of Tests Based on Kernel-Type Density Estimators.- 4. Nonparametric Estimation of the Regression Curve and Components of a Convolution.- 1. Some Asymptotic Properties of Nonparametric Estimators of Regression Curves.- 2. Strong Consistency of Regression Curve Estimators in the Norm of the Space C(a, b).- 3. Limiting Distribution of the Maximal Deviation of Estimators of Regression Curves.- 4. Limiting Distribution of Quadratic Deviation of Estimators of Regression Curves.- 5. Nonparametric Estimators of Components of a Convolution (S.N. Bernstein's Problem).- 5. Projection Type Nonparametric Estimation of Probability Density.- 1. Consistency of Projection-Type Probability Density Estimator in the Norms of Spaces C and L2.- 2. Limiting Distribution of the Squared Norm of a Projection-Type Density Estimator.- Addendum Limiting Distribution of Quadratic Deviation for a Wide Class of Probability Density Estimators.- 1. Limiting Distribution of Un.- 2. Kernel Density Estimators / Rosenblatt-Parzen Estimators.- 3. Projection Estimators of Probability Density / Chentsov Estimators.- 4. Histogram.- 5. Deviation of Kernel Estimators in the Sence of the Hellinger Distance.- References.- Author Index.

Conditionally Unbiased Bounded-Influence Estimation in General Regression Models, with Applications to Generalized Linear Models

Abstract: In this article robust estimation in generalized linear models for the dependence of a response y on an explanatory variable x is studied. A subclass of the class of M estimators is defined by imposing the restriction that the score function must be conditionally unbiased, given x. Within this class of conditionally Fisher-consistent estimators, optimal bounded-influence estimators of regression parameters are identified, and their asymptotic properties are studied. The estimators studied in this article and the efficient bounded-influence estimators studied by Stefanski, Carroll, and Ruppert (1986) depend on an auxiliary centering constant and nuisance matrix. The centering constant can be given explicitly for the conditionally Fisher-consistent estimators, and thus they are easier to compute than the estimators studied by Stefanski et al. (1986). In addition, estimation of the nuisance matrix has no effect on the asymptotic distribution of the conditionally Fisher-consistent estimators; the sam...

Entropy expressions and their estimators for multivariate distributions

Abstract: Entropy expressions for several continuous multivariate distributions are derived. Point estimation of entropy for the multinormal distribution and for the distribution of order statistics from D.G. Weinman's (Ph.D dissertation, Ariz. State Univ., Tempe, AZ, 1966) exponential distribution is considered. The asymptotic distribution of the uniformly minimum variance unbiased estimator for multinormal entropy is obtained. Simulation results on convergence of the means and variances of these estimators are provided. >

Statistical analysis of the performance of information theoretic criteria in the detection of the number of signals in array processing

Abstract: The performances of the Akaike (1974) information criterion and the minimum descriptive length criterion methods are examined. The events which lead to erroneous decisions are considered, and, on the basis of these events, the probabilities of error for the two criteria are derived. The probabilities of the first two events are derived based on the asymptotic distribution of the sample eigenvalues of an estimated Hermitian matrix. It is further shown that the probabilities of missing and false alarm for these two criteria can be evaluated to a close approximation. Although the derivation of the probabilities of error is based on an asymptotic analysis, the results are confirmed to be in very close agreement with computer simulation results. >

Asymptotic Properties of Statistical Estimators in Stochastic Programming

Abstract: The aim of this article is to investigate the asymptotic behaviour of estimators of the optimal value and optimal solutions of a stochastic program. These estimators are closely related to the $M$-estimators introduced by Huber (1964). The parameter set of feasible solutions is supposed to be defined by a number of equality and inequality constraints. It will be shown that in the presence of inequality constraints the estimators are not asymptotically normal in general. Maximum likelihood and robust regression methods will be discussed as examples.

Asymptotics with increasing dimension for robust regression with applications to the bootstrap

Abstract: A stochastic expansion for $M$-estimates in linear models with many parameters is derived under the weak condition $\kappa n^{1/3}(\log n)^{2/3} \rightarrow 0$, where $n$ is the sample size and $\kappa$ the maximal diagonal element of the hat matrix. The expansion is used to study the asymptotic distribution of linear contrasts and the consistency of the bootstrap. In particular, it turns out that bootstrap works in cases where the usual asymptotic approach fails.

Local and Global Testing of Linear and Nonlinear Inequality Constraints in Nonlinear Econometric Models

Abstract: This paper considers a general nonlinear econometric model framework that contains a large class of estimators defined as solutions to optimization problems. For this framework we derive several asymptotically equivalent forms of a test statistic for the local (in a way made precise in the paper) multivariate nonlinear inequality constraints test H: h(1) > 0 versus K: 1 E RK. We extend these results to consider local hypotheses tests of the form H: hI(1) ? 0 and h2(3) = 0 versus K: 1 E RK. For each test we derive the asymptotic distribution for any size test as a weighted sum of x2-distributions. We contrast local as opposed to global inequality constraints testing and give conditions on the model and constraints when each is possible. This paper also extends the well-known duality results in testing multivariate equality constraints to the case of nonlinear multivariate inequality constraints and combinations of nonlinear inequality and equality constraints. This paper develops three local (in a way to be made precise) asymptotic tests for a set of nonlinear inequality restrictions on the parameters of nonlinear econometric models from the general class of models considered by Burguete, Gallant, and Souza [10] and Gallant [19], henceforth abbreviated as the BGS class of estimators. Models contained in this class are all of the least mean distance estimators and method of moments estimators. See Gallant [19, chap. 3] for a listing of all of the estimators in this class. The results are extended to devising local large-sample tests for combinations of multivariate nonlinear inequality and equality constraints. For the sake of expositional ease and brevity, we present our results for one member of this class: the maximum likelihood (ML) model. Modifications necessary for these procedures to apply to the BGS class of models are stated later in the paper.

Testing for a unit root in the presence of moving average errors

Abstract: SUMMARY In this paper we propose a new approach to testing for unit roots in a time series {Yt} with moving average innovations based on an instrumental variable estimator. If {Yt} is a random walk with moving average innovations, we derive the limiting distribution of the instrumental variable estimator when the estimated model is either (i) the true model, (ii) a random walk with shift in mean, or (iii) a random walk with shift in mean and a linear time trend. These distributions are identical to those tabulated by Dickey & Fuller (1979, 1981) in some cases, and easily transformed, in the spirit of Phillips (1987), to the Dickey & Fuller distributions in others.

A Nonparametric Test For Independence Of A Multivariate Time Series

Abstract: This paper develops a general nonparametric test for the null hypothesis that the vector of time series under scrut/iny is temporally and cross sectionally independent. The null class of the models can be extended to include weak dependence. This test can be used to test the adequacy of a fitted model. As an application of the test we show how we test diagnostically a vector autoregressive model fitted to the given data. This procedure is legitmate because the first order asymptotic distribution of the test statistic is robust to the estimated residual vector.

Shape distributions for landmark data

Abstract: The paper obtains the exact distribution of Bookstein's shape variables under his plausible model for landmark data. We consider its properties including invariances, marginal distributions and the relationship with Kendall's uniform measure. Particular cases for triangles and quadrilaterals are highlighted. A normal approximation to the distribution is obtained, extending Bookstein's result for three landmarks. The adequacy of these approximations is also studied.

Asymptotic normality for deconvolving kernel density estimators

Abstract: Suppose that we have 11 observations from the convolution model Y = X + £, where X and £ are the independent unobservable random variables, and £ is measurement error with a known distribution. We will discuss the asymptotic normality for deconvolving kernel density estimators of the unknown density f x 0 of X by assuming either the tail of the characteristic function of £ behaves as

The maximum of a Gaussian process whose mean path has a maximum, with an application to the strength of bundles of fibres

Abstract: Daniels and Skyrme (1985) derived the joint distribution of the maximum, and the time at which it is attained, of a Brownian path superimposed on a parabolic curve near its maximum. In the present paper the results are extended to include Gaussian processes which behave locally like Brownian motion, or a process transformable to it, near the maximum of the mean path. This enables a wider class of practical problems to be dealt with. The results are used to obtain the asymptotic distribution of breaking load and extension of a bundle of fibres which can admit random slack or plastic yield, as suggested by Phoenix and Taylor (1973). Simulations confirm the approximations reasonably well. The method requires consideration not only of a Brownian bridge but also of an analogous process with covariance function t 1(1 + t 2), .

Bootstrap Bartlett Adjustment in Seemingly Unrelated Regression

Abstract: Seemingly unrelated regression (SUR) is a method introduced by Zellner (1962) for estimating several regression equations simultaneously, a common and important problem in econometrics. This article considers hypothesis testing in SUR using the log-likelihood ratio (LLR) test. Although the asymptotic distribution of this statistic is well known, substantial departures occur in samples of the size commonly employed by economists. Significance levels given by the use of this asymptotic distribution are too low, sometimes by an order of magnitude, leading to the rejection of too many true null hypotheses. A common approach to this problem is to use a Bartlett (1937) adjustment to the LLR statistic, in which the test statistic is multiplied by a factor derived from second-order asymptotics before it is referred to the asymptotic χ2(q) distribution. This article proposes that the Bartlett adjustment factor be computed using Efron's bootstrap (1979, 1982). In the cases examined, this leads to more accu...

Using U Statistics to Derive the Asymptotic Distribution of Fisher's Z Statistic

Abstract: A simple derivation of the asymptotic distribution of Fisher's Z statistic for general bivariate parent distributions F is obtained using U-statistic theory. This method easily reveals that the asymptotic variance of Z generally depends on the correlation ρ and on certain moments of F. It also reveals the particular structure of F that makes the asymptotic variance of Z independent of ρ, and shows that there are many distributions F with this property. The bivariate normal is only one such F.

Bootstrapping Empirical Functions

Abstract: In our previous sections we have established weak convergence of mean residual life, total time of test, Lorenz and Goldie processes to appropriate Gaussian processes. Apart from a few special cases (cf. Section 8), these limiting processes are functions of the underlying distributions. Consequently when testing for statistical hypotheses for example, one would have to compute the resulting limiting distribution for each F of interest. The same is true when trying to use our results for constructing confidence bands for the theoretical functionals of the said empirical processes. This type of problems can be solved by adapting the bootstrap method, proposed by Efron (1979, 1982), to the present situation. As we will see, the bootstrap method is simply a Monte Carlo simulation determined by the given observations.

A goodness-of-fit test using Moran's statistic with estimated parameters

Abstract: SUMMARY Moran's log spacings statistic for testing goodness of fit is shown to have the same asymptotic distribution when parameters must be estimated from the sample as it does when parameters are known, provided the estimation is done efficiently. For small samples, a chi-squared approximation is given and shown to be accurate when estimation is carried out by minimizing Moran's statistic itself regarded as a function of the unknown parameters. A numerical example is given to show the usefulness of Moran's statistic.