摘要:

The folklore on one-step estimation is that it inherits the breakdown point of the preliminary estimator and yet has the same large sample distribution as the fully iterated version as long as the preliminary estimate converges faster thann–1/4, wherenis the sample size. We investigate the extent to which this folklore is valid for one-step GM estimators and their associated standard errors in linear regression. We find that one-step GM estimates based on Newton-Raphson or Scoring inherit the breakdown point of high breakdown point initial estimates such as least median of squares provided the usual weights that limit the influence of extreme points in the design space are based on location and scatter estimates with high breakdown points. Moreover, these estimators have bounded influence functions, and their standard errors can have high breakdown points. The folklore concerning the large sample theory is correct assuming the regression errors are symmetrically distributed and homoscedastic. If the errors are asymmetric and homoscedastic, Scoring still provides root-nconsistent estimates of the slope parameters, but Newton-Raphson fails to improve on the rate of convergence of the preliminary estimates. If the errors are symmetric and heteroscedastic, Newton-Raphson provides root-nconsistent estimates, but Scoring fails to improve on the rate of convergence of the preliminary estimate. Our primary concern is with the stability of the inferences associated with the estimates, not merely with the point estimates themselves. To this end we define the notion of standard error breakdown, which occurs if the estimated standard deviations of the parameter estimates can be driven to zero or infinity, and study the large sample validity of the standard error estimates. A real data set from the literature illustrates the issues.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475224

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

In this article a method for obtaining tests for overdispersion with respect to a natural exponential family is derived. The tests are designed to be powerful against arbitrary alternative mixture models where only the first two moments of the mixed distribution are specified. Various tests for extra-Poisson and extra-binomial variation are obtained as special cases; the use of a particular test may be motivated by a consideration of the mechanism through which the overdispersion may arise. The common occurrence of extra-Poisson and extra-binomial variation has been noted by several authors. However, the Poisson and binomial models remain valid in many instances and, because of their simplicity and appeal, it is of real interest to ascertain when they apply. This paper develops a unifying theory for testing for overdispersion and generalizes tests previously derived, including those by Fisher (1950), Collings and Margolin (1985), and Prentice (1986). It also shows the Pearson statistic to be a score test for overdispersion in a certain situation.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475225

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

In this article, procedures analogous to Karl Pearson's well-known chi-squared goodness-of-fit test for a simple null hypothesis are developed under the random censorship model. It is shown that one straightforward analog of Pearson's statistic is diminished in applicability due to the form of its limiting distribution. This leads to the development of an asymptotically exact test based on a Wald-type statistic with a chi-squared limiting null distribution. This test is compared and contrasted theoretically and via a simulation with Akritas’ test with respect to significance levels, asymptotic local powers, and finite sample powers. The general conclusions from the simulation study are that the proposed test usually achieves the desired significance levels when the probability of observing a censored or an uncensored value in the last interval is not small, whereas Akritas’ test tends to be a bit anticonservative. On the other hand, Akritas’ test is more powerful than the proposed test in a model with Weibull lifetimes, but in models with exponential and normal lifetimes neither test dominates the other.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475226

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

Numerous goodness-of-fit tests with asymptotic chi-squared distributions have been proposed for discrete multivariate data, and there has been much discussion about using asymptotic results for computing critical values when there are small expected cell values. Although exact methods would be preferred in these situations, it generally is believed that such methods are computationally intractable. We propose methods for calculating exact distributions and significance levels for goodness-of-fit statistics that are computationally feasible over a wide range of models. In particular, the distribution for a simple multinomial model can be evaluated in polynomial time. For composite null hypotheses, we calculate the distribution conditional on the sufficient statistics for the nuisance parameters. We calculate the characteristic function of a distribution and invert the characteristic function using the fast Fourier transform (FFT). Our approach emphasizes the relationship between exact methods and probability formulas. Our technique, transforming the domain of the problem, is interesting for two reasons: First, algorithms that use the FFT and the convolution theorem are efficient for calculating the distribution of sums of independent statistics; and second, less storage is needed when working in the frequency domain than in the probability domain. The algorithms can be applied to general goodness-of-fit statistics and are parallelizable.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475227

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

Consider anr×ccontingency table under the full multinomial model in which each classification is ordered. The problem is to test the null hypothesis of independence against the alternative that all local log odds ratios are nonnegative with at least one local log odds ratio positive. A number of tests have been proposed for this problem, including the Goodman–Kruskal gamma test; a family of linear tests studied by Agresti, Mehta, and Patel; and a test based on“C–D,”the difference of concordant and discordant pairs in the table. In this article we show that all of these tests can be improved on in some sense for most cases. In fact the preceding tests sometimes are inadmissible in a strict sense. Furthermore, we show by example that in some cases improved tests can yield substantially improved power functions. We suggest a test based on a linear statistic similar to that presented by Agresti, Mehta, and Patel but that is followed up with a test that orders points by their probabilities on sample points where the linear test would randomize. This latter test compares favorably with competitors and has optimal theoretical properties. Exact tests, which entail auxiliary randomization, are discussed, as are thepvalues of the test procedures, which do not use auxiliary randomization.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475228

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

For latent class analysis, a widely known statistical method for the unmixing of an observed frequency table into several unobservable ones, a flexible model is presented in order to restrain the unknown class sizes (mixing weights) and the unknown latent response probabilities. Two systems of basic equations are stated such that they simultaneously allow parameter fixations, the equality of certain parameters as well as linear logistic constraints of each of the original parameters. The maximum likelihood equations for the parameters of this “linear logistic latent class analysis” are given, and their estimation by means of the EM algorithm is described. Further, the criteria for their local identifiability and statistical tests (Pearson- and likelihood-ratio-χ2) for goodness of fit are outlined. The practical applicability of linear logistic latent class analysis is demonstrated by three examples: mixed logistic regression, a mixed Bradley-Terry model for paired comparisons with ties, and a local dependence latent class model in which the departure from stochastic independence is covered by a single additional parameter per class.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475229

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

This article presents an efficient method for computing exact conditional distributions of the sufficient statistics for the parameters of four polytomous response models. For nominal response, two baseline category logit models and, for ordinal response, two adjacent categories logit models are considered. The method consists of recursive generation of the joint distribution of the sufficient statistics, augmented by a technique to slice off portions of the evolving sample space so as to eventually yield the required conditional distribution. Two actual data sets are analyzed to illustrate the method.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475230

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

A solution to multivariate state-space modeling, forecasting, and smoothing is discussed. We allow for the possibilities of nonnormal errors and nonlinear functionals in the state equation, the observational equation, or both. An adaptive Monte Carlo integration technique known as the Gibbs sampler is proposed as a mechanism for implementing a conceptually and computationally simple solution in such a framework. The methodology is a general strategy for obtaining marginal posterior densities of coefficients in the model or of any of the unknown elements of the state space. Missing data problems (including thek-step ahead prediction problem) also are easily incorporated into this framework. We illustrate the broad applicability of our approach with two examples: a problem involving nonnormal error distributions in a linear model setting and a one-step ahead prediction problem in a situation where both the state and observational equations are nonlinear and involve unknown parameters.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475231

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

A family of multivariate dynamic generalized linear models is introduced as a general framework for the analysis of time series with observations from the exponential family. Besides common conditionally Gaussian models, this article deals with univariate models for counted and binary data and, as the most interesting multivariate case, models for nonstationary multicategorical time series. For univariate responses, a related yet different class of models has been introduced in a Bayesian setting by West, Harrison and Migon. Assuming conjugate prior-posterior distributions for the natural parameter of the exponential family, they derive an approximate filter for estimation of time-varying states or parameters. However, their method raises some problems; in particular, in extending it to the multivariate case. A different approach to filtering and smoothing is chosen in this article. To avoid a full Bayesian analysis based on numerical integration, which becomes computationally critical for higher dimensions, we propose to estimate time-varying parameters by posterior modes. A generalization of the extended Kalman filter and smoother for conditionally Gaussian observations is suggested for approximate posterior mode estimation. For the purpose of comparison, it is applied to data sets analyzed by the authors mentioned earlier. The quality of approximation is also studied by simulation experiments, indicating good estimation behavior, and an application to multicategorical business test data is given.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475232

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

This article deals with a novel utility function to design experiments in a Bayesian framework. This utility function is a linear combination of the gain in Shannon information and of the total outcome of the experiment, defined as the sum of observed values in the dependent variable of a linear model. Thus the expected posterior utility to be maximized is a combination of the BayesD-optimality criterion and the posterior expectation of the total output. Earlier studies have shown that Bayesian parallels of the classicalD-, A-, andE-optimal designs can be obtained by considering utility or loss functions concerned with efficient estimation of the parameters of interest. A different requirement that might be desirable in applied problems is to combine the accuracy of parameter estimation with the maximization of experimental output. The utility function considered here does this. We look at the implications of using this utility in deriving designs in the context of hierarchical linear models. In particular, designs for the one-way ANOVA and the straight-line models are obtained. New designs are determined analytically in some cases; in other cases we show that they can be computed by simple algorithms. An example is given to illustrate the elicitation of the parameter controlling the relative weight given to the two components of the utility.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475233

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor