摘要:

When a population is assumed to be composed of a finite number of subpopulations, a natural model to choose is the finite mixture model. It will often be the case, however, that the number of component distributions is unknown and must be estimated. This problem can be difficult; for instance, the density of two mixed normals is not bimodal unless the means are separated by at least 2 standard deviations. Hence modality of the data per se can be an insensitive approach to component estimation. We demonstrate that a mixture of two normals divided by a normal density having the same mean and variance as the mixed density is always bimodal. This analytic result and other related results form the basis for a diagnostic and a test for the number of components in a mixture of normals. The density is estimated using a kernel density estimator. Under the null hypothesis, the proposed diagnostic can be approximated by a stationary Gaussian process. Under the alternative hypothesis, components in the mixture will express themselves as major modes in the diagnostic plot. A test for mixing is based on the amount of smoothing necessary to suppress these large deviations from a Gaussian process.

ISSN:0162-1459    

DOI:10.1080/01621459.1994.10476772

出版商:Taylor & Francis Group    

年代:1994

数据来源: Taylor

摘要:

A statistics(·) is computed on spatially indexed data {Xi:i∈D}, whereDis a finite subset of the integer lattice Z2. We propose a simple nonparametric method for estimating the moments (e.g., variance, skewness) ofs(D), using only the observed data at hand. The method uses “replicates” ofs(·) computed on smaller subsets ofD.No explicit knowledge of the underlying spatial dependence mechanism is needed, and the marginal distribution ofXimay also be unknown. The shape ofDcan be quite irregular, ands(·) is allowed to be a general statistic. The proposed estimator is shown to be consistent and asymptotically normal (under mild conditions ons(D) and “mixing” conditions on the strength of spatial dependence). As a numerical example, the estimator is used in assessing the geographic clumping of cancer deaths in the United States.

ISSN:0162-1459    

DOI:10.1080/01621459.1994.10476773

出版商:Taylor & Francis Group    

年代:1994

数据来源: Taylor

摘要:

Suppose the expected value of a response variableYmay be writtenh(Xβ +γ(T)) whereXandTare covariates, each of which may be vector-valued, β is an unknown parameter vector, γ is an unknown smooth function, andhis a known function. In this article, we outline a method for estimating the parameter β, γ of this type of semiparametric model using a quasi-likelihood function. Algorithms for computing the estimates are given and the asymptotic distribution theory for the estimators is developed. The generalization of this approach to the case in whichYis a multivariate response is also considered. The methodology is illustrated on two data sets and the results of a small Monte Carlo study are presented.

ISSN:0162-1459    

DOI:10.1080/01621459.1994.10476774

出版商:Taylor & Francis Group    

年代:1994

数据来源: Taylor

摘要:

Berliner considered Bayesian and likelihood-based approaches for estimation and prediction in a chaotic system with measurement error. This article proposes the use of estimating functions for this problem. Logistic and exponential maps are analyzed. Estimators are shown to be consistent and asymptotically normal. Small-sample behavior is studied with simulations.

ISSN:0162-1459    

DOI:10.1080/01621459.1994.10476775

出版商:Taylor & Francis Group    

年代:1994

数据来源: Taylor

摘要:

Density estimates, such as histograms and more sophisticated versions, are important in applied and theoretical statistics. In applied statistics, a density estimate gives the data analyst a graphical overview of the shape of the distribution. This overview allows the data analyst to arrive immediately at a qualitative impression of the location, scale, and various aspects of the extremes of the distribution. In theoretical statistics, the shape of the density allows the researcher to link the data to families of curves, perhaps indexed parametrically. By estimating a density nonparametrically, certain aspects of the data can be viewed without the restriction of a priori imposing limitations of a class of parametric curves. In this article we introduce density estimation for functions of observations. To motivate the study, one type of function used is the interpoint distance between observations arising in spatial statistics from the fields of biometry and regional science. A second type of function considered is the sum of observations as might occur in claims models in insurance. The nonparametric density estimates are introduced, and certain computational issues are discussed. A central limit theorem for the estimator is provided. This asymptotic result is interesting because, under certain mild conditions, the density estimate enjoys a rate of convergence similar to parametric estimates. This rate of convergence is much faster than the usual rate of convergence in nonparametric density estimation.

ISSN:0162-1459    

DOI:10.1080/01621459.1994.10476776

出版商:Taylor & Francis Group    

年代:1994

数据来源: Taylor

摘要:

A ranked set sample consists entirely of independently distributed order statistics and can occur naturally in many experimental settings, including problems in reliability. When each ranked set from which an order statistic is drawn is of the same size, and when the statistic of each fixed order is sampled the same number of times, the ranked set sample is said to be balanced. Stokes and Sager have shown that the edfFnof a balanced ranked set sample from the cdfFis an unbiased estimator ofFand is more precise than the edf of a simple random sample of the same size. The nonparametric maximum likelihood estimator (MLE)FofFis studied in this article. Its existence and uniqueness is demonstrated, and a general numerical procedure is presented and is shown to converge toF. If the ranked set sample is balanced, it is shown that the EM algorithm, withFnas a seed, converges to the unique solution (F) of the problem's self-consistency equations; the consistency of every iterate of the EM algorithm is also demonstrated. The modifications needed to obtain similar results in unbalanced cases are also discussed. Finally, the results of a simulation study are reported, which support the claim that the nonparametric maximum likelihood estimator, as approximated by an appropriate iterate of the EM algorithm, performs well in the unbalanced case whereFnis inapplicable and performs better thanFnin balanced cases where both estimators exist and can be compared.

ISSN:0162-1459    

DOI:10.1080/01621459.1994.10476777

出版商:Taylor & Francis Group    

年代:1994

数据来源: Taylor

摘要:

A study is made of some unusual location estimates first proposed in 1977 by J. S. Maritz, M. Wu, and R. G. Staudte, Jr., who established some strong robustness properties of these estimates, such as redescending influence functions and, in some cases, full efficiency at the Cauchy model. It is further shown here that the estimates, called M-W-S estimates, are derivable from a rank-based scheme founded on convex functions and hence are regular, are unique, and have associated tests and confidence intervals. The tests belong to a family of signed-rank-type tests, and their distributions have nice combinatorial properties. It is not the case, therefore, that a redescending influence function need imply all the computational irregularities possessed by redescending M estimates. In addition, M-W-S estimates are shown to have breakdown point of .5, a very strong robustness property.

ISSN:0162-1459    

DOI:10.1080/01621459.1994.10476778

出版商:Taylor & Francis Group    

年代:1994

数据来源: Taylor

摘要:

Three classes of testing procedures based on one-step high breakdown point bounded influence estimators, for testing subhypotheses in linear models are developed. These are drop-in-dispersion, Wald-type, and score-type tests. The asymptotic distributions of these testing procedures are obtained under the null hypothesis and under contiguous alternatives. Their stability properties are studied in terms of their influence functions and breakdown points. It is shown that the tests have bounded influence functions. For the Wald-type tests, the level and power breakdowns are determined by the breakdown point of the parameter estimate and the associated variance-covariance matrix. The drop-in-dispersion test exhibits high-level breakdown but not high power breakdown point. Similar behavior is exhibited by the score-type tests. But slight modifications can be made in the construction of the test statistics to ensure high breakdown points in terms of both level and power. An example is given to illustrate the usefulness of high-breakdown testing procedures.

ISSN:0162-1459    

DOI:10.1080/01621459.1994.10476779

出版商:Taylor & Francis Group    

年代:1994

数据来源: Taylor

摘要:

We present a robust version of Mallows'sCPfor regression models. It is defined byRCP=WPσ2- (UP-VP), whereWP= ωiω2ir2iis a weighted residual sum of squares computed from a robust fit of modelP, σ2is a robust and consistent estimator of σ2in the full model, andUPandVPare constants depending on the weight function and the number of parameters in modelP.Good subset models are those withRCPclose toVPor smaller thanVP. When the weights are identically 1,WPbecomes the residual sum of squares of a least squares fit, andRCPreduces to Mallows'sCP. The robust model selection procedure based onRCPallows us to choose the models that fit the majority of the data by taking into account the presence of outliers and possible departures from the normality assumption on the error distribution. Together with the classicalCP, the robust version suggests several models from which we can choose.

ISSN:0162-1459    

DOI:10.1080/01621459.1994.10476780

出版商:Taylor & Francis Group    

年代:1994

数据来源: Taylor

摘要:

Various statistical tools are available for modeling the relationship between response and covariate if the data are fully observable. In the situation of censored data, however, those tools are no longer directly applicable. This article provides an easily implemented methodology for modeling the association, based on censored data. The form of the regression relationship will be completely determined by the data; no assumptions are made about this form. Basic ideas behind the methodology are to transform the observed data in an appropriate simple way and then to apply a locally weighted least squares regression. The proposed estimator involves a variable bandwidth that automatically adapts to the design of the data points. That the methodology is very easy to implement is illustrated by several examples, including simulation studies and an analysis of the Stanford Heart Transplant Data and the Primary Biliary Cirrhosis Data. Several theoretical considerations are reflected in the examples. Finally, some basic asymptotic results are established.

ISSN:0162-1459    

DOI:10.1080/01621459.1994.10476781

出版商:Taylor & Francis Group    

年代:1994

数据来源: Taylor