摘要:

Consider a problem in which you and a group of other experts must report your individual predictive distributions for an observable random variableXto some decision maker. Suppose that the report of each expert is assigned a prior weight by the decision maker and that these weights are then updated based on the observed value ofX. In this situation you will try to maximize your updated, or posterior, weight by appropriately choosing the distribution that you report, rather than necessarily simply reporting your honest predictive distribution. We study optimal reporting strategies under various conditions regarding your knowledge and beliefs aboutXand the reports of the other experts, and under various utility functions for your posterior weight. We present the only utility functions for which it is always optimal to report your honest predictive distribution. Attention is restricted to problems in whichXcan take only a finite number of values.

ISSN:0162-1459    

DOI:10.1080/01621459.1989.10478758

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

Tests are proposed for detecting possible changes in parameters when the observations are obtained sequentially in time. While deriving the tests the alternative one has in mind specifies the parameter process as a martingale. The distribution theory of these tests relies on the large-sample results; that is, only the limiting null distributions are known (except in very special cases). The main tool in establishing these limiting distributions is weak convergence of stochastic processes. Suppose that we have vector-valued observationsx1, …,xnobtained sequentially in time (or ordered in some other linear fashion). Their joint distribution is described by determining the initial distribution forx1and the conditional distribution for eachxkgiven the past up toxk–1. Suppose further that these distributions depend on ap-dimensional parameter vectorθ. At least locally (i.e., in a short time period) this may be more or less legitimate. In the long run, however, the possibility of some changes in the observation-generating process should be taken into account. Specifically, it is assumed here that those changes occur through a parameter variation in the form of a martingale. The martingale specification has an advantage of covering several types of departure of constancy: for example, a single jump at an unknown time point (the so-called change-point model) or slow random variation (typically random walk). The tests are derived by first finding the locally most powerful test against a martingale-type alternative when the starting value of the parameter process is known. After some simplification a test having a known numerically tractable limiting distribution is developed. When the starting point is unknown an efficient estimate is substituted for it. In addition, the corresponding limiting distribution is established. The proposed tests turn out to be based on cumulative sums of the score function (the derivative of the log-likelihood).

ISSN:0162-1459    

DOI:10.1080/01621459.1989.10478759

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

The threshold autoregressive model is one of the nonlinear time series models available in the literature. It was first proposed by Tong (1978) and discussed in detail by Tong and Lim (1980) and Tong (1983). The major features of this class of models are limit cycles, amplitude dependent frequencies, and jump phenomena. Much of the original motivation of the model is concerned with limit cycles of a cyclical time series, and indeed the model is capable of producing asymmetric limit cycles. The threshold autoregressive model, however, has not received much attention in application. This is due to (a) the lack of a suitable modeling procedure and (b) the inability to identify the threshold variable and estimate the threshold values. The primary goal of this article, therefore, is to suggest a simple yet widely applicable model-building procedure for threshold autoregressive models. Based on some predictive residuals, a simple statistic is proposed to test for threshold nonlinearity and specify the threshold variable. Some supplementary graphic devices are then used to identify the number and locations of the potential thresholds. Finally, these statistics are used to build a threshold model. The test statistic and its properties are derived by simple linear regression. Its performance in the finite-sample case is evaluated by simulation and real-world data analysis. The statistic performs well as compared with an alternative test available in the literature. Further applications of threshold autoregressive models are also suggested, including handling heterogeneous time series and modeling random processes with periodic variances whose periodicity is not fixed. The latter phenomenon is commonly encountered in practice, especially in econometrics and biological sciences.

ISSN:0162-1459    

DOI:10.1080/01621459.1989.10478760

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

Intuition suggests that altering the covariance structure of a parametric model for repeated-measures data alters the variances of the model's estimated mean parameters. The purpose of this article is to sharpen such intuition for a family of growth-curve models with differing numbers of random effects for the individual sampling units and with a fixed structure on the mean. For every member of this family, the maximum likelihood (ML) estimator of the fixed effects is identical to the ordinary least squares (OLS) estimator. In addition, simple closed-form ML and restricted maximum likelihood estimators for the variance and covariance parameters exist for every member. As a consequence, closed-form expressions for the estimated variance-covariance matrix of the OLS estimator of the fixed effects also exist for the entire family. We derive explicit relationships between the variance and covariance parameter estimators from different members of the family and thereby extend some familiar results. For example, it is well known that for balanced and complete longitudinal designs the compound symmetry assumption for the covariance structure of the serial observations (i.e., assuming one random effect for each sampling unit) yields a more precise estimate of the slope of the population growth curve than of its intercept. It is also well known that for such designs the diagonal covariance structure assumption of OLS regression (i.e., no random effects for the units) yields a more precise estimate of the intercept than does the compound symmetry assumption, and a less precise estimate of the slope. We extend such relationships as these to growth-curve models whose covariance structures are of increasing linear complexity (i.e., assuming two or more random effects for each of the sampling units). We find that, in general, the variance of the OLS estimator depends strongly on assumed covariance structure. We also find that, specifically, the linear growth-curve assumption (i.e., two random effects for each sampling unit) is a conservative assumption, in that such will not give misleadingly small variance estimates for both intercept and slope even if a more complex covariance structure actually holds. We illustrate these points with a data example. In the conclusion, we extend our results by showing how they may apply to more general families of balanced growth-curve models that allow wider ranges of covariance structures and designs for the random effects, such as when different population subgroups require different covariance structures. We also address several computational issues involved in transformations of growth-curve models to canonical form.

ISSN:0162-1459    

DOI:10.1080/01621459.1989.10478761

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

This article provides a reconsideration of Kramer's (1980) results on least squares estimation in linear models with autocorrelated errors. Kramer's results are shown to be dependent on his measure of efficiency and to understate the advantages of correcting for autocorrelation.

ISSN:0162-1459    

DOI:10.1080/01621459.1989.10478762

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

In this article the bivariate location problem is treated. New appealing bivariate analogs of the univariate sign tests are proposed for testing the null hypothesis concerning the unknown symmetry center. These tests remain unaltered under any nonsingular linear transformation. From these promising findings a whole family of locally most powerful invariant sign tests is introduced. The tests proposed earlier (Blumen 1958; Hodges 1955) are specific members of this family. For example, Blumen's test appears to be optimal against bivariate normal (or any other elliptic) alternatives. The limiting distributions are derived both under the null hypothesis and under the contiguous alternatives. These limiting distributions are then used to derive asymptotic relative efficiencies. It is found that Blumen's test has the efficiency .785 relative to Hotelling's test against bivariate normal alternatives. For other locally most powerful sign tests the corresponding efficiency depends on the significance level and the power, but not too strongly. In fact, the value .785 also serves as an approximation for other sign tests. The lower bound for the efficiency of Blumen's test relative to Hotelling's test is established against elliptic alternatives. The restriction to unimodal elliptic alternatives increases the lower bound to ½. Finally, some results on Hodges's test are included. Bivariate sign tests can be applied, for example, in paired comparison situations with a bivariate response variable. The hypothesis of no difference between two treatments then implies that the paired response differences are symmetric about the origin, which can be tested by using bivariate sign tests. No extra assumptions concerning the unknown bivariate distributions are needed. In addition, other examples of applications are included. The proposed tests use the direction angles of the observations. Therefore, they can be used also in testing the uniformity (and a kind of symmetry) of a circular distribution. In Tables 1, 2, and 3 critical values for three different sign tests are given.

ISSN:0162-1459    

DOI:10.1080/01621459.1989.10478763

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

This article discusses efficiency properties of some common stratified estimators, in the context of a superpopulation model, relative to the greatest lower bound on the variance of the Horvitz—Thompson estimator. The estimators discussed use both Dalenius—Hodges and model-based survey sampling (MBSS) stratification and a variety of sample allocation methods, including optimum, proportionate, and uniform sample allocation. The main result is that both Dalenius—Hodges stratification with optimal allocation and MBSS stratification with uniform allocation yield approximately minimum variance estimators, with convergence to the lower bound at rateO(L–2), whereLis the number of strata. Since this lower bound has been shown to hold for many types of finite population estimators, the results derived here have broad implications. A series of examples is presented in which Dalenius—Hodges/optimum allocation is consistently more efficient than MBSS/uniform allocation.

ISSN:0162-1459    

DOI:10.1080/01621459.1989.10478764

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

In estimating means, totals, and other parameters for small domains of a finite population, the survey statistician is usually faced with a domain sample size that is random rather than controlled at the selection stage. Often a sensible approach is to make design-based inference conditionally on the realized sample size in the domain (nd). In this article, we suggest and analyze some small domain estimators and their design-based conditional confidence intervals. The conditional outlook leads us to some new small domain estimators that (a) are based on regression of pertinent auxiliary information and are therefore efficient to the extent that the auxiliary information is strong; (b) are nearly design unbiased, conditionally onnd, as well as unconditionally; and (c) give rise to design-based confidence intervals that are valid conditionally as well as unconditionally. The conditional properties of these new regression estimators are first derived theoretically, then confirmed through a Monte Carlo simulation based on Canadian business survey data. The well-known synthetic estimator has the attractive stability property that the variance is often very small, but it is avoided by many survey statisticians owing to the possibility of a substantial design bias. Several “improved” methods aim at reduced bias yet good stability by building the estimator as the sum of a synthetic estimator term and an “adjustment term.” In particular, the adjustment term can be made to eliminate the bias almost completely, as in the regression estimation method of Särndal (1981, 1984). However, such a bias-removing adjustment term adds a considerable variance component when the domain sample sizendis very small. This is viewed as the price to pay for “measurability”—that is, being able to calculate, from the sample itself, an estimate of the sampling error and a valid confidence interval. In this article we construct a new regression estimator in which the adjustment term is “dampened” for particularly smallnd. This approach trades a mild bias for a decreased variance and effectively removes the likelihood of “wild” estimates while still admitting a valid design-based confidence interval. It is recommended for practitioners who wish to be able to calculate design-based precision in the form of a valid confidence interval. In our Monte Carlo study, we compare our approach to other methods, including another “synthetic term plus adjustment term method” suggested by Fuller and Harter (1987), in which the emphasis is on obtaining good mean squared error performance, rather than on nearly complete bias removal through the adjustment term. The efficiency gains realized by the latter method over our dampened regression estimator method were not substantial in our Monte Carlo study.

ISSN:0162-1459    

DOI:10.1080/01621459.1989.10478765

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

Smoothing splines have a penalized likelihood motivation (Good and Gaskins 1971) allowing direct application to nonparametric regression in likelihood-based models. The notion of a weighted likelihood for the nonparametric kernel estimation of a regression function is proposed, generalizing the local likelihood theory of Tibshirani and Hastie (1987). Let the data be of the form (xi, Yi) (i= 1, …,n), wherexi∈[0, 1]dare lattice points and theYiare independent random variables from a family of distributions with parameterλi= g(xi), withghaving continuous partial derivatives of orderk≥ 2. The goal is to arrive at a nonparametric estimate λoof λo=g(xo) for a fixed pointxo∈[0, 1]d.We consider the estimator λothat maximizes the weighted likelihood functionW(λ) = Σni=1W[(xo– xi)/b] logf(Yi:λ), withfthe density ofYi, Wa symmetric kernel with compact support, andbthe bandwidth that controls the degree of smoothing. Sufficient conditions for consistency and asymptotic normality of λoare given. If theYiare normal random variables with meanλiand equal variance, then λois the kernel estimator of Priestly—Chao (1972). It is a weighted average ofYicorresponding toxiin a neighborhood ofxo. The kernel governs the weights and the bandwidth controls the size of the neighborhood. The kernel estimator of the relative risk function is developed for censored survival times under the assumption of the Cox proportional hazards model. The weighted likelihood approach based on the full likelihood is illustrated with real and simulated data.

ISSN:0162-1459    

DOI:10.1080/01621459.1989.10478766

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

A kernel estimate of a curve that uses an adaptive procedure for local selection of the bandwidth is considered here. A two-step procedure is proposed for estimating the local bandwidth that minimizes the mean squared error (MSE) of a kernel estimator for nonparametric regression. First, a consistent estimate of the exact MSE is constructed. Then the bandwidth that minimizes the estimate of the MSE is calculated. Sufficient conditions under which this bandwidth is asymptotically optimal and normally distributed are given. The local bandwidth selection procedure was implemented on some simulated data and compared to a global bandwidth selection procedure proposed by Rice (1984b). A 68%–91% reduction in the average MSE of a kernel estimator was realized with the local bandwidth selection procedure. Such a scheme was also studied by Müller (1985) and termed a direct pilot estimator approach. Müller derived sufficient conditions similar to those presented here, under which the direct pilot estimator approach provides a consistent estimator of the local bandwidth. His result is slightly more general in the specification of the interval that is searched for the optimizing bandwidth. The asymptotic normality of the optimizing bandwidth and the simulation results presented here have not to our knowledge appeared in the statistical literature. Müller and Stadtmüller (1987) proposed a local bandwidth selection procedure for kernel estimates that is based on the asymptotic expression for the MSE of the kernel estimator. Our procedure is based upon the finite sample expression for the MSE. It remains to be shown how the two procedures compare when applied to small data sets.

ISSN:0162-1459    

DOI:10.1080/01621459.1989.10478767

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor