摘要:

A cohort of individuals exposed to some risk is followed up to a point of timeM, and observations on two random variables (Y, Δ) are recorded for each individual. The variable Δ refers to one of the four possible events that can occur for an individual in the period [0,M]: (i) dies of a specific disease, say cancer, (ii) dies of a natural cause, (iii) withdraws from the study, and (iv) is alive and still under study at timeM. The variableYrefers to the time at which an event occurs. Based on such data fornindividuals, we consider the problem of estimation of a specific occurrence/exposure rate (SOER), which is a risk ratio defined as the ratio of probability of death due to cancer in the interval [0,M]to the mean lifetime of all individuals up to the time pointM. The asymptotic distribution of a nonparametric estimator of SOER is shown to be normal, and the asymptotic variance involves unknown parameters. Various ways of bootstrapping are discussed for construction of confidence intervals for SOER and compared. Some numerical illustrations are provided.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475178

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

This article extends the problem of variable selection to a nonparametric regression model with categorical covariates. Two selection criteria are considered: the cross-validation (CV) criterion and the accumulated prediction error (APE) criterion. We find that, asymptotically, the CV criterion performs well only when the true model is infinite-dimensional, while the APE criterion is appropriate when the true model is finite-dimensional. This is very similar to the case of linear regression model. A simulation study reveals some interesting small-sample properties of these criteria. To be more specific, suppose that we have observations (X1,Y1), …, (Xn, Yn) that are iid random vectors andX= (X(1),X(2), …), where theX(i)'s are categorical. We allowYto be of any type. Now a new observationXhas arrived and we want to predict the correspondingY. Such a framework is more appropriate than regressions with fixed covariates in situations where the covariates are observational rather than being controlled. For instance,Ycould be the time from HIV infection to developing clinical AIDS, and the covariates (mostly categorical or reducible to categorical) could be observations from blood tests, a physical examination, or further personal information, such as sexual practices obtained from an interview. Take another example:Ycould be the premium of an insurance policy with the covariates being the customer's general demographical information. Our goal is to select a subset of covariates that best predictY. We define the true model dimension asd0if the regression functionE(Y|X(1),X(2), …) is ad0-variate function. The main conclusions of the article are: (1) The popular CV criterion performs well only whend0= ∞. (2) There exist other criteria that are more appropriate than CV whend0< ∞. (3) There is no difference between conditional and unconditional prediction errors, as far as asymptotics are concerned. (4) The selection range has to depend on the sample size. In fact, we argue that, for a given sample sizen, we should only select models with the number of covariates not exceeding the order of magnitude ofo(logn). (5) Simulation study indicates that the CV criterion has nice small-sample properties.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475179

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

A common difficulty in regression problems with Poisson (or binomial or other exponential family) response variables is over-dispersion: the scatter around the fitted regression is too large by the standards of Poisson variability. This article concerns the description, estimation, and testing of various patterns of overdispersion, with particular emphasis on the Poisson case. Asymmetric maximum likelihood (AML) is a method of fitting regressions for the conditional percentiles of the response variable as a function of the predictors (e.g., the conditional 90th percentile ofygivenx). Distances between the various regression percentiles give a direct assessment of overdispersion. The discussion is carried through in terms of an archaeological data set, where we see that the counts are overdispersed by a factor of 1.35 in one part of the covariate space, but not at all in another. Moreover, the overdispersion is about 40% larger in the positive response direction than in the negative. The AML estimates are easy to compute and relate nicely to the usual maximum likelihood estimates for generalized linear regression.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475180

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

Estimation of the covariance structure of spatial processes is a fundamental prerequisite for problems of spatial interpolation and the design of monitoring networks. We introduce a nonparametric approach to global estimation of the spatial covariance structure of a random functionZ(x, t) observed repeatedly at timesti(i= 1, …,T) at a finite number of sampling stationsxi(i= 1, 2, …,N) in the plane. Our analyses assume temporal stationarity but do not assume spatial stationarity (or isotropy). We analyze thespatial dispersionsvar(Z(xi, t) −Z(xj, t)) as a natural metric for the spatial covariance structure and model these as a general smooth function of the geographic coordinates of station pairs (xi, xj). The model is constructed in two steps. First, using nonmetric multidimensional scaling (MDS) we compute a two-dimensional representation of the sampling stations for which a monotone function of interpoint distancesδijapproximates the spatial dispersions. MDS transforms the problem into one for which the covariance structure, expressed in terms of spatial dispersions, is stationary and isotropic. Second, we compute thin-plate splines to provide smooth mappings of the geographic representation of the sampling stations into their MDS representation. The composition of this mappingfand a monotone functiongderived from MDS yields a nonparametric estimator of var(Z(xa, t) −Z(xb, t)) for any two geographic locationsxaandxb(monitored or not) of the formg(|f(xa) −f(xb)|). By restricting the monotone functiongto a class of conditionally nonpositive definite variogram functions, we ensure that the resulting nonparametric model corresponds to a nonnegative definite covariance model. We usebiorthogonal grids, introduced by Bookstein in the field of morphometrics, to depict the thin-plate spline mappings that embody the nature of the anisotropy and nonstationarity in the sample covariance matrix. An analysis of mesoscale variability in solar radiation monitored in southwestern British Columbia demonstrates this methodology.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475181

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

A fast algorithm for calculating the nonparametric maximum likelihood estimate (MLE) of a mixing distribution,G(·). in a mixture model is discussed. The literature contains several methods for computing the nonparametric MLE of G, but these methods are slow. In this article we develop an algorithm for maximizing the log-likelihoodl(G) over the family ℊ, of all distribution functions, that yields the nonparametric MLE ofG. In some semiparametric problems, a structural or fixed parameterβis of interest, and we are interested in computing profile likelihood, supGl(G, β), for a grid of values ofβ. The algorithm that we propose is fast enough for this purpose. Examples illustrate and compare the algorithms; one taken from an article by Laird, the common mean problem, and one taken from the literature on optimal experimental design. It is also noted that routines from the literature on semi-infinite programming may be used to compute the profile log-likelihood.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475182

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

Brown and Hettmansperger proposed an affine invariant bivariate analogue of the sign test, the OS test, based on the generalized median of Oja. On the other hand, Oja and Nyblom introduced a family of locally most powerful affine invariant sign tests. In the case of elliptic distributions, the locally most powerful Blumen's test and the OS test are shown to be asymptotically equivalent. Formulas for calculating asymptotic relative efficiencies of the OS test and the Oja bivariate median are given. It is shown that if the contours of a distribution are of a similar shape, the relative efficiencies of the OS test and Blumen's test depend on the distribution only through the shape of the contours. For the power family of contours |x1|p+ |x2|p=c,p> 0, numerical calculations show that the efficiency of the OS test relative to Blumen's test attains its minimum 1 asp= 2 (spherical/elliptic case) and increases to infinity asp→ 0. In the bivariate Laplace case with independent marginals (p= 1) as well as in the casep= ∞ the relative efficiency is 1.028.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475183

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

For comparing responses in two groups of subjects observed repeatedly, we propose a group sequential procedure based on linear rank statistics. The asymptotic normality of the sequentially computed linear rank statistics is obtained, and construction of the group sequential boundaries is based on this distribution theory. By virtue of this asymptotic approximation, the proposed procedure can be applied to interim analyses with either continuous or discrete repeated measurements. Even for staggered patient entry, simulation results suggest the theory is approximately correct. It can also be useful for testing the equality of two changes and rates of change, as well as the equality of two means of the responses. This procedure is illustrated with a real example.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475184

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

In this article the notion of leverage of a design point when fitting a linear regression model is interpreted geometrically. In the case of least squares fitting, the leverage indicators based on the diagonal of the hat matrix are widely applied. By interpreting these hat matrix indicators geometrically, leverage can be generalized to groups of design points, as well as to other methods of fitting. The article introduces a leverage indicator that is appropriate forL1regression and discusses some aspects of this new diagnostic. It is shown that, in the case ofL1regression, the leverage indicators have a precise interpretation. They tell us about the breakdown and/or the exactness of fit. As an application, the article considers the maximal possible breakdown value ofL1regression and the choice of designs that achieve this maximum.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475185

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

A relationship between the Kruskal–Wallis nonparametric statistical test onksamples and the Borda positional method for voting is established and then exploited to gain a complete analysis of the counterintuitive results and statistical orderings that arise when a set of data is restricted to various subsets of theksamples. This is done by introducing and examining the “dictionary” of possible orderings of the samples occurring with the Kruskal-Wallis procedure. It is shown that using ranks, as opposed to any other weights, minimizes the number and kinds of paradoxes that can arise from examining subsets of the data—projection paradoxes. The idea of a dictionary for a statistical procedure is discussed. The dictionaries for the Deshpandé classℒof statistical procedures (including Bhapkar'sVtest and Deshpandé'sLtest) are computed. An estimate for the relative sizes of the number of paradoxes for the Kruskal-Wallis test and any other such test—by comparing relative sizes of their dictionaries—is given in a few cases. An analysis is given of the method of selecting the “best” of several samples by recursively narrowing down the size of the set of samples one is “reasonably certain” must contain the optimal choice. It is shown that the best sample may be dependent on which recursive method is chosen: For the same data, different recursive methods can yield different outcomes.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475186

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10475187

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor