Automatic Smoothing for Poisson Regression

Abstract

Adaptive choice of smoothing parameters for nonparametric Poisson regression (O'Sullivan et al., 1986 O'Sullivan , F. , Yandell , B. S. , Raynor , W. J., Jr. ( 1986 ). Automatic smoothing of regression functions in generalized linear models . J. Amer. Statist. Assoc. 81 : 96 – 103 . [CSA] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) is considered in this article. A computable approximation of the unbiased risk estimate (AUBR) for Poisson regression is introduced. This approximation can be used to automatically tune the smoothing parameter for the penalized likelihood estimator. An alternative choice is the generalized approximate cross validation (GACV) proposed by Xiang and Wahba (1996 Xiang , D. , Wahba , G. ( 1996 ). A generalized approximate cross validation for smoothing splines with non-Gaussian data . Statist. Sinica 6 (3): 675 – 692 .[Web of Science ®] , [Google Scholar]). Although GACV enjoys a great success in practice when applying for nonparametric logisitic regression, its performance for Poisson regression is not clear. Numerical simulations have been conducted to evaluate the GACV and AUBR based tuning methods. We found that GACV has a tendency to oversmooth the data when the intensity function is small. As a consequence, we suggest tuning the smoothing parameter using AUBR in practice.     

The Siable-Law Model of Stock Returns

This study investigates the tail shapes of empirical distributions of returns on an extensive group of common stocks. The tails of the return distributions are found to be thinner than those of infinite variance stable distributions. Therefore, although homogeneity is evident in general, economic and statistical inferences drawn from stable-law parameters estimated from samples of stock returns may be misleading. This is in spite of the apparent overall similarity (in shape) between empirical and stable distributions.     

Inference for Income Distributions Using Grouped Data

We develop a general approach to estimation and inference for income distributions using grouped or aggregate data that are typically available in the form of population shares and class mean incomes, with unknown group bounds. We derive generic moment conditions and an optimal weight matrix that can be used for generalized method-of-moments (GMM) estimation of any parametric income distribution. Our derivation of the weight matrix and its inverse allows us to express the seemingly complex GMM objective function in a relatively simple form that facilitates estimation. We show that our proposed approach, which incorporates information on class means as well as population proportions, is more efficient than maximum likelihood estimation of the multinomial distribution, which uses only population proportions. In contrast to the earlier work of Chotikapanich, Griffiths, and Rao, and Chotikapanich, Griffiths, Rao, and Valencia, which did not specify a formal GMM framework, did not provide methodology for obtaining standard errors, and restricted the analysis to the beta-2 distribution, we provide standard errors for estimated parameters and relevant functions of them, such as inequality and poverty measures, and we provide methodology for all distributions. A test statistic for testing the adequacy of a distribution is proposed. Using eight countries/regions for the year 2005, we show how the methodology can be applied to estimate the parameters of the generalized beta distribution of the second kind (GB2), and its special-case distributions, the beta-2, Singh–Maddala, Dagum, generalized gamma, and lognormal distributions. We test the adequacy of each distribution and compare predicted and actual income shares, where the number of groups used for prediction can differ from the number used in estimation. Estimates and standard errors for inequality and poverty measures are provided. Supplementary materials for this article are available online.     

Properties of Designs in Experiments on Spatial Lattices

ABSTRACT

When spatial variation is present in experiments, it is clearly sensible to use designs with favorable properties under both generalized and ordinary least squares. This will make the statistical analysis more robust to misspecification of the spatial model than would be the case if designs were based solely on generalized least squares. In this article, treatment information is introduced as a way of studying the ordinary least squares properties of designs. The treatment information is separated into orthogonal frequency or polynomial components which are assumed to be independent under the spatial model. The well-known trend-resistant designs are those with no treatment information at the very low order frequency or polynomial components which tend to have the higher variances under the spatial model. Ideally, designs would be chosen with all the treatment information distributed at the higher-order components. However, the results in this article show that there are limits on how much trend resistance can be achieved as there are many constraints on the treatment information. In addition, appropriately chosen Williams squares designs are shown to have favorable properties under both ordinary and generalized least squares. At all times, the ordinary least squares properties of the designs are balanced against the generalized least squares objectives of optimizing neighbor balance.     

Projection Mean–Variance Bounds on Expectations of kth Record Values from Restricted Families

We present sharp mean–variance bounds for expectations of kth record values based on distributions coming from restricted families of distributions. These families are defined in terms of convex or star ordering with respect to generalized Pareto distribution. The bounds for expectations of kth record values from DD, DFR, DDA, and DFRA families are special cases of our results. The bounds are derived by application of the projection method.     

Asymptotic Normality of Extreme Quantile Estimators Based on the Peaks-Over-Threshold Approach

The POT (Peaks-Over-Threshold) approach consists of using the generalized Pareto distribution (GPD) to approximate the distribution of excesses over thresholds. In this article, we establish the asymptotic normality of the well-known extreme quantile estimators based on this POT method, under very general assumptions. As an illustration, from this result, we deduce the asymptotic normality of the POT extreme quantile estimators in the case where the maximum likelihood (ML) or the generalized probability-weighted moments (GPWM) methods are used. Simulations are provided in order to compare the efficiency of these estimators based on ML or GPWM methods with classical ones proposed in the literature.     

Modeling Association Between Two or More Categorical Variables that Allow for Multiple Category Choices

Multiple-response (or pick any/c) categorical variables summarize responses to survey questions that ask “pick any” from a set of item responses. Extensions to loglinear model methodology are proposed to model associations between these variables across all their items simultaneously. Because individual item responses to a multiple-response categorical variable are likely to be correlated, the usual chi-square distributional approximations for model-comparison statistics are not appropriate. Adjusted statistics and a new bootstrap procedure are developed to facilitate distributional approximations. Odds ratio and standardized Pearson residual measures are also developed to estimate specific associations and examine deviations from a specified model.     

Bounds for Expectations of Differences of Generalized Order Statistics Based on General and Life Distributions

Let X _? ^(r), r ≥ 1, denote generalized order statistics based on an arbitrary distribution function F with finite pth absolute moment for some 1 ≤ p ≤ ∞. We present sharp upper bounds on E(X _? ^(s) ? X _? ^(r)), 1 ≤ r < s, for F being either general or life distribution. The bounds are expressed in various scale units generated by pth central absolute or raw moments of F, respectively. The distributions achieving the bounds are specified.     

Generalized Inferences on the Common Mean in MANOVA Models

This article presents procedures for testing hypothesis and interval estimation of the common mean vector in MANOVA models when the covariance matrices are unknown and unequal. The methods are based on the concepts of generalized p-value and generalized confidence interval. Some important statistical properties of the exact test and confidence region are given. For two multivariate normal populations, a minor modification to the combined tests given by Zhou and Mathew (1994a Zhou , L. P. , Mathew , T. ( 1994a ). Combining independent tests in multivariate linear models . J. Multivariate Anal. 51 : 265 – 276 . [Google Scholar]) is proposed. Some simulation results to compare the performance of the proposed tests with others are reported. The simulation results indicate that new tests have significant gain in the power.