Goodness‐of‐fit tests for parametric models based on biased samples

The authors study the problem of checking the adequacy of a parametric model for a distribution using several possibly censored weight biased samples. They discuss identifiability problems related to the underlying distribution and the distributions of the biased samples. They propose a test statistic based on the supremum of the weighted aggregated martingale residual processes from a number of such samples. Both numerical and graphical procedures are discussed, which the authors apply to do model checking for oil exploration drilling data.     

Semiparametric linear transformation models for current status data

Current status data arise when the death of every subject in a study cannot be determined precisely, but is known only to have occurred before or after a random monitoring time. The authors discuss the analysis of such data under semiparametric linear transformation models for which they propose a general inference procedure based on estimating functions. They determine the properties of the estimates they propose for the regression parameters of the model and illustrate their technique using tumorigenicity data.     

General mixed‐data model: Extension of general location and grouped continuous models

The authors propose a general model for the joint distribution of nominal, ordinal and continuous variables. Their work is motivated by the treatment of various types of data. They show how to construct parameter estimates for their model, based on the maximization of the full likelihood. They provide algorithms to implement it, and present an alternative estimation method based on the pairwise likelihood approach. They also touch upon the issue of statistical inference. They illustrate their methodology using data from a foreign language achievement study.     

The weighted likelihood

The authors consider a weighted version of the classical likelihood that applies when the need is felt to diminish the role of some of the data in order to trade bias for precision. They propose an axiomatic derivation of the weighted likelihood, for which they show that aspects of classical theory continue to obtain. They suggest a data‐based method of selecting the weights and show that it leads to the James‐Stein estimator in various contexts. They also provide applications.     

Small area estimation of poverty indicators

The authors propose to estimate nonlinear small area population parameters by using the empirical Bayes (best) method, based on a nested error model. They focus on poverty indicators as particular nonlinear parameters of interest, but the proposed methodology is applicable to general nonlinear parameters. They use a parametric bootstrap method to estimate the mean squared error of the empirical best estimators. They also study small sample properties of these estimators by model‐based and design‐based simulation studies. Results show large reductions in mean squared error relative to direct area‐specific estimators and other estimators obtained by “simulated” censuses. The authors also apply the proposed method to estimate poverty incidences and poverty gaps in Spanish provinces by gender with mean squared errors estimated by the mentioned parametric bootstrap method. For the Spanish data, results show a significant reduction in coefficient of variation of the proposed empirical best estimators over direct estimators for practically all domains. The Canadian Journal of Statistics 38: 369–385; 2010 © 2010 Statistical Society of Canada     

Bayesian prediction of waiting times in stochastic models

The authors show how saddlepoint techniques lead to highly accurate approximations for Bayesian predictive densities and cumulative distribution functions in stochastic model settings where the prior is tractable, but not necessarily the likelihood or the predictand distribution. They consider more specifically models involving predictions associated with waiting times for semi‐Markov processes whose distributions are indexed by an unknown parameter θ. Bayesian prediction for such processes when they are not stationary is also addressed and the inverse‐Gaussian based saddlepoint approximation of Wood, Booth & Butler (1993) is shown to accurately deal with the nonstationarity whereas the normal‐based Lugannani & Rice (1980) approximation cannot, Their methods are illustrated by predicting various waiting times associated with M/M/q and M/G/1 queues. They also discuss modifications to the matrix renewal theory needed for computing the moment generating functions that are used in the saddlepoint methods.     

Adaptive estimation in partially linear autoregressive models

The authors consider a partially linear autoregressive model and construct kernel‐based estimates for both the parametric and nonparametric components. They propose an estimation procedure for the model and illustrate it through simulated and real data. Their work shows that the proposed estimation procedure not only has good asymptotic properties but also works well numerically. It also suggests that a partially linear autoregression is more appropriate than a completely nonparametric autoregression for some sets of data.     

A robust nonparametric estimation of the autoregression function under an ergodic hypothesis

The authors propose a family of robust nonparametric estimators for regression or autoregression functions based on kernel methods. They show the strong uniform consistency of these estimators under a general ergodicity condition when the data are unbounded and range over suitably increasing sequences of compact sets. They give some implications of these results for stating the prediction in Markovian processes with finite order and show, through simulation, the efficiency of the predictors they propose.     

Robust sequential designs for nonlinear regression

The authors introduce the formal notion of an approximately specified nonlinear regression model and investigate sequential design methodologies when the fitted model is possibly of an incorrect parametric form. They present small‐sample simulation studies which indicate that their new designs can be very successful, relative to some common competitors, in reducing mean squared error due to model misspecifi‐cation and to heteroscedastic variation. Their simulations also suggest that standard normal‐theory inference procedures remain approximately valid under the sequential sampling schemes. The methods are illustrated both by simulation and in an example using data from an experiment described in the chemical engineering literature.     

Bayesian methods for generalized linear models with covariates missing at random

The authors propose methods for Bayesian inference for generalized linear models with missing covariate data. They specify a parametric distribution for the covariates that is written as a sequence of one‐dimensional conditional distributions. They propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. They examine the properties of the proposed prior and resulting posterior distributions. They also present a Bayesian criterion for comparing various models, and a calibration is derived for it. A detailed simulation is conducted and two real data sets are examined to demonstrate the methodology.     

Testing for lack of dependence in the functional linear model

The authors consider the linear model Y_n = ψX_n + ?_n relating a functional response with explanatory variables. They propose a simple test of the nullity of ψ based on the principal component decomposition. The limiting distribution of their test statistic is chi‐squared, but this distribution is also an excellent approximation in finite samples. The authors illustrate their method using data from terrestrial magnetic observatories.     

Stationary state space models for longitudinal data

The authors consider a class of state space models for the analysis of non‐normal longitudinal data whose latent process follows a stationary AR(1) model with exponential dispersion model margins. They propose to estimate parameters through an estimating equation approach based on the Kalman smoother. This allows them to carry out a straightforward analysis of a wide range of non‐normal data. They illustrate their approach via a simulation study and through analyses of Brazilian precipitation and US polio infection data.     

Local influence for generalized linear mixed models

The authors describe a method for assessing model inadequacy in maximum likelihood estimation of a generalized linear mixed model. They treat the latent random effects in the model as missing data and develop the influence analysis on the basis of a Q‐function which is associated with the conditional expectation of the complete‐data log‐likelihood function in the EM algorithm. They propose a procedure to detect influential observations in six model perturbation schemes. They also illustrate their methodology in a hypothetical situation and in two real cases.     

Bayesian estimation in Kibble's bivariate gamma distribution

The authors describe Bayesian estimation for the parameters of the bivariate gamma distribution due to Kibble (1941). The density of this distribution can be written as a mixture, which allows for a simple data augmentation scheme. The authors propose a Markov chain Monte Carlo algorithm to facilitate estimation. They show that the resulting chain is geometrically ergodic, and thus a regenerative sampling procedure is applicable, which allows for estimation of the standard errors of the ergodic means. They develop Bayesian hypothesis testing procedures to test both the dependence hypothesis of the two variables and the hypothesis of equal means. They also propose a reversible jump Markov chain Monte Carlo algorithm to carry out the model selection problem. Finally, they use sets of real and simulated data to illustrate their methodology.     

A new family of random graphs for testing spatial segregation

The authors discuss a graph‐based approach for testing spatial point patterns. This approach falls under the category of data‐random graphs, which have been introduced and used for statistical pattern recognition in recent years. The authors address specifically the problem of testing complete spatial randomness against spatial patterns of segregation or association between two or more classes of points on the plane. To this end, they use a particular type of parameterized random digraph called a proximity catch digraph (PCD) which is based on relative positions of the data points from various classes. The statistic employed is the relative density of the PCD, which is a U‐statistic when scaled properly. The authors derive the limiting distribution of the relative density, using the standard asymptotic theory of U‐statistics. They evaluate the finite‐sample performance of their test statistic by Monte Carlo simulations and assess its asymptotic performance via Pitman's asymptotic efficiency, thereby yielding the optimal parameters for testing. They further stress that their methodology remains valid for data in higher dimensions.     

Bayesian robust transformation and variable selection: A unified approach

The authors consider the problem of simultaneous transformation and variable selection for linear regression. They propose a fully Bayesian solution to the problem, which allows averaging over all models considered including transformations of the response and predictors. The authors use the Box‐Cox family of transformations to transform the response and each predictor. To deal with the change of scale induced by the transformations, the authors propose to focus on new quantities rather than the estimated regression coefficients. These quantities, referred to as generalized regression coefficients, have a similar interpretation to the usual regression coefficients on the original scale of the data, but do not depend on the transformations. This allows probabilistic statements about the size of the effect associated with each variable, on the original scale of the data. In addition to variable and transformation selection, there is also uncertainty involved in the identification of outliers in regression. Thus, the authors also propose a more robust model to account for such outliers based on a t‐distribution with unknown degrees of freedom. Parameter estimation is carried out using an efficient Markov chain Monte Carlo algorithm, which permits moves around the space of all possible models. Using three real data sets and a simulated study, the authors show that there is considerable uncertainty about variable selection, choice of transformation, and outlier identification, and that there is advantage in dealing with all three simultaneously. The Canadian Journal of Statistics 37: 361–380; 2009 © 2009 Statistical Society of Canada     

Goodness-of-fit tests for parametric models in censored regression

The authors propose a goodness-of-fit test for parametric regression models when the response variable is right-censored. Their test compares an estimation of the error distribution based on parametric residuals to another estimation relying on nonparametric residuals. They call on a bootstrap mechanism in order to approximate the critical values of tests based on Kolmogorov-Smirnov and Cramér-von Mises type statistics. They also present the results of Monte Carlo simulations and use data from a study about quasars to illustrate their work.     

Model‐free causal inference of binary experimental data

For binary experimental data, we discuss randomization‐based inferential procedures that do not need to invoke any modeling assumptions. In addition to the classical method of moments, we also introduce model‐free likelihood and Bayesian methods based solely on the physical randomization without any hypothetical super population assumptions about the potential outcomes. These estimators have some properties superior to moment‐based ones such as only giving estimates in regions of feasible support. Due to the lack of identification of the causal model, we also propose a sensitivity analysis approach that allows for the characterization of the impact of the association between the potential outcomes on statistical inference.     

Corrected local polynomial estimation in varying‐coefficient models with measurement errors

The authors study a varying‐coefficient regression model in which some of the covariates are measured with additive errors. They find that the usual local linear estimator (LLE) of the coefficient functions is biased and that the usual correction for attenuation fails to work. They propose a corrected LLE and show that it is consistent and asymptotically normal, and they also construct a consistent estimator for the model error variance. They then extend the generalized likelihood technique to develop a goodness of fit test for the model. They evaluate these various procedures through simulation studies and use them to analyze data from the Framingham Heart Study.     

Estimation of sex‐specific survival with uncertainty in sex assessment

In monomorphic species, determination of sex from behavior is prone to errors. The authors develop capture‐recapture survival models that account for uncertainty in the assessment of sex. They examine parameter redundancy for four basic models with constant or time‐dependent survival and encounter probabilities. They further develop a more refined and more appropriate model for an Audouin's gull data set where four distinct behavioral clues have been used. They examine how useful it is to incorporate the least reliable of the clues and the genetic determination of sex available for only a handful of individuals. They finally discuss the implications of their findings for the design of field studies.