A model selection criterion based on the BHHJ measure of divergence

The aim of this work is the discussion and investigation of measures of divergence and model selection criteria. A recently introduced measure of divergence, the so-called BHHJ measure (Basu, A., Harris, I.R., Hjort, N.L., Jones, M.C., 1998. Robust and efficient estimation by minimising a density power divergence. Biometrika 85, 549–559) is investigated and a new model selection criterion the divergence information criterion (DIC) based on this measure is proposed. Simulations are performed to check the appropriateness of the proposed criterion.     

Goodness-of-fit Procedures for Copula Models Based on the Probability Integral Transformation

Abstract.  Wang & Wells [ J. Amer. Statist. Assoc. 95 (2000) 62] describe a non-parametric approach for checking whether the dependence structure of a random sample of censored bivariate data is appropriately modelled by a given family of Archimedean copulas. Their procedure is based on a truncated version of the Kendall process introduced by Genest & Rivest [ J. Amer. Statist. Assoc. 88 (1993) 1034] and later studied by Barbe et al . [ J. Multivariate Anal. 58 (1996) 197]. Although Wang & Wells (2000) determine the asymptotic behaviour of their truncated process, their model selection method is based exclusively on the observed value of its L ²-norm. This paper shows how to compute asymptotic p -values for various goodness-of-fit test statistics based on a non-truncated version of Kendall's process. Conditions for weak convergence are met in the most common copula models, whether Archimedean or not. The empirical behaviour of the proposed goodness-of-fit tests is studied by simulation, and power comparisons are made with a test proposed by Shih [ Biometrika 85 (1998) 189] for the gamma frailty family.     

Robust estimation for zero-inflated poisson autoregressive models based on density power divergence

In this study, we consider a robust estimation for zero-inflated Poisson autoregressive models using the minimum density power divergence estimator designed by Basu et al. [Robust and efficient estimation by minimising a density power divergence. Biometrika. 1998;85:549–559]. We show that under some regularity conditions, the proposed estimator is strongly consistent and asymptotically normal. The performance of the estimator is evaluated through Monte Carlo simulations. A real data analysis using New South Wales crime data is also provided for illustration.     

Consistency of minimizing a penalized density power divergence estimator for mixing distribution

In this paper, we study the MDPDE (minimizing a density power divergence estimator), proposed by Basu et al. (Biometrika 85:549–559, 1998), for mixing distributions whose component densities are members of some known parametric family. As with the ordinary MDPDE, we also consider a penalized version of the estimator, and show that they are consistent in the sense of weak convergence. A simulation result is provided to illustrate the robustness. Finally, we apply the penalized method to analyzing the red blood cell SLC data presented in Roeder (J Am Stat Assoc 89:487–495, 1994). This research was supported (in part) by KOSEF through Statistical Research Center for Complex Systems at Seoul National University.     

Bayesian analysis of quasi-life tables

Quasi-life tables, in which the data arise from many concurrent, independent, discrete-time renewal processes, were defined by Baxter (1994, Biometrika 81:567–577), who outlined some methods for estimation. The processes are not observed individually; only the total numbers of renewals at each time point are observed. Crowder and Stephens (2003, Lifetime Data Anal 9:345–355) implemented a formal estimating-equation approach that invokes large-sample theory. However, these asymptotic methods fail to yield sensible estimates for smaller samples. In this paper, we implement a Bayesian analysis based on MCMC computation that works equally well for large and small sample sizes. We give three simulated examples, studying the Bayesian results, the impact of changing prior specification, and empirical properties of the Bayesian estimators of the lifetime distribution parameters. We also study the Baxter (1994, Biometrika 81:567–577) data, and uncover structure that has not been commented upon previously.     

Robust estimation in the normal mixture model

This paper focuses on the inference of the normal mixture model with unequal variances. A feature of the model is flexibility of density shape, but its flexibility causes the unboundedness of the likelihood function and excessive sensitivity of the maximum likelihood estimator to outliers. A modified likelihood approach suggested in Basu et al. [1998, Biometrika 85, 549–559] can overcome these drawbacks. It is shown that the modified likelihood function is bounded above under a mild condition on mixing proportions and the resultant estimator is robust to outliers. A relationship between robustness and efficiency is investigated and an adaptive method for selecting the tuning parameter of the modified likelihood is suggested, based on the robust model selection criterion and the cross-validation. An EM-like algorithm is also constructed. Numerical studies are presented to evaluate the performance. The robust method is applied to single nuleotide polymorphism typing for the purpose of outlier detection and clustering.     

Interval estimation of the joint survival function for successive duration times under left truncation and right censoring

In incident cohort studies, survival data often include subjects who have had an initiate event at recruitment and may potentially experience two successive events (first and second) during the follow-up period. Since the second duration process becomes observable only if the first event has occurred, left truncation and dependent censoring arise if the two duration times are correlated. To confront the two potential sampling biases, we propose two inverse-probability-weighted (IPW) estimators for the estimation of the joint survival function of two successive duration times. One of them is similar to the estimator proposed by Chang and Tzeng [Nonparametric estimation of sojourn time distributions for truncated serial event data – a weight adjusted approach, Lifetime Data Anal. 12 (2006), pp. 53–67]. The other is the extension of the nonparametric estimator proposed by Wang and Wells [Nonparametric estimation of successive duration times under dependent censoring, Biometrika 85 (1998), pp. 561–572]. The weak convergence of both estimators are established. Furthermore, the delete-one jackknife and simple bootstrap methods are used to estimate standard deviations and construct interval estimators. A simulation study is conducted to compare the two IPW approaches.     

Adjustments of the profile likelihood from a new perspective

Various modifications of the profile likelihood have been proposed over the past 20 years. Their main theoretical basis is higher-order approximation of some target likelihood, defined by a suitable model reduction via conditioning or marginalisation, where the reduced model is indexed only by the parameter of interest. However, an exact reduced target likelihood exists only for special classes of models. In this paper, a general target likelihood is defined through model restriction along the least favourable curve in the parameter space. The profile likelihood can be seen as a purely estimative counterpart of this least favourable target likelihood. We show that various modifications of the profile likelihood arise by refining the estimation process. In particular, we show that bias reduction of the profile loglikelihood as an estimate of the expectation of the least favourable target loglikelihood gives adjustments that agree to second order with the adjustment proposed by Severini [1998a. Biometrika 85, 403–411] as an easily computable approximation to the modified profile loglikelihood.     

Profile empirical-likelihood inferences for the single-index-coefficient regression model

This article deals with a new profile empirical-likelihood inference for a class of frequently used single-index-coefficient regression models (SICRM), which were proposed by Xia and Li (J. Am. Stat. Assoc. 94:1275–1285, 1999a). Applying the empirical likelihood method (Owen in Biometrika 75:237–249, 1988), a new estimated empirical log-likelihood ratio statistic for the index parameter of the SICRM is proposed. To increase the accuracy of the confidence region, a new profile empirical likelihood for each component of the relevant parameter is obtained by using maximum empirical likelihood estimators (MELE) based on a new and simple estimating equation for the parameters in the SICRM. Hence, the empirical likelihood confidence interval for each component is investigated. Furthermore, corrected empirical likelihoods for functional components are also considered. The resulting statistics are shown to be asymptotically standard chi-squared distributed. Simulation studies are undertaken to assess the finite sample performance of our method. A study of real data is also reported.     

Bayesian Estimators for Small Area Models when Auxiliary Information is Measured with Error

Small area estimators in linear models are typically expressed as a convex combination of direct estimators and synthetic estimators from a suitable model. When auxiliary information used in the model is measured with error, a new estimator, accounting for the measurement error in the covariates, has been proposed in the literature. Recently, for area‐level model, Ybarra & Lohr (Biometrika, 95, 2008, 919) suggested a suitable modification to the estimates of small area means based on Fay & Herriot (J. Am. Stat. Assoc., 74, 1979, 269) model where some of the covariates are measured with error. They used a frequentist approach based on the method of moments. Adopting a Bayesian approach, we propose to rewrite the measurement error model as a hierarchical model; we use improper non‐informative priors on the model parameters and show, under a mild condition, that the joint posterior distribution is proper and the marginal posterior distributions of the model parameters have finite variances. We conduct a simulation study exploring different scenarios. The Bayesian predictors we propose show smaller empirical mean squared errors than the frequentist predictors of Ybarra & Lohr (Biometrika, 95, 2008, 919), and they seem also to be more stable in terms of variability and bias. We apply the proposed methodology to two real examples.     

Semi-parametric hybrid empirical likelihood inference for two-sample comparison with censored data

Two-sample comparison problems are often encountered in practical projects and have widely been studied in literature. Owing to practical demands, the research for this topic under special settings such as a semiparametric framework have also attracted great attentions. Zhou and Liang (Biometrika 92:271–282, 2005) proposed an empirical likelihood-based semi-parametric inference for the comparison of treatment effects in a two-sample problem with censored data. However, their approach is actually a pseudo-empirical likelihood and the method may not be fully efficient. In this study, we develop a new empirical likelihood-based inference under more general framework by using the hazard formulation of censored data for two sample semi-parametric hybrid models. We demonstrate that our empirical likelihood statistic converges to a standard chi-squared distribution under the null hypothesis. We further illustrate the use of the proposed test by testing the ROC curve with censored data, among others. Numerical performance of the proposed method is also examined.     

Books on probability ideas are reviewed

In this paper, semiparametric methods are applied to estimate multivariate volatility functions, using a residual approach as in [J. Fan and Q. Yao, Efficient estimation of conditional variance functions in stochastic regression, Biometrika 85 (1998), pp. 645–660; F.A. Ziegelmann, Nonparametric estimation of volatility functions: The local exponential estimator, Econometric Theory 18 (2002), pp. 985–991; F.A. Ziegelmann, A local linear least-absolute-deviations estimator of volatility, Comm. Statist. Simulation Comput. 37 (2008), pp. 1543–1564], among others. Our main goal here is two-fold: (1) describe and implement a number of semiparametric models, such as additive, single-index and (adaptive) functional-coefficient, in volatility estimation, all motivated as alternatives to deal with the curse of dimensionality present in fully nonparametric models; and (2) propose the use of a variation of the traditional cross-validation method to deal with model choice in the class of adaptive functional-coefficient models, choosing simultaneously the bandwidth, the number of covariates in the model and also the single-index smoothing variable. The modified cross-validation algorithm is able to tackle the computational burden caused by the model complexity, providing an important tool in semiparametric volatility estimation. We briefly discuss model identifiability when estimating volatility as well as nonnegativity of the resulting estimators. Furthermore, Monte Carlo simulations for several underlying generating models are implemented and applications to real data are provided.     

An Exact Corrected Log-Likelihood Function for Cox's Proportional Hazards Model under Measurement Error and Some Extensions

Abstract.  This paper studies Cox's proportional hazards model under covariate measurement error. Nakamura's [ Biometrika 77 (1990) 127] methodology of corrected log-likelihood will be applied to the so-called Breslow likelihood, which is, in the absence of measurement error, equivalent to partial likelihood. For a general error model with possibly heteroscedastic and non-normal additive measurement error, corrected estimators of the regression parameter as well as of the baseline hazard rate are obtained. The estimators proposed by Nakamura [Biometrics 48 (1992) 829], Kong et al. [ Scand. J. Statist. 25 (1998) 573] and Kong & Gu [ Statistica Sinica 9 (1999) 953] are re-established in the special cases considered there. This sheds new light on these estimators and justifies them as exact corrected score estimators. Finally, the method will be extended to some variants of the Cox model.     

An exact two-sample test based on the baumgartner-weiss-schindler statistic and a modification of lepage's test

It is shown that the nonparametric two-saniDle test recently proposed by Baumgartner, WeiB, Schindler (1998, Biometrics, 54, 1129-1135) does not control the type I error rate in case of small sample sizes. We investigate the exact permutation test based on their statistic and demonstrate that this test is almost not conservative. Comparing exact tests, the procedure based on the new statistic has a less conservative size and is, according to simulation results, more powerful than the often employed Wilcoxon test. Furthermore, the new test is also powerful with regard to less restrictive settings than the location-shift model. For example, the test can detect location-scale alternatives. Therefore, we use the test to create a powerful modification of the nonparametric location-scale test according to Lepage (1971, Biometrika, 58, 213-217). Selected critical values for the proposed tests are given.     

GEEs for repeated categorical responses based on generalized residuals

Clustered or correlated samples of categorical response data arise frequently in many fields of application. The method of generalized estimating equations (GEEs) introduced in Liang and Zeger [Longitudinal data analysis using generalized linear models, Biometrika 73 (1986), pp. 13–22] is often used to analyse this type of data. GEEs give consistent estimates of the regression parameters and their variance based upon the Pearson residuals. Park et al. [Alternative GEE estimation procedures for discrete longitudinal data, Comput. Stat. Data Anal. 28 (1998), pp. 243–256] considered a modification of the GEE approach using the Anscombe residual and the deviance residual. In this work, we propose to extend this idea to a family of generalized residuals. A wide simulation study is conducted for binary and Poisson correlated outcomes and also two numerical illustrations are presented.     

Analysis of an European union election using principal component analysis

While studying the results from one European Parliament election, the question of principal component analysis (PCA) suitability for this kind of data was raised. Since multiparty data should be seen as compositional data (CD), the application of PCA is inadvisable and may conduct to ineligible results. This work points out the limitations of PCA to CD and presents a practical application to the results from the European Parliament election in 2004. We present a comparative study between the results of PCA, Crude PCA and Logcontrast PCA (Aitchison in: Biometrika 70:57–61, 1983; Kucera, Malmgren in: Marine Micropaleontology 34:117–120, 1998). As a conclusion of this study, and concerning the mentioned data set, the approach which produced clearer results was the Logcontrast PCA. Moreover, Crude PCA conducted to misleading results since nonlinear relations were presented between variables and the linear PCA proved, once again, to be inappropriate to analyse data which can be seen as CD.     

Cross-ratio estimation for bivariate failure times with left truncation

The cross-ratio is an important local measure that characterizes the dependence between bivariate failure times. To estimate the cross-ratio in follow-up studies where delayed entry is present, estimation procedures need to account for left truncation. Ignoring left truncation yields biased estimates of the cross-ratio. We extend the method of Hu et al., Biometrika 98:341–354 (2011) by modifying the risk sets and relevant indicators to handle left-truncated bivariate failure times, which yields the cross-ratio estimate with desirable asymptotic properties that can be shown by the same techniques used in Hu et al., Biometrika 98:341–354 (2011). Numerical studies are conducted.     

Goodness-of-fit tests for semiparametric biased sampling models

Consider an s-sample biased sampling model in which the distribution function for each of the first s−1 samples is related to the unknown distribution function G of the sth sample by a known parametric selection bias weight function. Gilbert et al. (Biometrika 86 (1999) 27) gave a procedure for semiparametric maximum likelihood estimation of the parameters in this model. In many applications, information are scarce for basing the choice of the parametric weight function(s), motivating the need for goodness-of-fit tests of the hypothesis that the weight functions are correctly specified. Cramér–von Mises-type, Anderson–Darling-type, and Kolmogorov–Smirnov-type test statistics are studied which compare discrepancies between the empirical distribution of G and the semiparametric maximum likelihood estimator of G. Finite-sample properties of the tests are evaluated with simulations and with a real example of HIV genetic sequence data.     

The limiting bound of Efron's W-formula for hypothesis testing when a nuisance parameter is present only under the alternative

When testing a hypothesis with a nuisance parameter present only under the alternative, the maximum of a test statistic over the nuisance parameter space has been proposed. Different upper bounds for the one-sided tail probabilities of the maximum tests were provided. Davies (1977. Biometrika 64, 247–254) studied the problem when the parameter space is an interval, while Efron (1997. Biometrika 84, 143–157) considered the problem with some finite points of the parameter space and obtained a W-formula. We study the limiting bound of Efron's W-formula when the number of points in the parameter space goes to infinity. The conditions under which the limiting bound of the W-formula is identical to that of Davies are given. The results are also extended to two-sided tests. Examples are used to illustrate the conditions, including case-control genetic association studies. Efficient calculations of upper bounds for the tail probability with finite points in the parameter space are described.     

A generalized ARFIMA process with Markov-switching fractional differencing parameter

We propose a general class of Markov-switching-ARFIMA (MS-ARFIMA) processes in order to combine strands of long memory and Markov-switching literature. Although the coverage of this class of models is broad, we show that these models can be easily estimated with the Durbin–Levinson–Viterbi algorithm proposed. This algorithm combines the Durbin–Levinson and Viterbi procedures. A Monte Carlo experiment reveals that the finite sample performance of the proposed algorithm for a simple mixture model of Markov-switching mean and ARFIMA(1, d, 1) process is satisfactory. We apply the MS-ARFIMA models to the US real interest rates and the Nile river level data, respectively. The results are all highly consistent with the conjectures made or empirical results found in the literature. Particularly, we confirm the conjecture in Beran and Terrin [J. Beran and N. Terrin, Testing for a change of the long-memory parameter. Biometrika 83 (1996), pp. 627–638.] that the observations 1 to about 100 of the Nile river data seem to be more independent than the subsequent observations, and the value of differencing parameter is lower for the first 100 observations than for the subsequent data.