On MSE of EBLUP

We consider Best Linear Unbiased Predictors (BLUPs) and Empirical Best Linear Unbiased Predictors (EBLUPs) under the general mixed linear model. The BLUP was proposed by Henderson (Ann Math Stat 21:309–310, 1950). The formula of this BLUP includes unknown elements of the variance-covariance matrix of random variables. If the elements in the formula of the BLUP proposed by Henderson (Ann Math Stat 21:309–310, 1950) are replaced by some type of estimators, we obtain the two-stage predictor called the EBLUP which is model-unbiased (Kackar and Harville in Commun Stat A 10:1249–1261, 1981). Kackar and Harville (J Am Stat Assoc 79:853–862, 1984) show an approximation of the mean square error (the MSE) of the predictor and propose an estimator of the MSE. The MSE and estimators of the MSE are also studied by Prasad and Rao (J Am Stat Assoc 85:163–171, 1990), Datta and Lahiri (Stat Sin 10:613–627, 2000) and Das et al. (Ann Stat 32(2):818–840, 2004). In the paper we consider the BLUP proposed by Royall (J Am Stat Assoc 71:657–473, 1976. Ża̧dło (On unbiasedness of some EBLU predictor. Physica-Verlag, Heidelberg, pp 2019–2026, 2004) shows that the BLUP proposed by Royall (J Am Stat Assoc 71:657–473, 1976) may be treated as a generalisation of the BLUP proposed by Henderson (Ann Math Stat 21:309–310, 1950) and proves model unbiasedness of the EBLUP based on the formula of the BLUP proposed by Royall (J Am Stat Assoc 71:657–473, 1976) under some assumptions. In this paper we derive the formula of the approximate MSE of the EBLUP and its estimators. We prove that the approximation of the MSE is accurate to terms o(D ⁻¹) and that the estimator of the MSE is approximately unbiased in the sense that its bias is o(D ⁻¹) under some assumptions, where D is the number of domains. The proof is based on the results obtained by Datta and Lahiri (Stat Sin 10:613–627, 2000). Using our results we show some EBLUP based on the special case of the general linear model. We also present the formula of its MSE and estimators of its MSE and their performance in Monte Carlo simulation study.      

On an exponential bound for the Kaplan–Meier estimator

We review limit theory and inequalities for the Kaplan–Meier Kaplan and Meier (J Am Stat Assoc 53:457–481, 1958) product limit estimator of a survival function on the whole line . Along the way we provide bounds for the constant in an interesting inequality due to Biotouzé et al. (Ann Inst H Poincaré Probab Stat 35:735–763, 1999), and provide some numerical evidence in support of one of their conjectures. Supported in part by NSF grant DMS-0503822 and by NI-AID grant 2R01 AI291968-04.     

Orthogonal decomposition of point-symmetry for multiway tables

For multiway contingency tables, Wall and Lienert (Biom. J. 18:259–264, 1976) considered the point-symmetry model. For square contingency tables, Tomizawa (Biom. J. 27:895–905, 1985) gave a theorem that the point-symmetry model holds if and only if both the quasi point-symmetry and the marginal point-symmetry models hold. This paper proposes some quasi point-symmetry models and marginal point-symmetry models for multiway tables, and extends Tomizawa’s (Biom. J. 27:895–905, 1985) theorem into multiway tables. We also show that for multiway tables the likelihood ratio statistic for testing goodness of fit of the point-symmetry model is asymptotically equivalent to the sum of those for testing the quasi point-symmetry model with some order and the marginal point-symmetry model with the corresponding order. An example is given.     

A new generalized Balakrishnan skew-normal distribution

We introduce a new family of skew-normal distributions that contains the skew-normal distributions introduced by Azzalini (Scand J Stat 12:171–178, 1985), Arellano-Valle et al. (Commun Stat Theory Methods 33(7):1465–1480, 2004), Gupta and Gupta (Test 13(2):501–524, 2008) and Sharafi and Behboodian (Stat Papers, 49:769–778, 2008). We denote this distribution by GBSN _n (λ₁, λ₂). We present some properties of GBSN _n (λ₁, λ₂) and derive the moment generating function. Finally, we use two numerical examples to illustrate the practical usefulness of this distribution.     

An alternative stochastic restricted Liu estimator in linear regression

In this paper, we introduce an alternative stochastic restricted Liu estimator for the vector of parameters in a linear regression model when additional stochastic linear restrictions on the parameter vector are assumed to hold. The new estimator is a generalization of the ordinary mixed estimator (OME) (Durbin in J Am Stat Assoc 48:799–808, 1953; Theil and Goldberger in Int Econ Rev 2:65–78, 1961; Theil in J Am Stat Assoc 58:401–414, 1963) and Liu estimator proposed by Liu (Commun Stat Theory Methods 22:393–402, 1993). Necessary and sufficient conditions for the superiority of the new stochastic restricted Liu estimator over the OME, the Liu estimator and the estimator proposed by Hubert and Wijekoon (Stat Pap 47:471–479, 2006) in the mean squared error matrix (MSEM) sense are derived. Furthermore, a numerical example based on the widely analysed dataset on Portland cement (Woods et al. in Ind Eng Chem 24:1207–1241, 1932) and a Monte Carlo evaluation of the estimators are also given to illustrate some of the theoretical results.     

A test for fractional cointegration using the sieve bootstrap

The concept of fractional cointegration (Cheung and Lai in J Bus Econ Stat 11:103–112, 1993) has been introduced to generalize traditional cointegration (Engle and Granger in Econometrica 55:251–276, 1987) to the long memory framework. In this work we propose a test for fractional cointegration with the sieve bootstrap and compare by simulations the performance of our proposal with other semiparametric methods existing in literature: the three steps technique of Marinucci and Robinson (J Econom 105:225–247, 2001) and the procedure to determine the fractional cointegration rank of Robinson and Yajima (J Econom 106:217–241, 2002).     

On population-based simulation for static inference

In this paper we present a review of population-based simulation for static inference problems. Such methods can be described as generating a collection of random variables {X _n } _n=1,…,N in parallel in order to simulate from some target density π (or potentially sequence of target densities). Population-based simulation is important as many challenging sampling problems in applied statistics cannot be dealt with successfully by conventional Markov chain Monte Carlo (MCMC) methods. We summarize population-based MCMC (Geyer, Computing Science and Statistics: The 23rd Symposium on the Interface, pp. 156–163, 1991; Liang and Wong, J. Am. Stat. Assoc. 96, 653–666, 2001) and sequential Monte Carlo samplers (SMC) (Del Moral, Doucet and Jasra, J. Roy. Stat. Soc. Ser. B 68, 411–436, 2006a), providing a comparison of the approaches. We give numerical examples from Bayesian mixture modelling (Richardson and Green, J. Roy. Stat. Soc. Ser. B 59, 731–792, 1997).     

Bayesian analysis of skew-t multivariate null intercept measurement error model

The multivariate skew-t distribution (J Multivar Anal 79:93–113, 2001; J R Stat Soc, Ser B 65:367–389, 2003; Statistics 37:359–363, 2003) includes the Student t, skew-Cauchy and Cauchy distributions as special cases and the normal and skew–normal ones as limiting cases. In this paper, we explore the use of Markov Chain Monte Carlo (MCMC) methods to develop a Bayesian analysis of repeated measures, pretest/post-test data, under multivariate null intercept measurement error model (J Biopharm Stat 13(4):763–771, 2003) where the random errors and the unobserved value of the covariate (latent variable) follows a Student t and skew-t distribution, respectively. The results and methods are numerically illustrated with an example in the field of dentistry.     

A note on the asymptotic distribution of a Perron-type innovational outlier unit root test with a break

This note provides the asymptotic distribution of a Perron-type innovational outlier unit root test developed by Popp (J Stat Comput Sim 78:1145–1161, 2008) in case of a shift in the intercept for non-trending data. In Popp (J Stat Comput Sim 78:1145–1161, 2008), only critical values for finite samples based on Monte Carlo techniques are tabulated. Using similar arguments as in Zivot and Andrews (J Bus Econ Stat 10:251–270, 1992), weak convergence is shown for the test statistics.     

Regression analysis of multivariate recurrent event data with a dependent terminal event

Recurrent event data occur in many clinical and observational studies (Cook and Lawless, Analysis of recurrent event data, 2007) and in these situations, there may exist a terminal event such as death that is related to the recurrent event of interest (Ghosh and Lin, Biometrics 56:554–562, 2000; Wang et al., J Am Stat Assoc 96:1057–1065, 2001; Huang and Wang, J Am Stat Assoc 99:1153–1165, 2004; Ye et al., Biometrics 63:78–87, 2007). In addition, sometimes there may exist more than one type of recurrent events, that is, one faces multivariate recurrent event data with some dependent terminal event (Chen and Cook, Biostatistics 5:129–143, 2004). It is apparent that for the analysis of such data, one has to take into account the dependence both among different types of recurrent events and between the recurrent and terminal events. In this paper, we propose a joint modeling approach for regression analysis of the data and both finite and asymptotic properties of the resulting estimates of unknown parameters are established. The methodology is applied to a set of bivariate recurrent event data arising from a study of leukemia patients.     

Sampling design proportional to order statistic of auxiliary variable

The sampling designs dependent on sample moments of auxiliary variables are well known. Lahiri (Bull Int Stat Inst 33:133–140, 1951) considered a sampling design proportionate to a sample mean of an auxiliary variable. Sing and Srivastava (Biometrika 67(1):205–209, 1980) proposed the sampling design proportionate to a sample variance while Wywiał (J Indian Stat Assoc 37:73–87, 1999) a sampling design proportionate to a sample generalized variance of auxiliary variables. Some other sampling designs dependent on moments of an auxiliary variable were considered e.g. in Wywiał (Some contributions to multivariate methods in, survey sampling. Katowice University of Economics, Katowice, 2003a); Stat Transit 4(5):779–798, 2000) where accuracy of some sampling strategies were compared, too.These sampling designs cannot be useful in the case when there are some censored observations of the auxiliary variable. Moreover, they can be much too sensitive to outliers observations. In these cases the sampling design proportionate to the order statistic of an auxiliary variable can be more useful. That is why such an unequal probability sampling design is proposed here. Its particular cases as well as its conditional version are considered, too. The sampling scheme implementing this sampling design is proposed. The inclusion probabilities of the first and second orders were evaluated. The well known Horvitz–Thompson estimator is taken into account. A ratio estimator dependent on an order statistic is constructed. It is similar to the well known ratio estimator based on the population and sample means. Moreover, it is an unbiased estimator of the population mean when the sample is drawn according to the proposed sampling design dependent on the appropriate order statistic.     

A numerical study for comparing two response-adaptive designs for continuous treatment effects

We study two sequential, response-adaptive randomized designs for clinical trials; one has been proposed in Bandyopadhyay and Biswas (Biometrika 88: 409–419, 2001) and in Biswas and Basu (Sankhya Ser B 63:27–42, 2001), the other stems from the randomly reinforced urn introduced and studied in Muliere et al. (J Stat Plan Inference 136:1853–1874, 2006a). Both designs can be used in clinical trials where the response from each patient is a continuous variable. Comparison is conducted through numerical studies and along a new guideline for the evaluation of a response-adaptive design.     

Bayesian inference for the Birnbaum–Saunders nonlinear regression model

We develop a Bayesian analysis for the class of Birnbaum–Saunders nonlinear regression models introduced by Lemonte and Cordeiro (Comput Stat Data Anal 53:4441–4452, 2009). This regression model, which is based on the Birnbaum–Saunders distribution (Birnbaum and Saunders in J Appl Probab 6:319–327, 1969a), has been used successfully to model fatigue failure times. We have considered a Bayesian analysis under a normal-gamma prior. Due to the complexity of the model, Markov chain Monte Carlo methods are used to develop a Bayesian procedure for the considered model. We describe tools for model determination, which include the conditional predictive ordinate, the logarithm of the pseudo-marginal likelihood and the pseudo-Bayes factor. Additionally, case deletion influence diagnostics is developed for the joint posterior distribution based on the Kullback–Leibler divergence. Two empirical applications are considered in order to illustrate the developed procedures.     

On estimation and local influence analysis for measurement errors models under heavy-tailed distributions

Scale mixtures of normal distributions form a class of symmetric thick-tailed distributions that includes the normal one as a special case. In this paper we consider local influence analysis for measurement error models (MEM) when the random error and the unobserved value of the covariates jointly follow scale mixtures of normal distributions, providing an appealing robust alternative to the usual Gaussian process in measurement error models. In order to avoid difficulties in estimating the parameter of the mixing variable, we fixed it previously, as recommended by Lange et al. (J Am Stat Assoc 84:881–896, 1989) and Berkane et al. (Comput Stat Data Anal 18:255–267, 1994). The local influence method is used to assess the robustness aspects of the parameter estimates under some usual perturbation schemes. However, as the observed log-likelihood associated with this model involves some integrals, Cook’s well–known approach may be hard to apply to obtain measures of local influence. Instead, we develop local influence measures following the approach of Zhu and Lee (J R Stat Soc Ser B 63:121–126, 2001), which is based on the EM algorithm. Results obtained from a real data set are reported, illustrating the usefulness of the proposed methodology, its relative simplicity, adaptability and practical usage.     

A projector oriented approach to the best linear unbiased estimator

The paper provides a projector based approach to the best linear unbiased estimator (BLUE). By revisiting the so called generalized projection operator, introduced in Rao (J R Stat Soc Ser B Stat Methodol 36:442–448, 1974), a number of new formulae for BLUE is established. Furthermore, some attention is paid to the coincidence of the BLUE and the ordinary least squares estimator.     

On pseudo-values for regression analysis in competing risks models

For regression on state and transition probabilities in multi-state models Andersen et al. (Biometrika 90:15–27, 2003) propose a technique based on jackknife pseudo-values. In this article we analyze the pseudo-values suggested for competing risks models and prove some conjectures regarding their asymptotics (Klein and Andersen, Biometrics 61:223–229, 2005). The key is a second order von Mises expansion of the Aalen-Johansen estimator which yields an appropriate representation of the pseudo-values. The method is illustrated with data from a clinical study on total joint replacement. In the application we consider for comparison the estimates obtained with the Fine and Gray approach (J Am Stat Assoc 94:496–509, 1999) and also time-dependent solutions of pseudo-value regression equations.     

Asymptotic theory for the Cox semi-Markov illness-death model

Irreversible illness-death models are used to model disease processes and in cancer studies to model disease recovery. In most applications, a Markov model is assumed for the multistate model. When there are covariates, a Cox (1972, J Roy Stat Soc Ser B 34:187–220) model is used to model the effect of covariates on each transition intensity. Andersen et al. (2000, Stat Med 19:587–599) proposed a Cox semi-Markov model for this problem. In this paper, we study the large sample theory for that model and provide the asymptotic variances of various probabilities of interest. A Monte Carlo study is conducted to investigate the robustness and efficiency of Markov/Semi-Markov estimators. A real data example from the PROVA (1991, Hepatology 14:1016–1024) trial is used to illustrate the theory.     

On the Breslow–Holubkov estimator

Breslow and Holubkov (J Roy Stat Soc B 59:447–461 1997a) developed semiparametric maximum likelihood estimation for two-phase studies with a case–control first phase under a logistic regression model and noted that, apart for the overall intercept term, it was the same as the semiparametric estimator for two-phase studies with a prospective first phase developed in Scott and Wild (Biometrica 84:57–71 1997). In this paper we extend the Breslow–Holubkov result to general binary regression models and show that it has a very simple relationship with its prospective first-phase counterpart. We also explore why the design of the first phase only affects the intercept of a logistic model, simplify the calculation of standard errors, establish the semiparametric efficiency of the Breslow–Holubkov estimator and derive its asymptotic distribution in the general case.     

Protection of privacy in efficient application of randomized response techniques

In estimating the proportion of people bearing a sensitive attribute A, say, in a given community, following Warner’s (J Am Stat Assoc 60:63–69, 1965) pioneering work, certain randomized response (RR) techniques are available for application. These are intended to ensure efficient and unbiased estimation protecting a respondent’s privacy when it touches a person’s socially stigmatizing feature like rash driving, tax evasion, induced abortion, testing HIV positive, etc. Lanke (Int Stat Rev 44:197–203, 1976), Leysieffer and Warner (J Am Stat Assoc 71:649–656, 1976), Anderson (Int Stat Rev 44:213–217, 1976, Scand J Stat 4:11–19, 1977) and Nayak (Commun Stat Theor Method 23:3303–3321, 1994) among others have discussed how maintenance of efficiency is in conflict with protection of privacy. In their RR-related activities the sample selection is traditionally by simple random sampling (SRS) with replacement (WR). In this paper, an extension of an essential similarity in case of general unequal probability sample selection even without replacement is reported. Large scale surveys overwhelmingly employ complex designs other than SRSWR. So extension of RR techniques to complex designs is essential and hence this paper principally refers to them. New jeopardy measures to protect revelation of secrecy presented here are needed as modifications of those in the literature covering SRSWR alone. Observing that multiple responses are feasible in addressing such a dichotomous situation especially with Kuk’s (Biometrika 77:436–438, 1990) and Christofides’ (Metrika 57:195–200, 2003) RR devices, an average of the response-specific jeopardizing measures is proposed. This measure which is device dependent, could be regarded as a technical characteristic of the device and it should be made known to the participants before they agree to use the randomization device. The views expressed are the authors’, not of the organizations they work for. Prof Chaudhuri’s research is partially supported by CSIR Grant No. 21(0539)/02/EMR-II.     

Regression analysis for cumulative incidence probability under competing risks and left-truncated sampling

The cumulative incidence function provides intuitive summary information about competing risks data. Via a mixture decomposition of this function, Chang and Wang (Statist. Sinca 19:391–408, 2009) study how covariates affect the cumulative incidence probability of a particular failure type at a chosen time point. Without specifying the corresponding failure time distribution, they proposed two estimators and derived their large sample properties. The first estimator utilized the technique of weighting to adjust for the censoring bias, and can be considered as an extension of Fine’s method (J R Stat Soc Ser B 61: 817–830, 1999). The second used imputation and extends the idea of Wang (J R Stat Soc Ser B 65: 921–935, 2003) from a nonparametric setting to the current regression framework. In this article, when covariates take only discrete values, we extend both approaches of Chang and Wang (Statist Sinca 19:391–408, 2009) by allowing left truncation. Large sample properties of the proposed estimators are derived, and their finite sample performance is investigated through a simulation study. We also apply our methods to heart transplant survival data.