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).     

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.     

Classification using distance nearest neighbours

This paper proposes a new probabilistic classification algorithm using a Markov random field approach. The joint distribution of class labels is explicitly modelled using the distances between feature vectors. Intuitively, a class label should depend more on class labels which are closer in the feature space, than those which are further away. Our approach builds on previous work by Holmes and Adams (J. R. Stat. Soc. Ser. B 64:295–306, 2002; Biometrika 90:99–112, 2003) and Cucala et al. (J. Am. Stat. Assoc. 104:263–273, 2009). Our work shares many of the advantages of these approaches in providing a probabilistic basis for the statistical inference. In comparison to previous work, we present a more efficient computational algorithm to overcome the intractability of the Markov random field model. The results of our algorithm are encouraging in comparison to the k-nearest neighbour algorithm.     

Tests of fit for the Laplace distribution based on correcting moments of entropy estimators

ABSTRACT

In this paper, we first consider the entropy estimators introduced by Vasicek [A test for normality based on sample entropy. J R Statist Soc, Ser B. 1976;38:54–59], Ebrahimi et al. [Two measures of sample entropy. Stat Probab Lett. 1994;20:225–234], Yousefzadeh and Arghami [Testing exponentiality based on type II censored data and a new cdf estimator. Commun Stat – Simul Comput. 2008;37:1479–1499], Alizadeh Noughabi and Arghami [A new estimator of entropy. J Iran Statist Soc. 2010;9:53–64], and Zamanzade and Arghami [Goodness-of-fit test based on correcting moments of modified entropy estimator. J Statist Comput Simul. 2011;81:2077–2093], and the nonparametric distribution functions corresponding to them. We next introduce goodness-of-fit test statistics for the Laplace distribution based on the moments of nonparametric distribution functions of the aforementioned estimators. We obtain power estimates of the proposed test statistics with Monte Carlo simulation and compare them with the competing test statistics against various alternatives. Performance of the proposed new test statistics is illustrated in real cases.     

The structured elastic net for quantile regression and?support vector classification

In view of its ongoing importance for a variety of practical applications, feature selection via ℓ ₁-regularization methods like the lasso has been subject to extensive theoretical as well empirical investigations. Despite its popularity, mere ℓ ₁-regularization has been criticized for being inadequate or ineffective, notably in situations in which additional structural knowledge about the predictors should be taken into account. This has stimulated the development of either systematically different regularization methods or double regularization approaches which combine ℓ ₁-regularization with a second kind of regularization designed to capture additional problem-specific structure. One instance thereof is the ‘structured elastic net’, a generalization of the proposal in Zou and Hastie (J. R. Stat. Soc. Ser. B 67:301–320, 2005), studied in Slawski et al. (Ann. Appl. Stat. 4(2):1056–1080, 2010) for the class of generalized linear models.     

A two-stage estimation in the Clayton–Oakes model with marginal linear transformation models for multivariate failure time data

This paper considers the analysis of multivariate survival data where the marginal distributions are specified by semiparametric transformation models, a general class including the Cox model and the proportional odds model as special cases. First, consideration is given to the situation where the joint distribution of all failure times within the same cluster is specified by the Clayton–Oakes model (Clayton, Biometrika 65:141–151, l978; Oakes, J R Stat Soc B 44:412–422, 1982). A two-stage estimation procedure is adopted by first estimating the marginal parameters under the independence working assumption, and then the association parameter is estimated from the maximization of the full likelihood function with the estimators of the marginal parameters plugged in. The asymptotic properties of all estimators in the semiparametric model are derived. For the second situation, the third and higher order dependency structures are left unspecified, and interest focuses on the pairwise correlation between any two failure times. Thus, the pairwise association estimate can be obtained in the second stage by maximizing the pairwise likelihood function. Large sample properties for the pairwise association are also derived. Simulation studies show that the proposed approach is appropriate for practical use. To illustrate, a subset of the data from the Diabetic Retinopathy Study is used.     

A partially adaptive estimator for the censored regression model based on a mixture of normal distributions

The goal of this paper is to introduce a partially adaptive estimator for the censored regression model based on an error structure described by a mixture of two normal distributions. The model we introduce is easily estimated by maximum likelihood using an EM algorithm adapted from the work of Bartolucci and Scaccia (Comput Stat Data Anal 48:821–834, 2005). A Monte Carlo study is conducted to compare the small sample properties of this estimator to the performance of some common alternative estimators of censored regression models including the usual tobit model, the CLAD estimator of Powell (J Econom 25:303–325, 1984), and the STLS estimator of Powell (Econometrica 54:1435–1460, 1986). In terms of RMSE, our partially adaptive estimator performed well. The partially adaptive estimator is applied to data on wife’s hours worked from Mroz (1987). In this application we find support for the partially adaptive estimator over the usual tobit model.     

A stable estimator of the information matrix under EM for dependent data

This article develops a new and stable estimator for information matrix when the EM algorithm is used in maximum likelihood estimation. This estimator is constructed using the smoothed individual complete-data scores that are readily available from running the EM algorithm. The method works for dependent data sets and when the expectation step is an irregular function of the conditioning parameters. In comparison to the approach of Louis (J. R. Stat. Soc., Ser. B 44:226–233, 1982), this new estimator is more stable and easier to implement. Both real and simulated data are used to demonstrate the use of this new estimator.     

An improved class of estimators for the population mean

Starting from the Rao (Commun Stat Theory Methods 20:3325–3340, 1991) regression estimator, we propose a class of estimators for the unknown mean of a survey variable when auxiliary information is available. The bias and the mean square error of the estimators belonging to the class are obtained and the expressions for the optimum parameters minimizing the asymptotic mean square error are given in closed form. A simple condition allowing us to improve the classical regression estimator is worked out. Finally, in order to compare the performance of some estimators with the regression one, a simulation study is carried out when some population parameters are supposed to be unknown.     

More Powerful Control of the False Discovery Rate Under Dependence

In a breakthrough paper, Benjamini and Hochberg (J Roy Stat Soc Ser B 57:289–300, 1995) proposed a new error measure for multiple testing, the FDR; and developed a distribution-free procedure to control it under independence among the test statistics. In this paper we argue by extensive simulation and theoretical considerations that the assumption of independence is not needed. Along the lines of (Ann Stat 32:1035–1061, 2004b), we moreover provide a more powerful method, that exploits an estimator of the number of false nulls among the tests. We propose a whole family of iterative estimators that prove robust under dependence and independence between the test statistics. These estimators can be used to improve also classical multiple testing procedures, and in general to estimate the weight of a known component in a mixture distribution. Innovations are illustrated by simulations.     

On the entropy estimators

The paper introduces an estimator of the entropy of a continuous random variable. The estimator is obtained by modifying the estimator proposed by Ebrahimi et al. [Two measures of sample entropy, Statist. Probab. Lett. 20 (1994), pp. 225–234]. The consistency of the estimator is proved and comparisons are made with Vasicek's estimator [A test for normality based on sample entropy, J. R. Stat. Soc. Ser. B 38 (1976), pp. 54–59], van Es estimator [Estimating functionals related to a density by class of statistics based on spacings, Scand. J. Statist. 19 (1992), pp. 61–72], Ebrahimi et al. estimator and Correa estimator [A new estimator of entropy, Comm. Statist. Theory Methods 24 (1995), pp. 2439–2449]. The results indicate that the proposed estimator has smaller mean-squared error than above estimators. A real example is presented and analysed.     

Robust second-order least-squares estimator for regression models

The second-order least-squares estimator (SLSE) was proposed by Wang (Statistica Sinica 13:1201–1210, 2003) for measurement error models. It was extended and applied to linear and nonlinear regression models by Abarin and Wang (Far East J Theor Stat 20:179–196, 2006) and Wang and Leblanc (Ann Inst Stat Math 60:883–900, 2008). The SLSE is asymptotically more efficient than the ordinary least-squares estimator if the error distribution has a nonzero third moment. However, it lacks robustness against outliers in the data. In this paper, we propose a robust second-order least squares estimator (RSLSE) against X-outliers. The RSLSE is highly efficient with high breakdown point and is asymptotically normally distributed. We compare the RSLSE with other estimators through a simulation study. Our results show that the RSLSE performs very well.     

A Bayesian latent variable approach to functional principal components analysis with binary and count data

Recently, van der Linde (Comput. Stat. Data Anal. 53:517–533, 2008) proposed a variational algorithm to obtain approximate Bayesian inference in functional principal components analysis (FPCA), where the functions were observed with Gaussian noise. Generalized FPCA under different noise models with sparse longitudinal data was developed by Hall et al. (J. R. Stat. Soc. B 70:703–723, 2008), but no Bayesian approach is available yet. It is demonstrated that an adapted version of the variational algorithm can be applied to obtain a Bayesian FPCA for canonical parameter functions, particularly log-intensity functions given Poisson count data or logit-probability functions given binary observations. To this end a second order Taylor expansion of the log-likelihood, that is, a working Gaussian distribution and hence another step of approximation, is used. Although the approach is conceptually straightforward, difficulties can arise in practical applications depending on the accuracy of the approximation and the information in the data. A modified algorithm is introduced generally for one-parameter exponential families and exemplified for binary and count data. Conditions for its successful application are discussed and illustrated using simulated data sets. Also an application with real data is presented.     

Estimation of 2D jump location curve and 3D jump location surface in nonparametric regression

A new procedure is proposed to estimate the jump location curve and surface in the two-dimensional (2D) and three-dimensional (3D) nonparametric jump regression models, respectively. In each of the 2D and 3D cases, our estimation procedure is motivated by the fact that, under some regularity conditions, the ridge location of the rotational difference kernel estimate (RDKE; Qiu in Sankhyā Ser. A 59, 268–294, 1997, and J. Comput. Graph. Stat. 11, 799–822, 2002; Garlipp and Müller in Sankhyā Ser. A 69, 55–86, 2007) obtained from the noisy image is asymptotically close to the jump location of the true image. Accordingly, a computational procedure based on the kernel smoothing method is designed to find the ridge location of RDKE, and the result is taken as the jump location estimate. The sequence relationship among the points comprising our jump location estimate is obtained. Our jump location estimate is produced without the knowledge of the range or shape of jump region. Simulation results demonstrate that the proposed estimation procedure can detect the jump location very well, and thus it is a useful alternative for estimating the jump location in each of the 2D and 3D cases.     

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.      

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.     

Efficiency improvement in a class of survival models through model-free covariate incorporation

In randomized clinical trials, we are often concerned with comparing two-sample survival data. Although the log-rank test is usually suitable for this purpose, it may result in substantial power loss when the two groups have nonproportional hazards. In a more general class of survival models of Yang and Prentice (Biometrika 92:1–17, 2005), which includes the log-rank test as a special case, we improve model efficiency by incorporating auxiliary covariates that are correlated with the survival times. In a model-free form, we augment the estimating equation with auxiliary covariates, and establish the efficiency improvement using the semiparametric theories in Zhang et al. (Biometrics 64:707–715, 2008) and Lu and Tsiatis (Biometrics, 95:674–679, 2008). Under minimal assumptions, our approach produces an unbiased, asymptotically normal estimator with additional efficiency gain. Simulation studies and an application to a leukemia study show the satisfactory performance of the proposed method.     

Goodness-of-fit test based on correcting moments of modified entropy estimator

In this paper, we first propose a new estimator of entropy for continuous random variables. Our estimator is obtained by correcting the coefficients of Vasicek's [A test for normality based on sample entropy, J. R. Statist. Soc. Ser. B 38 (1976), pp. 54–59] entropy estimator. We prove the consistency of our estimator. Monte Carlo studies show that our estimator is better than the entropy estimators proposed by Vasicek, Ebrahimi et al. [Two measures of sample entropy, Stat. Probab. Lett. 20 (1994), pp. 225–234] and Correa [A new estimator of entropy, Commun. Stat. Theory Methods 24 (1995), pp. 2439–2449] in terms of root mean square error. We then derive the non-parametric distribution function corresponding to our proposed entropy estimator as a piece-wise uniform distribution. We also introduce goodness-of-fit tests for testing exponentiality and normality based on the said distribution and compare its performance with their leading competitors.     

On the existence of unbiased estimators for the portfolio weights obtained by maximizing the Sharpe ratio

We consider the problem of estimating the portfolio weights obtained by maximizing the Sharpe ratio. Assuming that the underlying asset returns are independent and multivariate normally distributed, Okhrin and Schmid (J. Econom. 134:235–256, 2006) showed that the frequently used sample estimators of these weights do not have a first moment. This paper proves that an unbiased estimator of the Sharpe ratio portfolio weights does not exist at all. Moreover, we show that there is no asymptotically unbiased estimator of these weights within the family of estimators which are bounded by cylinder functions.     

Double ASN Minimax sampling plans by variables when the standard deviation is unknown

We deal with the double sampling plans by variables proposed by Bowker and Goode (Sampling Inspection by Variables, McGraw–Hill, New York, 1952) when the standard deviation is unknown. Using the procedure for the calculation of the OC given by Krumbholz and Rohr (Allg. Stat. Arch. 90:233–251, 2006), we present an optimization algorithm allowing to determine the ASN Minimax plan. This plan, among all double plans satisfying the classical two-point-condition on the OC, has the minimal ASN maximum.