On computing standard errors for marginal structural Cox models

In recent decades, marginal structural models have gained popularity for proper adjustment of time-dependent confounders in longitudinal studies through time-dependent weighting. When the marginal model is a Cox model, using current standard statistical software packages was thought to be problematic because they were not developed to compute standard errors in the presence of time-dependent weights. We address this practical modelling issue by extending the standard calculations for Cox models with case weights to time-dependent weights and show that the coxph procedure in R can readily compute asymptotic robust standard errors. Through a simulation study, we show that the robust standard errors are rather conservative, though corresponding confidence intervals have good coverage. A second contribution of this paper is to introduce a Cox score bootstrap procedure to compute the standard errors. We show that this method is efficient and tends to outperform the non-parametric bootstrap in small samples.     

Derivatives and Fisher information of bivariate copulas

Data sets with complex relationships between random variables are increasingly studied in statistical applications. A popular approach to model their dependence is the use of copula functions. Our contribution is to derive expressions for the observed and expected information for several bivariate copula families, in particular for the Student’s $t$ -copula. Further likelihood derivatives which are required for numerical implementations are computed and a numerically stable implementation is provided in the R-package VineCopula. Using a real world data set of stock returns, we demonstrate the applicability of our approach for the routinely calculation of standard errors. In particular, we illustrate how this prevents overestimating the time-variation of dependence parameters in a rolling window analysis.     

Likelihood-free parallel tempering

Approximate Bayesian Computational (ABC) methods, or likelihood-free methods, have appeared in the past fifteen years as useful methods to perform Bayesian analysis when the likelihood is analytically or computationally intractable. Several ABC methods have been proposed: MCMC methods have been developed by Marjoram et al. (2003) and by Bortot et al. (2007) for instance, and sequential methods have been proposed among others by Sisson et al. (2007), Beaumont et al. (2009) and Del Moral et al. (2012). Recently, sequential ABC methods have appeared as an alternative to ABC-PMC methods (see for instance McKinley et al., 2009; Sisson et al., 2007). In this paper a new algorithm combining population-based MCMC methods with ABC requirements is proposed, using an analogy with the parallel tempering algorithm (Geyer 1991). Performance is compared with existing ABC algorithms on simulations and on a real example.     

An adaptive sequential Monte Carlo method for approximate Bayesian computation

Approximate Bayesian computation (ABC) is a popular approach to address inference problems where the likelihood function is intractable, or expensive to calculate. To improve over Markov chain Monte Carlo (MCMC) implementations of ABC, the use of sequential Monte Carlo (SMC) methods has recently been suggested. Most effective SMC algorithms that are currently available for ABC have a computational complexity that is quadratic in the number of Monte Carlo samples (Beaumont et al., Biometrika 86:983?C990, 2009; Peters et al., Technical report, 2008; Toni et al., J.?Roy. Soc. Interface 6:187?C202, 2009) and require the careful choice of simulation parameters. In this article an adaptive SMC algorithm is proposed which admits a computational complexity that is linear in the number of samples and adaptively determines the simulation parameters. We demonstrate our algorithm on a toy example and on a birth-death-mutation model arising in epidemiology.     

Statistische Analyse ausgewählter Schätzer im kontemporär korrelierten Regressionsmodell mit prädeterminiertem verzögerten Regressor und autokorrelierten Fehlern

Under a weaker assumption of independency apartial analysis of single equations is possiblewithout the specification of a simultaneous equation model. In the extended simple regression model with the weaker independency assumption the OLS-estimator turns out to bequite robust, even in extreme variations, whereas the GLS-estimator shows agreat sensitivity with regard to the modification of the independency. A Monte Carlo study confirms the results concerning the asymptotic bias and indicates a higher variance for the GLS- than for the OLS-estimator. In small samples the standard deviation of the OLS-estimator is smaller than the deviation of a consistent instrumental variables estimator which asymptotic efficiency loss compared to the efficient GLS-estimator in the stochastically independent regression is small as long as the regressor has a high autocorrelation.     

Parametric survival densities from phase-type models

After a brief historical survey of parametric survival models, from actuarial, biomedical, demographical and engineering sources, this paper discusses the persistent reasons why parametric models still play an important role in exploratory statistical research. The phase-type models are advanced as a flexible family of latent-class models with interpretable components. These models are now supported by computational statistical methods that make numerical calculation of likelihoods and statistical estimation of parameters feasible in theory for quite complicated settings. However, consideration of Fisher Information and likelihood-ratio type tests to discriminate between model families indicates that only the simplest phase-type model topologies can be stably estimated in practice, even on rather large datasets. An example of a parametric model with features of mixtures, multiple stages or ‘hits’, and a trapping-state is given to illustrate simple computational tools in R, both on simulated data and on a large SEER 1992–2002 breast-cancer dataset.     

On the joint distribution of success runs of several lengths in the sequence of MBT and its applications

Let \(X_1 ,X_2 ,\ldots ,X_n \) be a sequence of Markov Bernoulli trials (MBT) and \(\underline{X}_n =( {X_{n,k_1 } ,X_{n,k_2 } ,\ldots ,X_{n,k_r } })\) be a random vector where \(X_{n,k_i } \) represents the number of occurrences of success runs of length \(k_i \,( {i=1,2,\ldots ,r})\) . In this paper the joint distribution of \(\underline{X}_n \) in the sequence of \(n\) MBT is studied using method of conditional probability generating functions. Five different counting schemes of runs namely non-overlapping runs, runs of length at least \(k\) , overlapping runs, runs of exact length \(k\) and \(\ell \) -overlapping runs (i.e. \(\ell \) -overlapping counting scheme), \(0\le \ell are considered. The pgf of joint distribution of \(\underline{X}_n \) is obtained in terms of matrix polynomial and an algorithm is developed to get exact probability distribution. Numerical results are included to demonstrate the computational flexibility of the developed results. Various applications of the joint distribution of \(\underline{X}_n \) such as in evaluation of the reliability of \(( {n,f,k})\!\!:\!\!G\) and \(\!:\!\!G\) system, in evaluation of quantities related to start-up demonstration tests, acceptance sampling plans are also discussed.     

Order selection in finite mixtures of linear regressions

Finite mixture models can adequately model population heterogeneity when this heterogeneity arises from a finite number of relatively homogeneous clusters. An example of such a situation is market segmentation. Order selection in mixture models, i.e. selecting the correct number of components, however, is a problem which has not been satisfactorily resolved. Existing simulation results in the literature do not completely agree with each other. Moreover, it appears that the performance of different selection methods is affected by the type of model and the parameter values. Furthermore, most existing results are based on simulations where the true generating model is identical to one of the models in the candidate set. In order to partly fill this gap we carried out a (relatively) large simulation study for finite mixture models of normal linear regressions. We included several types of model (mis)specification to study the robustness of 18 order selection methods. Furthermore, we compared the performance of these selection methods based on unpenalized and penalized estimates of the model parameters. The results indicate that order selection based on penalized estimates greatly improves the success rates of all order selection methods. The most successful methods were \(MDL2\) , \(MRC\) , \(MRC_k\) , \(ICL\) – \(BIC\) , \(ICL\) , \(CAIC\) , \(BIC\) and \(CLC\) but not one method was consistently good or best for all types of model (mis)specification.     

Bayesian parameter inference for partially observed stopped processes

We consider Bayesian parameter inference associated to partially-observed stochastic processes that start from a set B ₀ and are stopped or killed at the first hitting time of a known set A. Such processes occur naturally within the context of a wide variety of applications. The associated posterior distributions are highly complex and posterior parameter inference requires the use of advanced Markov chain Monte Carlo (MCMC) techniques. Our approach uses a recently introduced simulation methodology, particle Markov chain Monte Carlo (PMCMC) (Andrieu et al. 2010), where sequential Monte Carlo (SMC) (Doucet et al. 2001; Liu 2001) approximations are embedded within MCMC. However, when the parameter of interest is fixed, standard SMC algorithms are not always appropriate for many stopped processes. In Chen et al. (2005), Del Moral (2004), the authors introduce SMC approximations of multi-level Feynman-Kac formulae, which can lead to more efficient algorithms. This is achieved by devising a sequence of sets from B ₀ to A and then performing the resampling step only when the samples of the process reach intermediate sets in the sequence. The choice of the intermediate sets is critical to the performance of such a scheme. In this paper, we demonstrate that multi-level SMC algorithms can be used as a proposal in PMCMC. In addition, we introduce a flexible strategy that adapts the sets for different parameter proposals. Our methodology is illustrated on the coalescent model with migration.     

Equalities between OLSE,BLUE and BLUP in the linear model

We consider equalities between the ordinary least squares estimator ( $\mathrm {OLSE} $ ), the best linear unbiased estimator ( $\mathrm {BLUE} $ ) and the best linear unbiased predictor ( $\mathrm {BLUP} $ ) in the general linear model $\{ \mathbf y , \mathbf X \varvec{\beta }, \mathbf V \}$ extended with the new unobservable future value $ \mathbf y _{*}$ of the response whose expectation is $ \mathbf X _{*}\varvec{\beta }$ . Our aim is to provide some new insight and new proofs for the equalities under consideration. We also collect together various expressions, without rank assumptions, for the $\mathrm {BLUP} $ and provide new results giving upper bounds for the Euclidean norm of the difference between the $\mathrm {BLUP} ( \mathbf y _{*})$ and $\mathrm {BLUE} ( \mathbf X _{*}\varvec{\beta })$ and between the $\mathrm {BLUP} ( \mathbf y _{*})$ and $\mathrm {OLSE} ( \mathbf X _{*}\varvec{\beta })$ . A remark is made on the application to small area estimation.     

Let us do the twist again

Krämer (Sankhy $\bar{\mathrm{a }}$ 42:130–131, 1980) posed the following problem: “Which are the $\mathbf{y}$ , given $\mathbf{X}$ and $\mathbf{V}$ , such that OLS and Gauss–Markov are equal?”. In other words, the problem aimed at identifying those vectors $\mathbf{y}$ for which the ordinary least squares (OLS) and Gauss–Markov estimates of the parameter vector $\varvec{\beta }$ coincide under the general Gauss–Markov model $\mathbf{y} = \mathbf{X} \varvec{\beta } + \mathbf{u}$ . The problem was later called a “twist” to Kruskal’s Theorem, which provides conditions necessary and sufficient for the OLS and Gauss–Markov estimates of $\varvec{\beta }$ to be equal. The present paper focuses on a similar problem to the one posed by Krämer in the aforementioned paper. However, instead of the estimation of $\varvec{\beta }$ , we consider the estimation of the systematic part $\mathbf{X} \varvec{\beta }$ , which is a natural consequence of relaxing the assumption that $\mathbf{X}$ and $\mathbf{V}$ are of full (column) rank made by Krämer. Further results, dealing with the Euclidean distance between the best linear unbiased estimator (BLUE) and the ordinary least squares estimator (OLSE) of $\mathbf{X} \varvec{\beta }$ , as well as with an equality between BLUE and OLSE are also provided. The calculations are mostly based on a joint partitioned representation of a pair of orthogonal projectors.     

On the central limit theorem along subsequences of sums of i.i.d. random variables

Let \(\mathbb{N } = \{1, 2, 3, \ldots \}\) . Let \(\{X, X_{n}; n \in \mathbb N \}\) be a sequence of i.i.d. random variables, and let \(S_{n} = \sum _{i=1}^{n}X_{i}, n \in \mathbb N \) . Then \( S_{n}/\sqrt{n} \Rightarrow N(0, \sigma ^{2})\) for some \(\sigma ^{2} < \infty \) whenever, for a subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/\sqrt{n_{k}} \Rightarrow N(0, \sigma ^{2})\) . Motivated by this result, we study the central limit theorem along subsequences of sums of i.i.d. random variables when \(\{\sqrt{n}; n \in \mathbb N \}\) is replaced by \(\{\sqrt{na_{n}};n \in \mathbb N \}\) with \(\lim _{n \rightarrow \infty } a_{n} = \infty \) . We show that, for given positive nondecreasing sequence \(\{a_{n}; n \in \mathbb N \}\) with \(\lim _{n \rightarrow \infty } a_{n} = \infty \) and \(\lim _{n \rightarrow \infty } a_{n+1}/a_{n} = 1\) and given nondecreasing function \(h(\cdot ): (0, \infty ) \rightarrow (0, \infty )\) with \(\lim _{x \rightarrow \infty } h(x) = \infty \) , there exists a sequence \(\{X, X_{n}; n \in \mathbb N \}\) of symmetric i.i.d. random variables such that \(\mathbb E h(|X|) = \infty \) and, for some subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/\sqrt{n_{k}a_{n_{k}}} \Rightarrow N(0, 1)\) . In particular, for given \(0 < p < 2\) and given nondecreasing function \(h(\cdot ): (0, \infty ) \rightarrow (0, \infty )\) with \(\lim _{x \rightarrow \infty } h(x) = \infty \) , there exists a sequence \(\{X, X_{n}; n \in \mathbb N \}\) of symmetric i.i.d. random variables such that \(\mathbb E h(|X|) = \infty \) and, for some subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/n_{k}^{1/p} \Rightarrow N(0, 1)\) .     

Testing for cross-sectional dependence in a panel factor model using the wild bootstrap F test

This paper considers testing for cross-sectional dependence in a panel factor model. Based on the model considered by Bai (Econometrica 71: 135–171, 2003), we investigate the use of a simple $F$ test for testing for cross-sectional dependence when the factor may be known or unknown. The limiting distributions of these $F$ test statistics are derived when the cross-sectional dimension and the time-series dimension are both large. The main contribution of this paper is to propose a wild bootstrap $F$  test which is shown to be consistent and which performs well in Monte Carlo simulations especially when the factor is unknown.     

A unified method for constructing expectation tolerance intervals

Given a random sample of size \(n\) with mean \(\overline{X} \) and standard deviation \(s\) from a symmetric distribution \(F(x; \mu , \sigma ) = F_{0} (( x- \mu ) / \sigma ) \) with \(F_0\) known, and \(X \sim F(x;\; \mu , \sigma )\) independent of the sample, we show how to construct an expansion \( a_n^{\prime } = \sum _{i=0}^\infty \ c_i \ n^{-i} \) such that \(\overline{X} - s a_n^{\prime } < X < \overline{X} + s a_n^{\prime } \) with a given probability \(\beta \) . The practical value of this result is illustrated by simulation and using a real data set.     

Computation for intrinsic variable selection in normal regression models via expected-posterior prior

In this paper, we focus on the variable selection problem in normal regression models using the expected-posterior prior methodology. We provide a straightforward MCMC scheme for the derivation of the posterior distribution, as well as Monte Carlo estimates for the computation of the marginal likelihood and posterior model probabilities. Additionally, for large spaces, a model search algorithm based on $\mathit{MC}^{3}$ is constructed. The proposed methodology is applied in two real life examples, already used in the relevant literature of objective variable selection. In both examples, uncertainty over different training samples is taken into consideration.     

Conditional MSE-based discrimination of the sample mean and the post-stratification estimator in population sampling

Whenever a random sample is drawn from a stratified population, the post-stratification estimator $\tilde X$ usually is preferred to the sample mean $\tilde X$ , when the population mean is to be estimated. This is due to the fact that the variance of $\tilde X$ is asymptotically smaller than that of $\tilde X$ , while both estimators are asymptotically unbiased. However, this only holds looking at post-stratification unconditionally, when strata sample sizes are random. Conditioned on the realized sample sizes, the MSE of $\tilde X$ can be higher than that of $\tilde X$ which means that $\tilde X$ should be preferred to $\tilde X$ , even if it is biased. The conditional MSE difference of $\tilde X$ and $\tilde X$ is estimated, and using this estimation and its variance a heuristic test based on the Vysochanskiî-Petunin inequality is derived.     

On exact and optimal single-sampling plans by variables

We deal with sampling by variables with two-way protection in the case of a $N\>(\mu ,\sigma ^2)$ distributed characteristic with unknown $\sigma $ . The LR sampling plan proposed by Lieberman and Resnikoff (JASA 50: 457 ${-}$ 516, 1955) and the BSK sampling plan proposed by Bruhn-Suhr and Krumbholz (Stat. Papers 31: 195–207, 1990) are based on the UMVU and the plug-in estimator, respectively. For given $p_1$ (AQL), $p_2$ (RQL) and $\alpha ,\beta $ (type I and II errors) we present an algorithm allowing to determine the optimal LR and BSK plans having minimal sample size among all plans satisfying the corresponding two-point condition on the OC. An R (R: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org/ 2012) package, ExLiebeRes‘ (Krumbholz and Steuer ExLiebeRes: calculating exact LR- and BSK-plans, R-package version 0.9.9. http://exlieberes.r-forge.r-project.org 2012) implementing that algorithm is provided to the public.     

Structured orthogonal families of one and two strata prime basis factorial models

The models in structured families correspond to the treatments of a fixed effects base design \(\pi \) . Then the action of factors in \(\pi \) , on the fixed effects parameters of the models, is studied. Analyzing such a families enables the study of the action of nesting factors on the effects and interactions of nested factors. When \(\pi \) has an orthogonal structure, the family of models is said to be orthogonal. The models in the family can have one, two or more strata. Models with more than one stratum are obtained through nesting of one stratum models. A general treatment of the case in which the base design has orthogonal structure is presented and a special emphasis is given to the families of prime basis factorials models. These last models are, as it is well known, widely used in fertilization trials.     

EWMA charts: ARL considerations in case of changes in location and scale

Widely spread tools within the area of Statistical Process Control are control charts of various designs. Control chart applications are used to keep process parameters (e.g., mean \(\mu \) , standard deviation \(\sigma \) or percent defective \(p\) ) under surveillance so that a certain level of process quality can be assured. Well-established schemes such as exponentially weighted moving average charts (EWMA), cumulative sum charts or the classical Shewhart charts are frequently treated in theory and practice. Since Shewhart introduced a \(p\) chart (for attribute data), the question of controlling the percent defective was rarely a subject of an analysis, while several extensions were made using more advanced schemes (e.g., EWMA) to monitor effects on parameter deteriorations. Here, performance comparisons between a newly designed EWMA \(p\) control chart for application to continuous types of data, \(p=f(\mu ,\sigma )\) , and popular EWMA designs ( \(\bar{X}\) , \(\bar{X}\) - \(S^2\) ) are presented. Thus, isolines of the average run length are introduced for each scheme taking both changes in mean and standard deviation into account. Adequate extensions of the classical EWMA designs are used to make these specific comparisons feasible. The results presented are computed by using numerical methods.     

Principal points for an allometric extension model

A set of \(n\) -principal points of a \(p\) -dimensional distribution is an optimal \(n\) -point-approximation of the distribution in terms of a squared error loss. It is in general difficult to derive an explicit expression of principal points. Hence, we may have to search the whole space \(R^p\) for \(n\) -principal points. Many efforts have been devoted to establish results that specify a linear subspace in which principal points lie. However, the previous studies focused on elliptically symmetric distributions and location mixtures of spherically symmetric distributions, which may not be suitable to many practical situations. In this paper, we deal with a mixture of elliptically symmetric distributions that form an allometric extension model, which has been widely used in the context of principal component analysis. We give conditions under which principal points lie in the linear subspace spanned by the first several principal components.