Inferring large graphs using $$\ell _1$$-penalized likelihood

We address the issue of recovering the structure of large sparse directed acyclic graphs from noisy observations of the system. We propose a novel procedure based on a specific formulation of the \(\ell _1\)-norm regularized maximum likelihood, which decomposes the graph estimation into two optimization sub-problems: topological structure and node order learning. We provide convergence inequalities for the graph estimator, as well as an algorithm to solve the induced optimization problem, in the form of a convex program embedded in a genetic algorithm. We apply our method to various data sets (including data from the DREAM4 challenge) and show that it compares favorably to state-of-the-art methods. This algorithm is available on CRAN as the R package GADAG.     

Stability of feature selection in classification issues for high-dimensional correlated data

Handling dependence or not in feature selection is still an open question in supervised classification issues where the number of covariates exceeds the number of observations. Some recent papers surprisingly show the superiority of naive Bayes approaches based on an obviously erroneous assumption of independence, whereas others recommend to infer on the dependence structure in order to decorrelate the selection statistics. In the classical linear discriminant analysis (LDA) framework, the present paper first highlights the impact of dependence in terms of instability of feature selection. A second objective is to revisit the above issue using a flexible factor modeling for the covariance. This framework introduces latent components of dependence, conditionally on which a new Bayes consistency is defined. A procedure is then proposed for the joint estimation of the expectation and variance parameters of the model. The present method is compared to recent regularized diagonal discriminant analysis approaches, assuming independence among features, and regularized LDA procedures, both in terms of classification performance and stability of feature selection. The proposed method is implemented in the R package FADA, freely available from the R repository CRAN.     

Doubly reweighted estimators for the parameters of the multivariate t-distribution

The t-distribution (univariate and multivariate) has many useful applications in robust statistical analysis. The parameter estimation of the t-distribution is carried out using maximum likelihood (ML) estimation method, and the ML estimates are obtained via the Expectation-Maximization (EM) algorithm. In this article, we will use the maximum Lq-likelihood (MLq) estimation method introduced by Ferrari and Yang (2010 Ferrari, D., and Y. Yang. 2010. Maximum lq-likelihood estimation. The Annals of Statistics 38 (2):753–83.[Crossref], [Web of Science ®] , [Google Scholar]) to estimate all the parameters of the multivariate t-distribution. We modify the EM algorithm to obtain the MLq estimates. We provide a simulation study and a real data example to illustrate the performance of the MLq estimators over the ML estimators.     

Bayes estimation for exponential distributions with common location parameter and applications to multi-state reliability models

This paper considers the estimation of the stress–strength reliability of a multi-state component or of a multi-state system where its states depend on the ratio of the strength and stress variables through a kernel function. The article presents a Bayesian approach assuming the stress and strength as exponentially distributed with a common location parameter but different scale parameters. We show that the limits of the Bayes estimators of both location and scale parameters under suitable priors are the maximum likelihood estimators as given by Ghosh and Razmpour [15 M. Ghosh and A. Razmpour, Estimation of the common location parameter of several exponentials, Sankhyā, Ser. A 46 (1984), pp. 383–394. [Google Scholar]]. We use the Bayes estimators to determine the multi-state stress–strength reliability of a system having states between 0 and 1. We derive the uniformly minimum variance unbiased estimators of the reliability function. Interval estimation using the bootstrap method is also considered. Under the squared error loss function and linex loss function, risk comparison of the reliability estimators is carried out using extensive simulations.     

Semi-parametric Efficient Inference for Heteroscedastic Semivarying-coefficient Models

Semivarying-coefficient models with heteroscedastic errors are frequently used in statistical modeling. When the error is conditional heteroskedastic, Ahmad, et al. (2005 Ahmad, I., Leelahanon, S., Li, Q. (2005). Efficient estimation of a semiparametric partially linear varying coefficient model. Ann. Statist. 33(1):258–283.[Crossref], [Web of Science ®] , [Google Scholar]) proposed a general series method to obtain an efficient estimation. In this article we study the heteroscedastic semi-varying coefficient models with a nonparametric variance function, not only use the semi-parametric efficient normal approximation method to derive a family of semi-parametric efficient estimator, but also use the semi-parametric efficient empirical likelihood method to construct the efficient empirical likelihood confidence regions. The proposed estimators retain the double robustness feature of semi-parametric efficient estimator.     

On Maximum Likelihood Estimation for the Two Parameter Burr XII Distribution

This article considers three related aspects of maximum likelihood estimation of parameters in the two-parameter Burr XII distribution. Specifically, we first provide further clarification to some limiting results in Wingo (1993 Wingo , D. R. ( 1993 ). Maximum likelihood estimation of Burr XII distribution parameters under Type II censoring . Microelectron. Reliab. 33 : 1251 – 1257 .[Crossref], [Web of Science ®] , [Google Scholar]). We then focus on details in a proof of the uniqueness of the maximum likelihood estimators. Finally, we consider using the likelihood approach for data which does not satisfy Wingo's criterion, and show that this results in fitting either a Pareto distribution or an intuitively sensible degenerate distribution to the data. The discussion here is completely general, and not restricted to data obtained under Type II censoring.     

On Improvement in Variance Estimation Using Auxiliary Information

Kadilar and Cingi (2006 Kadilar , C. , Cingi , H. ( 2006 ). Improvement in variance estimation using auxiliary information . Hacett. J. Math. Statist. 35 ( 1 ): 111 – 115 . [Google Scholar]) have introduced an estimator for the population variance using an auxiliary variable in simple random sampling. We propose a new ratio-type exponential estimator for population variance which is always more efficient than usual ratio and regression estimators suggested by Isaki (1983 Isaki , C. T. ( 1983 ). Variance estimation using auxiliary information . J. Amer. Statist. Assoc. 78 : 117 – 123 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and by Kadilar and Cingi (2006 Kadilar , C. , Cingi , H. ( 2006 ). Improvement in variance estimation using auxiliary information . Hacett. J. Math. Statist. 35 ( 1 ): 111 – 115 . [Google Scholar]). Efficiency comparison is carried out both mathematically and numerically.     

Confidence Intervals for the Scale Parameter of the Power-Law Process

The power-law process (PLP) is a two-parameter model widely used for modeling repairable system reliability. Results on exact point estimation for both parameters as well as exact interval estimation for the shape parameter are well known. In this paper, we investigate the interval estimation for the scale parameter. Asymptotic confidence intervals are derived using Fisher information matrix and theoretical results by Cocozza-Thivent (1997 Cocozza-Thivent , C. ( 1997 ). Processus Stochastiques et Fiabilité des Systèmes . Berlin : Springer-Verlag . [Google Scholar]). The accuracy of the interval estimation for finite samples is studied by simulation methods.     

Estimation Methods for an Autoregressive Familial Correlation Structure

In this article, we apply an autoregressive correlation structure to the analysis of balanced familial clustered data in the one-parent case with homogeneous intra-class variance. We use the quasi-least squares procedure to derive estimators of the correlation parameters and compare them with maximum likelihood and moment estimators. Asymptotically, the quasi-least squares estimators are nearly as efficient as the maximum likelihood estimators. The small-sample case is analyzed through simulation, and the quasi-least squares estimators are found to be more robust than the maximum likelihood estimators. To show the application of the estimation procedures, data provided in Katapa (1993 Katapa , R. S. ( 1993 ). A test of hypothesis on familial correlations . Biometrics 49 : 569 – 576 . [Google Scholar]) are re-analyzed. For non stationary unbalanced familial data, we outline general correlation models which are natural extensions of the structure studied in this article.     

On generalized hyper-Poisson distribution of order k and its application

ABSTRACT

Here we introduce a new class of distributions namely the generalized hyper-Poisson distribution of order k (GHPD(k)) as an order k version of the alpha-generalized hyper-Poisson distribution of Kumar and Nair (Statistica, 2014b Kumar, C.S., Nair, B.U. (2014b). A three parameter hyper-Poisson distribution and some of its properties. Statistica. 74(2):183–198. [Google Scholar]). Several properties of the GHPD(k) are derived and the estimation of the parameters of the distribution by the method of mixed moments and the method of maximum likelihood is discussed. Certain testing procedures are suggested and all these estimation and testing procedures are illustrated with the help of a real-life data set. Further a simulation study is conducted.     

Prognostic assessment of repeatedly measured time-dependent biomarkers,with application to dilated cardiomyopathy

We propose new time-dependent sensitivity, specificity, ROC curves and net reclassification indices that can take into account biomarkers or scores that are repeatedly measured at different time-points. Inference proceeds through inverse probability weighting and resampling. The newly proposed measures exploit the information contained in biomarkers measured at different visits, rather than using only the measurements at the first visits. The contribution is illustrated via simulations and an original application on patients affected by dilated cardiomiopathy. The aim is to evaluate if repeated binary measurements of right ventricular dysfunction bring additive prognostic information on mortality/urgent heart transplant. It is shown that taking into account the trajectory of the new biomarker improves risk classification, while the first measurement alone might not be sufficiently informative. The methods are implemented in an R package (longROC), freely available on CRAN.     

Fitting spatial regressions to large datasets using unilateral approximations

Maximum likelihood estimation of a spatial model typically requires a sizeable computational capacity, even in relatively small samples, and becomes unfeasible in very large datasets. The unilateral approximation approach to spatial model estimation (suggested in Besag 1974 Besag, J. E. 1974. Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society. Series B (Methodological) 36 (2):192–236.[Crossref], [Web of Science ®] , [Google Scholar]) provides a viable alternative to maximum likelihood estimation that reduces substantially the computing time and the storage required. In this article, we extend the method, originally proposed for conditionally specified processes, to simultaneous and to general bilateral spatial processes over rectangular lattices. We prove the estimators’ consistency and study their finite-sample properties via Monte Carlo simulations.     

A variance shift model for detection of outliers in the linear mixed measurement error models

ABSTRACT

In this paper, we extend a variance shift model, previously considered in the linear mixed models, to the linear mixed measurement error models using the corrected likelihood of Nakamura (1990 Nakamura, T. (1990). Corrected score function for errors in variables models: methodology and application to generalized linear models. Biometrika 77:127–137.[Crossref], [Web of Science ®] , [Google Scholar]). This model assumes that a single outlier arises from an observation with inflated variance. We derive the score test and the analogue of the likelihood ratio test, to assess whether the ith observation has inflated variance. A parametric bootstrap procedure is implemented to obtain empirical distributions of the test statistics. Finally, results of a simulation study and an example of real data are presented to illustrate the performance of proposed tests.     

A Realized Stochastic Volatility Model With Box–Cox Transformation

This article presents a new class of realized stochastic volatility model based on realized volatilities and returns jointly. We generalize the traditionally used logarithm transformation of realized volatility to the Box–Cox transformation, a more flexible parametric family of transformations. A two-step maximum likelihood estimation procedure is introduced to estimate this model on the basis of Koopman and Scharth (2013 Koopman, S.J., Scharth, M. (2013), The Analysis of Stochastic Volatility in the Presence of Daily Realised Measures, Journal of Financial Econometrics, 11, 76–115.[Crossref], [Web of Science ®] , [Google Scholar]). Simulation results show that the two-step estimator performs well, and the misspecified log transformation may lead to inaccurate parameter estimation and certain excessive skewness and kurtosis. Finally, an empirical investigation on realized volatility measures and daily returns is carried out for several stock indices.     

A Marginalized Combined Gamma Frailty and Normal Random-effects Model for Repeated,Overdispersed, Time-to-event Outcomes

This article proposes a marginalized model for repeated or otherwise hierarchical, overdispersed time-to-event outcomes, adapting the so-called combined model for time-to-event outcomes of Molenberghs et al. (in press Molenberghs, G., Verbeke, G., Efendi, A., Braekers, R., Demétrio, C. G.B. (in press). A combined gamma frailty and normal random-effects model for repeated, overdispersed time-to-event data. In press. [Google Scholar]), who combined gamma and normal random effects. The two sets of random effects are used to accommodate simultaneously correlation between repeated measures and overdispersion. The proposed version allows for a direct marginal interpretation of all model parameters. The outcomes are allowed to be censored. Two estimation methods are proposed: full likelihood and pairwise likelihood. The proposed model is applied to data from a so-called comet assay and to data from recurrent asthma attacks in children. Both estimation methods perform very well. From simulation results, it follows that the marginalized combined model behaves similarly to the ordinary combined model in terms of point estimation and precision. It is also observed that the pairwise likelihood required more computation time on the one hand but is less sensitive to starting values and stabler in terms of bias with increasing sample size and censoring percentage than full likelihood, on the other, leaving room for both in practice.     

Density estimation via the random forest method

The problem of density estimation arises naturally in many contexts. In this paper, we consider the approach using a piecewise constant function to approximate the underlying density. We present a new density estimation method via the random forest method based on the Bayesian Sequential Partition (BSP) (Lu, Jiang, and Wong 2013 Lu, L., H. Jiang, and W. H. Wong, 2013. Multivariate density estimation by Bayesian Sequential Partitioning. Journal of the American Statistical Association 108(504):1402–10.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). Extensive simulations are carried out with comparison to the kernel density estimation method, BSP method, and four local kernel density estimation methods. The experiment results show that the new method is capable of providing accurate and reliable density estimation, even at the boundary, especially for i.i.d. data. In addition, the likelihood of the out-of-bag density estimation, which is a byproduct of the training process, is an effective hyperparameter selection criterion.     

Deviation Inequalities for the Estimator of Linear Parameter in Stochastic Processes

In this article, we obtain some deviation inequalities by non asymptotic approach, which extend the work of Liptser and Spokoiny (2000 Liptser , R. , Spokoiny , V. ( 2000 ). Deviation probability bound for martingales with applications to statistical estimation . Statist. Probab. Lett. 46 ( 4 ): 347 – 357 .[Crossref], [Web of Science ®] , [Google Scholar]) in some sense, and apply them to the problem of the parameter estimation for more general regression models and Itô equations without any ergodic assumptions. In particular, for some special parameter estimation problems of discrete-time stochastic processes, we also give the more precise estimation.     

Autocovariance Function of the Fractionally Integrated Separable Spatial ARMA (FISSARMA) Models

Spatial modeling is important in many fields and there are various kinds of spatial models. One of such models is known as the fractionally integrated separable spatial ARMA (FISSARMA) model. In the area of time series analysis, Sowell (1992 Sowell, F. (1992). Maximum likelihood estimation of stationary univariate fractionally integrated time series models. J. Econ. 53:165–188.[Crossref], [Web of Science ®] , [Google Scholar]) has established the autocovariance function of the long-memory models using hypergeometric function. In this paper we will extend Sowell’s work for FISSARMA models.     

Confidence Interval of the Difference of Two Independent Binomial Proportions Using Weighted Profile Likelihood

Interval estimation of the difference of two independent binomial proportions is an important problem in many applied settings. Newcombe (1998 Newcombe , R. G. ( 1998 ). Interval estimation for the difference between independent proportions: comparison of seven methods . Statistics in Medicine 17 : 873 – 890 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) compared the performance of several existing asymptotic methods, and based on the results obtained, recommended a method known as Wilson's method, a modified version of a method originally proposed for single binomial proportion. In this article, we propose a method based on profile likelihood, where the likelihood is weighted by noninformative Jeffrey' prior. By doing extensive simulations, we find that the proposed method performs well compared to Wilson's method. A SAS/IML program implementing this method is also given with this article.     

Large sample confidence intervals for the skewness parameter of the skew-normal distribution based on Fisher's transformation

The skew-normal model is a class of distributions that extends the Gaussian family by including a skewness parameter. This model presents some inferential problems linked to the estimation of the skewness parameter. In particular its maximum likelihood estimator can be infinite especially for moderate sample sizes and is not clear how to calculate confidence intervals for this parameter. In this work, we show how these inferential problems can be solved if we are interested in the distribution of extreme statistics of two random variables with joint normal distribution. Such situations are not uncommon in applications, especially in medical and environmental contexts, where it can be relevant to estimate the distribution of extreme statistics. A theoretical result, found by Loperfido [7 Loperfido, N. 2002. Statistical implications of selectively reported inferential results. Statist. Probab. Lett., 56: 13–22. [Crossref], [Web of Science ®] , [Google Scholar]], proves that such extreme statistics have a skew-normal distribution with skewness parameter that can be expressed as a function of the correlation coefficient between the two initial variables. It is then possible, using some theoretical results involving the correlation coefficient, to find approximate confidence intervals for the parameter of skewness. These theoretical intervals are then compared with parametric bootstrap intervals by means of a simulation study. Two applications are given using real data.