Likelihood-based risk estimation for variance-gamma models

Although the variance-gamma distribution is a flexible model for log-returns of financial assets, so far it has found rather limited applications in finance and risk management. One of the reasons is that maximum likelihood estimation of its parameters is not straightforward. We develop an EM-type algorithm based on Nitithumbundit and Chan (An ECM algorithm for skewed multivariate variance gamma distribution in normal mean–variance representation, arXiv:1504.01239, 2015) that bypasses the evaluation of the full likelihood, which may be difficult because the density is not in closed form and is unbounded for small values of the shape parameter. Moreover, we study the relative efficiency of our approach with respect to the maximum likelihood estimation procedures implemented in the VarianceGamma and ghyp R packages. Extensive simulation experiments and real-data analyses suggest that the multicycle ECM algorithm gives the best results in terms of root-mean-squared-error, for both parameter and value-at-risk estimation. The performance of the routines in the ghyp R package is similar but not as good, whereas the VarianceGamma package produces worse results, especially when the shape parameter is small.     

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.     

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.     

The Dantzig Discriminant Analysis with High Dimensional Data

It is well known that linear discriminant analysis (LDA) works well and is asymptotically optimal under fixed-p-large-n situations. But Bickel and Levina (2004 Bickel, P.J., Levina, E. (2004). Some theory for Fishers linear discriminant function, naive Bayes, and some alternatives when there are many more variables than observations. Bernoulli 10:989–1010.[Crossref], [Web of Science ®] , [Google Scholar]) showed that the LDA is as bad as random guessing when p > n. This article studies the sparse discriminant analysis via Dantzig penalized least squares. Our method avoids estimating the high-dimensional covariance matrix and does not need the sparsity assumption on the inverse of the covariance matrix. We show that the new discriminant analysis is asymptotically optimal theoretically. Simulation and real data studies show that the classifier performs better than the existing sparse methods.     

Quasi-random numbers for copula models

The present work addresses the question how sampling algorithms for commonly applied copula models can be adapted to account for quasi-random numbers. Besides sampling methods such as the conditional distribution method (based on a one-to-one transformation), it is also shown that typically faster sampling methods (based on stochastic representations) can be used to improve upon classical Monte Carlo methods when pseudo-random number generators are replaced by quasi-random number generators. This opens the door to quasi-random numbers for models well beyond independent margins or the multivariate normal distribution. Detailed examples (in the context of finance and insurance), illustrations and simulations are given and software has been developed and provided in the R packages copula and qrng.     

Discriminant analysis under the common principal components model

For two or more populations of which the covariance matrices have a common set of eigenvectors, but different sets of eigenvalues, the common principal components (CPC) model is appropriate. Pepler et al. (2015 Pepler, P. T., Uys, D. W. and Nel, D. G. (2015). Regularised covariance matrix estimation under the common principal components model. Communications in Statistics: Simulation and Computation. (In press). [Google Scholar]) proposed a regularized CPC covariance matrix estimator and showed that this estimator outperforms the unbiased and pooled estimators in situations, where the CPC model is applicable. This article extends their work to the context of discriminant analysis for two groups, by plugging the regularized CPC estimator into the ordinary quadratic discriminant function. Monte Carlo simulation results show that CPC discriminant analysis offers significant improvements in misclassification error rates in certain situations, and at worst performs similar to ordinary quadratic and linear discriminant analysis. Based on these results, CPC discriminant analysis is recommended for situations, where the sample size is small compared to the number of variables, in particular for cases where there is uncertainty about the population covariance matrix structures.     

The Asymptotic Approximation of EPMC for Linear Discriminant Rules Using a Moore-Penrose Inverse Matrix in High Dimension

We consider the discriminant rule in a high-dimensional setting, i.e., when the number of feature variables p is comparable to or larger than the number of observations N. The discriminant rule must be modified in order to cope with singular sample covariance matrix in high-dimension. One way to do so is by considering the Moor-Penrose inverse matrix. Recently, Srivastava (2006 Srivastava , M. S. ( 2006 ). Minimum distance classification rules for high dimensional data . J. Multivariate Anal. 97 : 2057 – 2070 .[Crossref], [Web of Science ®] , [Google Scholar]) proposed maximum likelihood ratio rule by using Moor-Penrose inverse matrix of sample covariance matrix. In this article, we consider the linear discriminant rule by using Moor-Penrose inverse matrix of sample covariance matrix (LDRMP). With the discriminant rule, the expected probability of misclassification (EPMC) is commonly used as measure of the classification accuracy. We investigate properties of EPMC for LDRMP in high-dimension as well as the one of the maximum likelihood rule given by Srivastava (2006 Srivastava , M. S. ( 2006 ). Minimum distance classification rules for high dimensional data . J. Multivariate Anal. 97 : 2057 – 2070 .[Crossref], [Web of Science ®] , [Google Scholar]). From our asymptotic results, we show that the classification accuracy of LDRMP depends on new distance. Additionally, our asymptotic result is verified by using the Monte Carlo simulation.     

Objective Bayesian transformation and variable selection using default Bayes factors

In this work, the problem of transformation and simultaneous variable selection is thoroughly treated via objective Bayesian approaches by the use of default Bayes factor variants. Four uniparametric families of transformations (Box–Cox, Modulus, Yeo-Johnson and Dual), denoted by T, are evaluated and compared. The subjective prior elicitation for the transformation parameter \(\lambda _T\), for each T, is not a straightforward task. Additionally, little prior information for \(\lambda _T\) is expected to be available, and therefore, an objective method is required. The intrinsic Bayes factors and the fractional Bayes factors allow us to incorporate default improper priors for \(\lambda _T\). We study the behaviour of each approach using a simulated reference example as well as two real-life examples.     

A Model Selection Criterion for Discriminant Analysis of Several Groups When the Dimension is Larger than the Total Sample Size

This article deals with a criterion for selection of variables for the multiple group discriminant analysis in high-dimensional data. The variable selection models considered for discriminant analysis in Fujikoshi (1985 Fujikoshi , Y. ( 1985 ). Selection of variables in discriminant analysis and canonical correlation analysis . In: Krishnaiah , P. R. , ed. Multivariate Analysis . Vol. VI. Amsterdam : North-Holland , pp. 219 – 236 . [Google Scholar], 2002 Fujikoshi , Y. ( 2002 ). Selection of variables for discriminant analysis in a high-dimensional case . Sankhya Ser. A 64 : 256 – 257 . [Google Scholar]) are the ones based on additional information due to Rao (1948 Rao , C. R. ( 1948 ). Tests of significance in multivariate analysis . Biometrika 35 : 58 – 79 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar], 1970 Rao , C. R. ( 1970 ). Inference on discriminant function coefficients . In: Bose , R. C. , ed. Essays in Probability and Statistics . Chapel Hill , NC : University of North Carolina Press , pp. 537 – 602 . [Google Scholar]). Our criterion is based on Akaike information criterion (AIC) for this model. The AIC has been successfully used in the literature in model selection when the dimension p is smaller than the sample size N. However, the case when p > N has not been considered in the literature, because MLE can not be estimated corresponding to singularity of the within-group covariance matrix. A popular method used to address the singularity problem in high-dimensional classification is the regularized method, which replaces the within-group sample covariance matrix with a ridge-type covariance estimate to stabilize the estimate. In this article, we propose AIC-type criterion by replacing MLE of the within-group covariance matrix with ridge-type estimator. This idea follows Srivastava and Kubokawa (2008 Srivastava , M. S. , Kubokawa , T. ( 2008 ). Akaike information criterion for selecting components of the mean vector in high dimensional data with fewer observations . J. Japan Statist. Soc. 38 : 259 – 283 . [Google Scholar]). Simulations revealed that our proposed criterion performs well.     

Selecting the best exponential population under Type-II progressive censoring scheme via empirical Bayes approach

The problem of selecting a population according to “selection and ranking” is an important statistical problem. The ideas in selecting the best populations with some demands having optimal criterion have been suggested originally by Bechhofer (1954 Bechhofer, R. E. (1954). A single-sample multiple-decision procedure for ranking means of normal populations with known variances. The Annals of Mathematical Statistics 25:16–39. [Google Scholar]) and Gupta (1956 Gupta, S. S. (1956). On a decision rule for a problem in ranking means. Mimeograph Series No. 150. Chapel Hill, North Carolina: University of North Carolina. [Google Scholar], 1965 Gupta, S. S. (1965). On some multiple decision (selection and ranking) rules. Technometrics 7:225–245. [Google Scholar]). In the area of ranking and selection, the large part of literature is connected with a single criterion. However, this may not satisfy the experimenter’s demand. We follow methodology of Huang and Lai (1999 Huang, W. T., Lai, Y. T. (1999). Empirical Bayes procedures for selecting the best population with multiple criteria. Annals of the Institute of Statistical Mathematics 51:281–299. [Google Scholar]) and the main focus of this article is to select a best population under Type-II progressively censored data for the case of right tail exponential distributions with a bounded and unbounded supports for μ_i. We formulate the problem and develop a Bayesian setup with two kinds of bounded and unbounded prior for μ_i. We introduce an empirical Bayes procedure and study the large sample behavior of the proposed rule. It is shown that the proposed empirical Bayes selection rule is asymptotically optimal.     

Variable selection for survival data with a class of adaptive elastic net techniques

The accelerated failure time (AFT) models have proved useful in many contexts, though heavy censoring (as for example in cancer survival) and high dimensionality (as for example in microarray data) cause difficulties for model fitting and model selection. We propose new approaches to variable selection for censored data, based on AFT models optimized using regularized weighted least squares. The regularized technique uses a mixture of \(\ell _1\) and \(\ell _2\) norm penalties under two proposed elastic net type approaches. One is the adaptive elastic net and the other is weighted elastic net. The approaches extend the original approaches proposed by Ghosh (Adaptive elastic net: an improvement of elastic net to achieve oracle properties, Technical Reports 2007) and Hong and Zhang (Math Model Nat Phenom 5(3):115–133 2010), respectively. We also extend the two proposed approaches by adding censoring observations as constraints into their model optimization frameworks. The approaches are evaluated on microarray and by simulation. We compare the performance of these approaches with six other variable selection techniques-three are generally used for censored data and the other three are correlation-based greedy methods used for high-dimensional data.     

Expected Posterior Priors in Selecting the Largest Mean of the Exponential Distributions

In reliability theory or survival analysis, selecting the largest mean among many exponential distributions is an important issue. Such a problem can also be viewed as a model selection problem via the Bayesian approach. It is well known that Bayes factors under proper priors have been very successful in Bayesian model selection or testing problems. However, Bayes factors are typically invalid with respect to improper noninformative priors. Objective Bayesian criteria are thus desired. In this work, we consider to use the expected posterior priors originally proposed by Pérez and Berger (2002 Pérez , J. M. , Berger , J. ( 2002 ). Expected posterior prior distributions for model selection . Biometrika 89 : 491 – 512 .[Crossref], [Web of Science ®] , [Google Scholar]) to select the largest exponential mean. Specific expected posterior priors are derived in recursive formulas. Some simulation results are also given to illustrate the method.     

APPLYING THE SAVAGE-DICKEY DENSITY RATIO TO DEFAULT BAYES FACTORS,WITH AN ILLUSTRATION TO OUTLIER DETECTION IN RANDOM EFFECTS MODELS

When no information is available and hence improper noninformative priors should be used, Bayes factor includes the unspecified constants and can not be calibrated. To solve this problem, we modify the intrinsic Bayes factor (IBF) of Berger and Pericchi 1-2 Berger, J. O. and Pericchi, L. R. 1996. The Intrinsic Bayes Factor for Model Selection and Prediction. Journal of the American Statistical Association, 91: 109–122. Berger, J. O. and Pericchi, L. R. 1998. Accurate and Stable Bayesian Model Selection: The Median Intrinsic Bayes Factor. Sankhya, Series B, 60: 1–18.   and the fractional Bayes factor (FBF) of O'Hagan [3] O'Hagan, A. 1995. Fractional Bayes Factors for Model Comparison. Journal of the Royal Statistical Society, Series B, 57: 99–138.  [Google Scholar] with the generalized Savage-Dickey density ratio of Verdinelli and Wasserman [4] Verdinelli, I. and Wasserman, L. 1995. Computing Bayes Factors Using a Generalization of Savage-Dickey Density Ratio. Journal of the American Statistical Association, 90: 614–618. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]. These modified IBF and FBF are applied to detecting outliers in random effects models with a mean-shift structure. The proposed methodology is exemplified by a simulation experiment with a generated data set and also applied to a real data set, Dyestuff data in Box and Tiao [5] Box, G. E.P. and Tiao, G. C. 1973. Bayesian Inference in Statistical Analysis U.S.A.: Addison-Wesley Publishing Co..  [Google Scholar]     

A NOTE ON ‘A COMPARISON OF BAYES AND MAXIMUM LIKELIHOOD ESTIMATION OF THE INTRACLASS CORRELATION COEFFICIENT’

Palmer and Broemeling [1] Palmer, J. L. and Broemeling, L. D. 1990. A Comparison of Bayes and Maximum Likelihood Estimation of the Intraclass Correlation Coefficient. Comm. Statist.-Theory Meth, 19: 953–975. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] compare Bayes and maximum likelihood estimates of the intraclass correlation (ICC). The prior information in their derivation of the Bayes estimator is placed on the variance components instead of the ICC itself. This paper finds a Bayes estimator of the ICC with the prior placed on the ICC. Bayes estimates based on three different priors are then compared to method of moments estimate.     

SOME RESULTS ON TRUNCATION DEPENDENCE INVARIANT CLASS OF COPULAS

ABSTRACT

In this paper, m-dimensional distribution functions with truncation invariant dependence structure are studied. Some of the properties of generalized Archimedean class of copulas under this dependence structure are presented including some results on the conditions of compatibility. It has been shown that Archimedean copula generalized as it is described by Jouini and Clemen[1] Jouini, M.N. and Clemen, R.T. 1996. Copula Models for Aggregating Expert Opinions. Operations Research, 44(3): 444–457.  [Google Scholar] which has the truncation invariant dependence structure has to have the form of independence or Cook-Johnson copula. We also consider a multi-parameter class of copulas derived from one-parameter Archimedean copulas. It has been shown that this class has a probabilistic meaning as a connecting copula of the truncated random pair with a right truncation region on the third variable. Multi-parameter copulas generated in this paper stays in the Archimedean class. We provide formulas to compute Kendall's tau and explore the dependence behavior of this multi-parameter class through examples.     

Selection of Binary Variables and Classification by Boosting

We adopt boosting for classification and selection of high-dimensional binary variables for which classical methods based on normality and non singular sample dispersion are inapplicable. Boosting seems particularly well suited for binary variables. We present three methods of which two combine boosting with the relatively classical variable selection methods developed in Wilbur et al. (2002 Wilbur , J. D. , Ghosh , J. K. , Nakatsu , C. H. , Brouder , S. M. , Doerge , R. W. ( 2002 ). Variable selection in high-dimensional multivariate binary data with application to the analysis of microbial community DNA fingerprints . Biometrics 58 : 378 – 386 . [Google Scholar]). Our primary interest is variable selection in classification with small misclassification error being used as validation of proposed method for variable selection. Two of the new methods perform uniformly better than Wilbur et al. (2002 Wilbur , J. D. , Ghosh , J. K. , Nakatsu , C. H. , Brouder , S. M. , Doerge , R. W. ( 2002 ). Variable selection in high-dimensional multivariate binary data with application to the analysis of microbial community DNA fingerprints . Biometrics 58 : 378 – 386 . [Google Scholar]) in one set of simulated and three real life examples.     

Bayes minimax ridge regression estimators

The problem of estimating of the vector β of the linear regression model y = Aβ + ? with ? ～ N_p(0, σ²I_p) under quadratic loss function is considered when common variance σ² is unknown. We first find a class of minimax estimators for this problem which extends a class given by Maruyama and Strawderman (2005 Maruyama, Y., and W. E. Strawderman. 2005. A new class of generalized Bayes minimax ridge regression estimators. Annals of Statistics 33:1753–70.[Crossref], [Web of Science ®] , [Google Scholar]) and using these estimators, we obtain a large class of (proper and generalized) Bayes minimax estimators and show that the result of Maruyama and Strawderman (2005 Maruyama, Y., and W. E. Strawderman. 2005. A new class of generalized Bayes minimax ridge regression estimators. Annals of Statistics 33:1753–70.[Crossref], [Web of Science ®] , [Google Scholar]) is a special case of our result. We also show that under certain conditions, these generalized Bayes minimax estimators have greater numerical stability (i.e., smaller condition number) than the least-squares estimator.     

Normality of Posterior Distribution Under Misspecification and Nonsmoothness,and Bayes Factor for Davies' Problem

We examine the large sample properties of Bayes procedures in a general framework, where data may be dependent and models may be misspecified and nonsmooth. The posterior distribution of parameters is shown to be asymptotically normal, centered at the quasi maximum likelihood estimator, under mild conditions. In this framework, the Bayes factor for the test problem of Davies (1997, 1987 Davies , R. B. ( 1987 ). Hypothesis testing when a nuisance parameter is present only under the alternative . Biometrika 74 : 33 – 43 .[Web of Science ®] , [Google Scholar]), where a parameter is unidentified under the null hypothesis, is analyzed. The probability that the Bayes factor leads to a correct conclusion about the hypotheses in Davies’ problem is shown to approach to one.     

Records Properties of Non Stationary Time Series

This article proposes an asymptotic expansion for the Studentized linear discriminant function using two-step monotone missing samples under multivariate normality. The asymptotic expansions related to discriminant function have been obtained for complete data under multivariate normality. The result derived by Anderson (1973 Anderson , T. W. ( 1973 ). An asymptotic expansion of the distribution of the Studentized classification statistic W . The Annals of Statistics 1 : 964 – 972 .[Crossref], [Web of Science ®] , [Google Scholar]) plays an important role in deciding the cut-off point that controls the probabilities of misclassification. This article provides an extension of the result derived by Anderson (1973 Anderson , T. W. ( 1973 ). An asymptotic expansion of the distribution of the Studentized classification statistic W . The Annals of Statistics 1 : 964 – 972 .[Crossref], [Web of Science ®] , [Google Scholar]) in the case of two-step monotone missing samples under multivariate normality. Finally, numerical evaluations by Monte Carlo simulations were also presented.     

The smooth colonel and the reverend find common ground

A semiparametric regression estimator that exploits categorical (i.e., discrete-support) kernel functions is developed for a broad class of hierarchical models including the pooled regression estimator, the fixed-effects estimator familiar from panel data, and the varying coefficient estimator, among others. Separate shrinking is allowed for each coefficient. Regressors may be continuous or discrete. The estimator is motivated as an intuitive and appealing generalization of existing methods. It is then supported by demonstrating that it can be realized as a posterior mean in the Lindley and Smith (1972 Lindley, D. V., Smith, A. F. M. (1972). Bayes estimates for the linear model. Journal of the Royal Statistical Society 34:1–41. [Google Scholar]) framework. As a demonstration of the flexibility of the proposed approach, the model is extended to nonparametric hierarchical regression based on B-splines.