Extended BIC for linear regression models with diverging number of relevant features and high or ultra-high feature spaces

In many conventional scientific investigations with high or ultra-high dimensional feature spaces, the relevant features, though sparse, are large in number compared with classical statistical problems, and the magnitude of their effects tapers off. It is reasonable to model the number of relevant features as a diverging sequence when sample size increases. In this paper, we investigate the properties of the extended Bayes information criterion (EBIC) (Chen and Chen, 2008) for feature selection in linear regression models with diverging number of relevant features in high or ultra-high dimensional feature spaces. The selection consistency of the EBIC in this situation is established. The application of EBIC to feature selection is considered in a SCAD cum EBIC procedure. Simulation studies are conducted to demonstrate the performance of the SCAD cum EBIC procedure in finite sample cases.     

Kernel density estimation for heavy-tailed distributions using the champernowne transformation

When estimating loss distributions in insurance, large and small losses are usually split because it is difficult to find a simple parametric model that fits all claim sizes. This approach involves determining the threshold level between large and small losses. In this article, a unified approach to the estimation of loss distributions is presented. We propose an estimator obtained by transforming the data set with a modification of the Champernowne cdf and then estimating the density of the transformed data by use of the classical kernel density estimator. We investigate the asymptotic bias and variance of the proposed estimator. In a simulation study, the proposed method shows a good performance. We also present two applications dealing with claims costs in insurance.     

A note on Stein-type shrinkage estimator in partial linear models

In this paper, an exact sufficient condition for the dominance of the Stein-type shrinkage estimator over the usual unbiased estimator in a partial linear model is exhibited. Comparison result is then done under the balanced loss function. It is assumed that the vector of disturbances is typically distributed according to the law belonging to the sub-class of elliptically contoured models. It is also shown that the dominance condition is robust. Furthermore, a nonparametric estimation after estimation of the linear part is added for detecting the efficiency of the obtained results.     

Bayes sequential estimation for a Poisson process under a LINEX loss function

In this paper, within the framework of a Bayesian model, we consider the problem of sequentially estimating the intensity parameter of a homogeneous Poisson process with a linear exponential (LINEX) loss function and a fixed cost per unit time. An asymptotically pointwise optimal (APO) rule is proposed. It is shown to be asymptotically optimal for the arbitrary priors and asymptotically non-deficient for the conjugate priors in a similar sense of Bickel and Yahav [Asymptotically pointwise optimal procedures in sequential analysis, in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, University of California Press, Berkeley, CA, 1967, pp. 401–413; Asymptotically optimal Bayes and minimax procedures in sequential estimation, Ann. Math. Statist. 39 (1968), pp. 442–456] and Woodroofe [A.P.O. rules are asymptotically non-deficient for estimation with squared error loss, Z. Wahrsch. verw. Gebiete 58 (1981), pp. 331–341], respectively. The proposed APO rule is illustrated using a real data set.     

A note on Pitman's measure of closeness with balanced loss function

In this note, we consider the problem of estimating an unknown parameter θ in the sense of the Pitman's measure of closeness (PMC) using the balanced loss function (BLF). We show that the PMC comparison of estimators under the BLF can be reduced to the PMC comparison under the usual absolute error loss. The Pitman-closest estimators of the location and scale parameters under BLF are also characterized. Illustrative examples are given to show the broad range applications of the obtained results.     

Model averaging for M-estimation

M-estimation is a widely used technique for robust statistical inference. In this paper, we study model selection and model averaging for M-estimation to simultaneously improve the coverage probability of confidence intervals of the parameters of interest and reduce the impact of heavy-tailed errors or outliers in the response. Under general conditions, we develop robust versions of the focused information criterion and a frequentist model average estimator for M-estimation, and we examine their theoretical properties. In addition, we carry out extensive simulation studies as well as two real examples to assess the performance of our new procedure, and find that the proposed method produces satisfactory results.     

On second-order improved estimation of a gamma scale parameter

This paper concludes our comprehensive study on point estimation of model parameters of a gamma distribution from a second-order decision theoretic point of view. It should be noted that efficient estimation of gamma model parameters for samples ‘not large’ is a challenging task since the exact sampling distributions of the maximum likelihood estimators and its variants are not known. Estimation of a gamma scale parameter has received less attention from the earlier researchers compared to shape parameter estimation. What we have observed here is that improved estimation of the shape parameter does not necessarily lead to improved scale estimation if a natural moment condition (which is also the maximum likelihood restriction) is satisfied. Therefore, this work deals with the gamma scale parameter estimation as a separate new problem, not as a by-product of the shape parameter estimation, and studies several estimators in terms of second-order risk.     

Modified minimax estimation of regression coefficients

Working with the linear regression model (1.1) and having the extraneous information (1.2) about regression coefficients the problem exists how to build estimators (1.3) with the risk (1.4) which enable to utilize the known information in order to reduce their risk as compared with the risk (1.6) of the LSE (1.5). Solution of this problem is known for the positive definite matrix T, namely in form for estimators (1.8) and (1.10).First, it is shown that the proposed estimators (2.6),(2.9) and (2.16) based on psedoinversions of the matrix L represent the solution of the problem of the positive semidefinite matrix T=L'L.Further, the problem of interpretability of estimators in the sense of the inequality (3.1) exists; it is shown that all mentioned estimators are at least partially interpretable in the sense of requirements (3.2) or (3.10).     

Cointegration Detection Using Dynamic Factor Models

A method of information-criterion-based cointegration detection using dynamic factor models is proposed. The results of the data-based and non data-based Monte Carlo simulations suggest that this method is as effective as conventional hypothesis-testing methods. In the proposed method, an observed multivariate time series is described in terms of common stochastic trends plus stationary autoregressive cycles. Then the best model is selected from among alternative models obtained by changing the number of common stochastic trends, on the basis of information criteria. Consequently, the cointegration rank is determined on the basis of the selected model. Two advantages of the proposed method are also discussed.     

Estimation and Prediction for Exponentiated Family of Distributions Based on Records

In this article, a family of distributions, namely the exponentiated family of distributions, is defined and for the unknown parameters, different point estimates are derived based on record statistics. Prediction for future record values is presented from a Bayesian view point. Two numerical examples and a Monte Carlo simulation study are presented to illustrate the results.