Robust signed-rank estimation and variable selection for semi-parametric additive partial linear models

A fully nonparametric model may not perform well or when the researcher wants to use a parametric model but the functional form with respect to a subset of the regressors or the density of the errors is not known. This becomes even more challenging when the data contain gross outliers or unusual observations. However, in practice the true covariates are not known in advance, nor is the smoothness of the functional form. A robust model selection approach through which we can choose the relevant covariates components and estimate the smoothing function may represent an appealing tool to the solution. A weighted signed-rank estimation and variable selection under the adaptive lasso for semi-parametric partial additive models is considered in this paper. B-spline is used to estimate the unknown additive nonparametric function. It is shown that despite using B-spline to estimate the unknown additive nonparametric function, the proposed estimator has an oracle property. The robustness of the weighted signed-rank approach for data with heavy-tail, contaminated errors, and data containing high-leverage points are validated via finite sample simulations. A practical application to an economic study is provided using an updated Canadian household gasoline consumption data.     

Semiparametric jump-preserving estimation for single-index models

Estimation of the single-index model with a discontinuous unknown link function is considered in this paper. Existed refined minimum average variance estimation (rMAVE) method can estimate the single-index parameter and unknown link function simultaneously by minimising the average pointwise conditional variance, where the conditional variance can be estimated using the local linear fit method with centred kernel function. When there are jumps in the link function, big biases around jumps can appear. For this reason, we embed the jump-preserving technique in the rMAVE method, then propose an adaptive jump-preserving estimation procedure for the single-index model. Concretely speaking, the conditional variance is obtained by the one among local linear fits with centred, left-sided and right-sided kernel functions who has minimum weighted residual mean squares. The resulting estimators can preserve the jumps well and also give smooth estimates of the continuity parts. Asymptotic properties are established under some mild conditions. Simulations and real data analysis show the proposed method works well.     

Umvu estimation for the cumulative hazard function in a pareto distribution of the first kind

The uniformly minimum variance unbiased estimator of the cumulative hazard function in the Pareto distribution of the first kind is derived. The variance of the estimator is also obtained in an analytic form, and for some cases its values are compared numerically with mean square errors of the maximum likelihood estimator.     

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.     

A novel regularization method for estimation and variable selection in multi-index models

Multi-index models have attracted much attention recently as an approach to circumvent the curse of dimensionality when modeling high-dimensional data. This paper proposes a novel regularization method, called MAVE-glasso, for simultaneous parameter estimation and variable selection in multi-index models. The advantages of the proposed method include transformation invariance, automatic variable selection, automatic removal of noninformative observations, and row-wise shrinkage. An efficient row-wise coordinate descent algorithm is proposed to calculate the estimates. Simulation and real examples are used to demonstrate the excellent performance of MAVE-glasso.     

Bandwidth selection through cross-validation for semi-parametric varying-coefficient partially linear models

We study bandwidth selection for a class of semi-parametric models. The proper choice of optimal bandwidth minimizes the prediction errors of the model. We provide detailed derivation of our procedure and the corresponding computation algorithms. Our proposed method simplifies the computation of the cross-validation criteria and facilitates more complicated inference and analysis in practice. A data set from Wisconsin Diabetes Registry has been analysed as an illustration.     

Robust estimation for order of hidden Markov models based on density power divergences

In this paper, we study the robust estimation for the order of hidden Markov model (HMM) based on a penalized minimum density power divergence estimator, which is obtained by utilizing the finite mixture marginal distribution of HMM. For this task, we adopt the locally conic parametrization method used in [D. Dacunha-Castelle and E. Gassiate, Testing in locally conic models and application to mixture models. ESAIM Probab. Stat. (1997), pp. 285–317; D. Dacunha-Castelle and E. Gassiate, Testing the order of a model using locally conic parametrization: population mixtures and stationary arma processes, Ann. Statist. 27 (1999), pp. 1178–1209; T. Lee and S. Lee, Robust and consistent estimation of the order of finite mixture models based on the minimizing a density power divergence estimator, Metrika 68 (2008), pp. 365–390] to avoid the difficulties that arise in handling mixture marginal models, such as the non-identifiability of the parameter space and the singularity problem with the asymptotic variance. We verify that the estimated order is consistent and simulation results are provided for illustration.     

Projectors and linear estimation in general linear models

The paper gives a self-contained account of minimum disper­sion linear unbiased estimation of the expectation vector in a linear model with the dispersion matrix belonging to some, rather arbitrary, set of nonnegative definite matrices. The approach to linear estimation in general linear models recommended here is a direct generalization of some ideas and results presented by Rao (1973, 19 74) for the case of a general Gauss-Markov model

A new insight into the nature of some estimation problems originaly arising in the context of a general Gauss-Markov model as well as the correspondence of results known in the literature to those obtained in the present paper for general linear models are also given. As preliminary results the theory of projectors defined by Rao (1973) is extended.     

On estimation of the PDF and CDF of the Lindley distribution

This article addresses two methods of estimation of the probability density function (PDF) and cumulative distribution function (CDF) for the Lindley distribution. Following estimation methods are considered: uniformly minimum variance unbiased estimator (UMVUE) and maximum likelihood estimator (MLE). Since the Lindley distribution is more flexible than the exponential distribution, the same estimators have been found out for the exponential distribution and compared. Monte Carlo simulations and a real data analysis are performed to compare the performances of the proposed methods of estimation.     

A link-free sparse group variable selection method for single-index model

For regression problems with grouped covariates, we adapt the idea of sparse group lasso (SGL) [10 J. Friedman, T. Hastie, and R. Tibshirani, A note on the group lasso and a sparse group lasso, Tech. Rep., Statistics Department, Stanford University, 2010. [Google Scholar]] to the framework of the sufficient dimension reduction. Assuming that the regression falls into a single-index structure, we propose a method called the sparse group sufficient dimension reduction to conduct group and within-group variable selections simultaneously without assuming a specific link function. Simulation studies show that our method is comparable to the SGL under the regular linear model setting and outperforms SGL with higher true positive rates and substantially lower false positive rates when the regression function is nonlinear. One immediate application of our method is to the gene pathway data analysis where genes naturally fall into groups (pathways). An analysis of a glioblastoma microarray data is included for illustration of our method.     

Stress-strength reliability estimation for exponentially distributed system with common minimum guarantee time

Abstract

In this article, we study the problem of estimating the stress-strength reliability, where the stress and strength variables follow independent exponential distributions with a common location parameter but different scale parameters. All parameters are assumed to be unknown. We derive the MLE, the UMVUE of the reliability parameter. We also derive the Bayes estimators considering conjugate prior distributions for the scale parameters and a dependent prior for the common location parameter. Monte Carlo simulations have been carried out to compare among the proposed estimators with respect to different loss functions.     

A semi-parametric additive model for variance heterogeneity

This paper presents a flexible model for variance heterogeneity in a normal error model. Specifically, both the mean and variance are modelled using semi-parametric additive models. We call this model a Mean And Dispersion Additive Model (MADAM). A successive relaxation algorithm for fitting the model is described and justified as maximizing a penalized likelihood function with penalties for lack of smoothness in the additive non-parametric functions in both mean and variance models. The algorithm is implemented in GLIM4, allowing flexible and interactive modelling of variance heterogeneity. Two data sets are used for demonstration.     

Robust estimation for zero-inflated poisson autoregressive models based on density power divergence

In this study, we consider a robust estimation for zero-inflated Poisson autoregressive models using the minimum density power divergence estimator designed by Basu et al. [Robust and efficient estimation by minimising a density power divergence. Biometrika. 1998;85:549–559]. We show that under some regularity conditions, the proposed estimator is strongly consistent and asymptotically normal. The performance of the estimator is evaluated through Monte Carlo simulations. A real data analysis using New South Wales crime data is also provided for illustration.     

Multiparameter estimation of discrete exponential distributions

Let X₁, …, X_p be independent random variables, all having the same distribution up to a possibly varying unspecified parameter, where each of the p distributions belongs to the family of one parameter discrete exponential distributions. The problem is to estimate the unknown parameters simultaneously. Hudson (1978) shows that the minimum variance unbiased estimator (MVUE) of the parameters is inadmissible under squared error loss, and estimators better than the MVUE are proposed. Essentially, these estimators shrink the MVUE towards the origin. In this paper, we indicate that estimators shifting the MVUE towards a point different from the origin or a point determined by the observations can be obtained.     

Semiparametric estimation in copula models

The author recalls the limiting behaviour of the empirical copula process and applies it to prove some asymptotic properties of a minimum distance estimator for a Euclidean parameter in a copula model. The estimator in question is semiparametric in that no knowledge of the marginal distributions is necessary. The author also proposes another semiparametric estimator which he calls “rank approximate Z‐estimator” and whose asymptotic normality he derives. He further presents Monte Carlo simulation results for the comparison of various estimators in four well‐known bivariate copula models.     

Regularization and variable selection for infinite variance autoregressive models

Autoregressive models with infinite variance are of great importance in modeling heavy-tailed time series and have been well studied. In this paper, we propose a penalized method to conduct model selection for autoregressive models with innovations having Pareto-like distributions with index α∈(0,2)$\alpha \in (\mathrm{0,2})$. By combining the least absolute deviation loss function and the adaptive lasso penalty, the proposed method is able to consistently identify the true model and at the same time produce efficient estimators with a convergence rate of n^−1/α$n^{-1/\alpha }$. In addition, our approach provides a unified way to conduct variable selection for autoregressive models with finite or infinite variance. A simulation study and a real data analysis are conducted to illustrate the effectiveness of our method.     

Inferences in dynamic logit models in semi-parametric setup for repeated binary data

Binary dynamic fixed and mixed logit models are extensively studied in the literature. These models are developed to examine the effects of certain fixed covariates through a parametric regression function as a part of the models. However, there are situations where one may like to consider more covariates in the model but their direct effect is not of interest. In this paper we propose a generalization of the existing binary dynamic logit (BDL) models to the semi-parametric longitudinal setup to address this issue of additional covariates. The regression function involved in such a semi-parametric BDL model contains (i) a parametric linear regression function in some primary covariates, and (ii) a non-parametric function in certain secondary covariates. We use a simple semi-parametric conditional quasi-likelihood approach for consistent estimation of the non-parametric function, and a semi-parametric likelihood approach for the joint estimation of the main regression and dynamic dependence parameters of the model. The finite sample performance of the estimation approaches is examined through a simulation study. The asymptotic properties of the estimators are also discussed. The proposed model and the estimation approaches are illustrated by reanalysing a longitudinal infectious disease data.     

Simultaneous equivariant estimation of the parameters of matrix scale and matrix location-scale models

Eaton and Olkin (1987) discussed the problem of best equivariant estimator of the matrix scale parameter with respect to different scalar loss functions. Edwin Prabakaran and Chandrasekar (1994) developed simultaneous equivariant estimation approach and illustrated the method with examples. The problems considered in this paper are simultaneous equivariant estimation of the parameters of (i) a matrix scale model and (ii) a multivariate location-scale model. By considering matrix loss function (Klebanov, Linnik and Ruhin, 1971) a characterization of matrix minimum risk equivariant (MMRE) estimator of the matrix parameter is obtained in each case. Illustrative examples are provided in which MMRE estimators are obtained with respect to two matrix loss functions.     

A multi-index model for quantile regression with ordinal data

In this paper, we propose a quantile approach to the multi-index semiparametric model for an ordinal response variable. Permitting non-parametric transformation of the response, the proposed method achieves a root-n rate of convergence and has attractive robustness properties. Further, the proposed model allows additional indices to model the remaining correlations between covariates and the residuals from the single-index, considerably reducing the error variance and thus leading to more efficient prediction intervals (PIs). The utility of the model is demonstrated by estimating PIs for functional status of the elderly based on data from the second longitudinal study of aging. It is shown that the proposed multi-index model provides significantly narrower PIs than competing models. Our approach can be applied to other areas in which the distribution of future observations must be predicted from ordinal response data.