Merging information for semiparametric density estimation

Summary.  The density ratio model specifies that the likelihood ratio of m −1 probability density functions with respect to the m th is of known parametric form without reference to any parametric model. We study the semiparametric inference problem that is related to the density ratio model by appealing to the methodology of empirical likelihood. The combined data from all the samples leads to more efficient kernel density estimators for the unknown distributions. We adopt variants of well-established techniques to choose the smoothing parameter for the density estimators proposed.     

Non parametric mixture priors based on an exponential random scheme

We propose a general procedure for constructing nonparametric priors for Bayesian inference. Under very general assumptions, the proposed prior selects absolutely continuous distribution functions, hence it can be useful with continuous data. We use the notion ofFeller-type approximation, with a random scheme based on the natural exponential family, in order to construct a large class of distribution functions. We show how one can assign a probability to such a class and discuss the main properties of the proposed prior, namedFeller prior. Feller priors are related to mixture models with unknown number of components or, more generally, to mixtures with unknown weight distribution. Two illustrations relative to the estimation of a density and of a mixing distribution are carried out with respect to well known data-set in order to evaluate the performance of our procedure. Computations are performed using a modified version of an MCMC algorithm which is briefly described.     

Statistical Inferences Based on Extended Weighted Log-Rank Estimating Function for Censored Regression Model

In the regression model with censored data, it is not straightforward to estimate the covariances of the regression estimators, since their asymptotic covariances may involve the unknown error density function and its derivative. In this article, a resampling method for making inferences on the parameter, based on some estimating functions, is discussed for the censored regression model. The inference procedures are associated with a weight function. To find the best weight functions for the proposed procedures, extensive simulations are performed. The validity of the approximation to the distribution of the estimator by a resampling technique is also examined visually. Implementation of the procedures is discussed and illustrated in a real data example.     

M-estimation Under a Two-Sample Semiparametric Model

We consider M -estimation under a two-sample semiparametric model in which the log ratio of two unknown density functions has a known parametric form. This two-sample semiparametric model, arising naturally from case-control studies and logistic discriminant analysis, can be regarded as a biased sampling model. A new class of M -estimators are constructed on the basis of the maximum semiparametric likelihood estimator of the underlying distribution function. It is shown that the proposed M -estimators are consistent and asymptotically normally distributed. A simulation study is presented to demonstrate the performance of the proposed M -estimators.     

Estimation for an inverted exponentiated Rayleigh distribution under type II progressive censoring

In this paper, we consider estimation of unknown parameters of an inverted exponentiated Rayleigh distribution under type II progressive censored samples. Estimation of reliability and hazard functions is also considered. Maximum likelihood estimators are obtained using the Expectation–Maximization (EM) algorithm. Further, we obtain expected Fisher information matrix using the missing value principle. Bayes estimators are derived under squared error and linex loss functions. We have used Lindley, and Tiernery and Kadane methods to compute these estimates. In addition, Bayes estimators are computed using importance sampling scheme as well. Samples generated from this scheme are further utilized for constructing highest posterior density intervals for unknown parameters. For comparison purposes asymptotic intervals are also obtained. A numerical comparison is made between proposed estimators using simulations and observations are given. A real-life data set is analyzed for illustrative purposes.     

Nonparametric Bayes methods for directional data

A model for directional data in q dimensions is studied. The data are assumed to arise from a distribution with a density on a sphere of q — 1 dimensions. The density is unimodal and rotationally symmetric, but otherwise of unknown form. The posterior distribution of the unknown mode (mean direction) is derived, and small-sample posterior inference is discussed. The posterior mean of the density is also given. A numerical method for evaluating posterior quantities based on sampling a Markov chain is introduced. This method is generally applicable to problems involving unknown monotone functions.     

On progressively censored inverted exponentiated Rayleigh distribution

In this paper, we discuss a progressively censored inverted exponentiated Rayleigh distribution. Estimation of unknown parameters is considered under progressive censoring using maximum likelihood and Bayesian approaches. Bayes estimators of unknown parameters are derived with respect to different symmetric and asymmetric loss functions using gamma prior distributions. An importance sampling procedure is taken into consideration for deriving these estimates. Further highest posterior density intervals for unknown parameters are constructed and for comparison purposes bootstrap intervals are also obtained. Prediction of future observations is studied in one- and two-sample situations from classical and Bayesian viewpoint. We further establish optimum censoring schemes using Bayesian approach. Finally, we conduct a simulation study to compare the performance of proposed methods and analyse two real data sets for illustration purposes.     

Robust Bayesian estimation of a two-parameter exponential distribution under generalized Type-I progressive hybrid censoring

A generalized Type-I progressive hybrid censoring scheme was proposed recently to overcome the limitations of the progressive hybrid censoring scheme. In this article, we provide a robust Bayesian method to estimate the unknown parameters of the two-parameter exponential distribution of a generalized Type-I progressive hybrid censored sample. For each parameter, we derive the marginal posterior density functions and the corresponding Bayesian estimators under the squared error loss function. To assess the proposed method, Monte Carlo simulations are performed using a real dataset.     

Inference on a progressive type I interval-censored truncated normal distribution

In this paper, we consider the problem of making statistical inference for a truncated normal distribution under progressive type I interval censoring. We obtain maximum likelihood estimators of unknown parameters using the expectation-maximization algorithm and in sequel, we also compute corresponding midpoint estimates of parameters. Estimation based on the probability plot method is also considered. Asymptotic confidence intervals of unknown parameters are constructed based on the observed Fisher information matrix. We obtain Bayes estimators of parameters with respect to informative and non-informative prior distributions under squared error and linex loss functions. We compute these estimates using the importance sampling procedure. The highest posterior density intervals of unknown parameters are constructed as well. We present a Monte Carlo simulation study to compare the performance of proposed point and interval estimators. Analysis of a real data set is also performed for illustration purposes. Finally, inspection times and optimal censoring plans based on the expected Fisher information matrix are discussed.     

Methods for estimating proportions of convex combinations of normals using linear feature selection

Let X be a random n-vector whose density function is given by a mixtur.e of two density functions, h₁ and h₂ with unknown mixture proportions, Y₁ and Y₂ We assume that each of h₁ and h₂ is a convex combination of known multivariate normal density functions whose corresponding mixture proportions are also unknown. We present three numerically tractable methods for estimating Y₁ and Y₂ related to the technique of Guseman and Walton (1977), and based on the linear feature selection technique of Guseman, Peters and Walker (1975).     

An application of linear feature selection to estimation of proportions

Let X be a random n-vector whose density function is given by a mixture of known multivariate normal density functions where the corresponding mixture proportions (a priori probabilities) are unknown. We present a numerically tractable method for obtaining estimates of the mixture proportions based on the linear feature selection technique of Guseman, Peters and Walker (1975).     

Quasi-Maximum Likelihood Estimation of GARCH Models With Heavy-Tailed Likelihoods

The non-Gaussian maximum likelihood estimator is frequently used in GARCH models with the intention of capturing heavy-tailed returns. However, unless the parametric likelihood family contains the true likelihood, the estimator is inconsistent due to density misspecification. To correct this bias, we identify an unknown scale parameter η_f that is critical to the identification for consistency and propose a three-step quasi-maximum likelihood procedure with non-Gaussian likelihood functions. This novel approach is consistent and asymptotically normal under weak moment conditions. Moreover, it achieves better efficiency than the Gaussian alternative, particularly when the innovation error has heavy tails. We also summarize and compare the values of the scale parameter and the asymptotic efficiency for estimators based on different choices of likelihood functions with an increasing level of heaviness in the innovation tails. Numerical studies confirm the advantages of the proposed approach.     

Estimation in a semiparametric panel data model with nonstationarity

In this paper, we consider a partially linear panel data model with nonstationarity and certain cross-sectional dependence. Accounting for the explosive feature of the nonstationary time series, we particularly employ Hermite orthogonal functions in this study. Under a general spatial error dependence structure, we then establish some consistent closed-form estimates for both the unknown parameters and the unknown functions for the cases where N and T go jointly to infinity. Rates of convergence and asymptotic normalities are established for the proposed estimators. Both the finite sample performance and the empirical applications show that the proposed estimation methods work well.     

The extended two-sample problem: nonparametric case

The classical two-sample problem is extended here to the case where the distribution functions of the observable random variables are specified functions of unknown distribution functions and the null hypotheses to be tested or the parameters to be estimated relate to these unknown distributions. Various properties of the proposed rank tests and derived estimates are studied.     

Statistical Inference and Applications of Mixture of Varying Coefficient Models

In this paper, we consider a new mixture of varying coefficient models, in which each mixture component follows a varying coefficient model and the mixing proportions and dispersion parameters are also allowed to be unknown smooth functions. We systematically study the identifiability, estimation and inference for the new mixture model. The proposed new mixture model is rather general, encompassing many mixture models as its special cases such as mixtures of linear regression models, mixtures of generalized linear models, mixtures of partially linear models and mixtures of generalized additive models, some of which are new mixture models by themselves and have not been investigated before. The new mixture of varying coefficient model is shown to be identifiable under mild conditions. We develop a local likelihood procedure and a modified expectation–maximization algorithm for the estimation of the unknown non‐parametric functions. Asymptotic normality is established for the proposed estimator. A generalized likelihood ratio test is further developed for testing whether some of the unknown functions are constants. We derive the asymptotic distribution of the proposed generalized likelihood ratio test statistics and prove that the Wilks phenomenon holds. The proposed methodology is illustrated by Monte Carlo simulations and an analysis of a CO₂‐GDP data set.     

Exact tests and confidence regions in nonlinear regression

Nonlinear regression models with spherically symmetric error vectors and a single nonlinear parameter are considered. On the basis of a new geometric approach, exact one- and two-sided tests and confidence regions for the nonlinear parameter are derived in the cases of known and unknown error variances. A geometric measure representation formula is used to determine the power functions of the tests if the error variance is known and to derive different lower bounds for the power function of a one-sided test in the case of an unknown error variance. The latter can be done quite effectively by constructing and measuring several balls inside the critical region. A numerical study compares the results for different density generating functions of the error distribution.     

Robust estimation for functional coefficient regression models with spatial data

A global smoothing procedure is developed using B-spline function approximation for estimating the unknown functions of a functional coefficient regression model with spatial data. A general formulation is used to treat mean regression, median regression, quantile regression and robust mean regression in one setting. The global convergence rates of the estimators of unknown coefficient functions are established. Various applications of the main results, including estimating conditional quantile coefficient functions and robustifying the mean regression coefficient functions are given. Finite sample properties of our procedures are studied through Monte Carlo simulations. A housing data example is used to illustrate the proposed methodology.     

Bayesian and maximum likelihood estimations of the inverse Weibull parameters under progressive type-II censoring

In this paper, the statistical inference of the unknown parameters of a two-parameter inverse Weibull (IW) distribution based on the progressive type-II censored sample has been considered. The maximum likelihood estimators (MLEs) cannot be obtained in explicit forms, hence the approximate MLEs are proposed, which are in explicit forms. The Bayes and generalized Bayes estimators for the IW parameters and the reliability function based on the squared error and Linex loss functions are provided. The Bayes and generalized Bayes estimators cannot be obtained explicitly, hence Lindley's approximation is used to obtain the Bayes and generalized Bayes estimators. Furthermore, the highest posterior density credible intervals of the unknown parameters based on Gibbs sampling technique are computed, and using an optimality criterion the optimal censoring scheme has been suggested. Simulation experiments are performed to see the effectiveness of the different estimators. Finally, two data sets have been analysed for illustrative purposes.     

Optimal kernels when estimating non-smooth densities

The derivation of new kernel functions for the kernel estimator of an unknown density function is given. These kernels are shown to be optimal in some sense when the underlying density f is continuous but its derivative f′ is not, and consequently a solu tion is presented for an unsolved problem which was stated by van Eeden (1985). Other attractive features of these kernels are also discussed and a number of graphs are listed.     

Partially Linear Additive Models with Unknown Link Functions

In this paper, we consider partially linear additive models with an unknown link function, which include single‐index models and additive models as special cases. We use polynomial spline method for estimating the unknown link function as well as the component functions in the additive part. We establish that convergence rates for all nonparametric functions are the same as in one‐dimensional nonparametric regression. For a faster rate of the parametric part, we need to define appropriate ‘projection’ that is more complicated than that defined previously for partially linear additive models. Compared to previous approaches, a distinct advantage of our estimation approach in implementation is that estimation directly reduces estimation in the single‐index model and can thus deal with much larger dimensional problems than previous approaches for additive models with unknown link functions. Simulations and a real dataset are used to illustrate the proposed model.