Optimality of quasi-score in the multivariate mean–variance model with an application to the zero-inflated Poisson model with measurement errors

In a multivariate mean–variance model, the class of linear score (LS) estimators based on an unbiased linear estimating function is introduced. A special member of this class is the (extended) quasi-score (QS) estimator. It is ‘extended’ in the sense that it comprises the parameters describing the distribution of the regressor variables. It is shown that QS is (asymptotically) most efficient within the class of LS estimators. An application is the multivariate measurement error model, where the parameters describing the regressor distribution are nuisance parameters. A special case is the zero-inflated Poisson model with measurement errors, which can be treated within this framework.     

Distribution of the mean reversion estimator in the Ornstein–Uhlenbeck process

ABSTRACT

We derive the exact distribution of the maximum likelihood estimator of the mean reversion parameter (κ) in the Ornstein–Uhlenbeck process using numerical integration through analytical evaluation of a joint characteristic function. Different scenarios are considered: known or unknown drift term, fixed or random start-up value, and zero or positive κ. Monte Carlo results demonstrate the remarkably reliable performance of our exact approach across all the scenarios. In comparison, misleading results may arise under the asymptotic distributions, including the advocated infill asymptotic distribution, which performs poorly in the tails when there is no intercept in the regression and the starting value of the process is nonzero.     

Variable selection in joint mean and variance models of Box–Cox transformation

In many applications, a single Box–Cox transformation cannot necessarily produce the normality, constancy of variance and linearity of systematic effects. In this paper, by establishing a heterogeneous linear regression model for the Box–Cox transformed response, we propose a hybrid strategy, in which variable selection is employed to reduce the dimension of the explanatory variables in joint mean and variance models, and Box–Cox transformation is made to remedy the response. We propose a unified procedure which can simultaneously select significant variables in the joint mean and variance models of Box–Cox transformation which provide a useful extension of the ordinary normal linear regression models. With appropriate choice of the tuning parameters, we establish the consistency of this procedure and the oracle property of the obtained estimators. Moreover, we also consider the maximum profile likelihood estimator of the Box–Cox transformation parameter. Simulation studies and a real example are used to illustrate the application of the proposed methods.     

On convergence of the distributions of random sequences with independent random indexes to variance–mean mixtures

We prove a transfer theorem for random sequences with independent random indexes in the double array limit setting under relaxed conditions. We also prove its partial inverse providing the necessary and sufficient conditions for the convergence of randomly indexed random sequences. Special attention is paid to the case where the elements of the basic double array are formed as cumulative sums of independent not necessarily identically distributed random variables. Using simple moment-type conditions we prove the theorem on convergence of the distributions of such sums to normal variance–mean mixtures.     

Rate of convergence for asymptotic variance of the Horvitz–Thompson estimator

Drawing distinct units without replacement and with unequal probabilities from a population is a problem often considered in the literature (e.g. Hanif and Brewer, 1980, Int. Statist. Rev. 48, 317–355). In such a case, the sample mean is a biased estimator of the population mean. For this reason, we use the unbiased Horvitz–Thompson estimator (1951). In this work, we focus our interest on the variance of this estimator. The variance is cumbersome to compute because it requires the calculation of a large number of second-order inclusion probabilities. It would be helpful to use an approximation that does not need heavy calculations. The Hájek (1964) variance approximation provides this advantage as it is free of second-order inclusion probabilities. Hájek (1964) proved that this approximation is valid under restrictive conditions that are usually not fulfilled in practice. In this paper, we give more general conditions and we show that this approximation remains acceptable for most practical problems.     

Tests for the variance parameter in the Fay–Herriot model

The Fay–Herriot model is a linear mixed model that plays a relevant role in small area estimation (SAE). Under the SAE set-up, tools for selecting an adequate model are required. Applied statisticians are often interested on deciding if it is worthwhile to use a mixed effect model instead of a simpler fixed-effect model. This problem is not standard because under the null hypothesis the random effect variance is on the boundary of the parameter space. The likelihood ratio test and the residual likelihood ratio test are proposed and their finite sample distributions are derived. Finally, we analyse their behaviour under simulated scenarios and we also apply them to real data.     

A Monte Carlo comparison of some variance estimators of the Horvitz–Thompson estimator

At the design and estimation stage of a survey, large survey organization often uses auxiliary information. This article discusses various procedures for improving variance estimation of the Horvitz–Thompson estimator of a finite population total with the aid of auxiliary information. To study the design-based properties of the proposed variance estimators relative to the standard one, a small scale Monte Carlo study is performed.     

Equilibrium excess-of-loss reinsurance–investment strategy for a mean–variance insurer under stochastic volatility model

This article considers an optimal excess-of-loss reinsurance–investment problem for a mean–variance insurer, and aims to develop an equilibrium reinsurance–investment strategy. The surplus process is assumed to follow the classical Cramér–Lundberg model, and the insurer is allowed to purchase excess-of-loss reinsurance and invest her surplus in a risk-free asset and a risky asset. The market price of risk depends on a Markovian, affine-form and square-root stochastic factor process. Under the mean–variance criterion, equilibrium reinsurance–investment strategy and the corresponding equilibrium value function are derived by applying a game theoretic framework. Finally, numerical examples are presented to illustrate our results.     

Statistical inference procedure for the mean–variance efficient frontier with estimated parameters

We suggest finite sample tests for the location of the efficient frontier with the estimated parameters in mean–variance space. The exact densities of the test statistics are derived. We implement the introduced testing procedure empirically by considering monthly returns of ten developed stock markets. It is shown that ignoring the uncertainty about the estimated parameters leads to a more frequent reconstruction of the efficient frontier.     

Approximate estimations in capture–recapture model with heteroscedastic measurement errors

In a capture–recapture experiment, the number of measurements for individual covariates usually equals the number of captures. This creates a heteroscedastic measurement error problem and the usual surrogate condition does not hold in the context of a measurement error model. This study adopts a small measurement error assumption to approximate the conventional estimating functions and the population size estimator. This study also investigates the biases of the resulting estimators. In addition, modifications for two common approximation methods, regression calibration and simulation extrapolation, to accommodate heteroscedastic measurement error are also discussed. These estimation methods are examined through simulations and illustrated by analysing a capture–recapture data set.     

Nonparametric estimator for the survival function of quality adjusted lifetime (QAL) in a three-state illness–death model

In this work, we consider nonparametric estimation of QAL distribution in a three-state illness–death model. In our approach, we first write down the expression for the distribution of QAL in terms of the joint distribution of the sojourn times in the three states. The estimate of the QAL distribution is obtained by substituting the estimates of sojourn time distributions in the expression of the QAL distribution. The proposed nonparametric estimate, assuming independence between time to illness and sojourn time in the state of illness, is uniformly consistent. Asymptotic normality has also been established. An estimate of asymptotic variance has been obtained. The performance of the proposed estimator is investigated by simulation. A data set of the Stanford Heart Transplant program has been analyzed for illustration.     

r − k Class estimator in the linear regression model with correlated errors

Autocorrelation in errors and multicollinearity among the regressors are serious problems in regression analysis. The aim of this paper is to examine multicollinearity and autocorrelation problems concurrently and to compare the r ? k class estimator to the generalized least squares estimator, the principal components regression estimator and the ridge regression estimator by the scalar and matrix mean square error criteria in the linear regression model with correlated errors.     

The mad estimator: when is it non-linear? applications to two-way design models

The mean absolute deviation (MAD) estimator has recently received a great deal of attention as applied to full-rank linear regression models. This paper provides a necessary and sufficient condition for the MAD estimator to be a non-linear estimator, in which case conditions for the variance of the MAD estimator to be larger or smaller than those for OLS are, in general, unknown. The non-linearity of the MAD estimator is examined for several two-way designs; in particular (1) randomized block design (2) two-way nested design (3) two-way classification with interaction and (4) partially balanced incomplete block design     

Phase type stress–strength models with reliability applications

The stress–strength model has attracted a great deal of attention in reliability analysis, and it has been studied under various modeling assumptions. In this article, the stress–strength reliability is studied for both single unit and multicomponent systems when stress and strength distributions are of phase type. Phase-type distributions, besides their analytical tractability, are a versatile tool for modeling a wide range of real life systems/processes. In particular, matrix-based expressions are obtained for the stress–strength reliability, and mean residual strength for an operating system. The results are illustrated for Erlang-type stress–strength distributions for a single unit system and a system having a general coherent structure. An example on the comparison of two multi-state units in stress–strength ordering is also presented.     

The admissible minimax estimator in Gauss–Markov model under a balanced loss function

In this paper, we investigate the admissible minimax estimator (AME) of regression coefficient in Gauss–Markov model under a balanced loss function. In the class of homogeneous linear estimators, we obtain the AME under two occasions, respectively. We also prove that the AME is a shrinkage estimator of the best linear unbiased estimator (BLUE). Furthermore, we prove that the AME dominates the BLUE under certain conditions.     

Joint mean–covariance model in generalized partially linear varying coefficient models for longitudinal data

In longitudinal data analysis, efficient estimation of regression coefficients requires a correct specification of certain covariance structure, and efficient estimation of covariance matrix requires a correct specification of mean regression model. In this article, we propose a general semiparametric model for the mean and the covariance simultaneously using the modified Cholesky decomposition. A regression spline-based approach within the framework of generalized estimating equations is proposed to estimate the parameters in the mean and the covariance. Under regularity conditions, asymptotic properties of the resulting estimators are established. Extensive simulation is conducted to investigate the performance of the proposed estimator and in the end a real data set is analysed using the proposed approach.     

A new method for the estimation of variance matrix with prescribed zeros in nonlinear mixed effects models

We propose a new method for the Maximum Likelihood Estimator (MLE) of nonlinear mixed effects models when the variance matrix of Gaussian random effects has a prescribed pattern of zeros (PPZ). The method consists of coupling the recently developed Iterative Conditional Fitting (ICF) algorithm with the Expectation Maximization (EM) algorithm. It provides positive definite estimates for any sample size, and does not rely on any structural assumption concerning the PPZ. It can be easily adapted to many versions of EM.     

Asymptotic expansion of the risk of maximum likelihood estimator with respect to α-divergence

For a given parametric probability model, we consider the risk of the maximum likelihood estimator with respect to α-divergence, which includes the special cases of Kullback–Leibler divergence, the Hellinger distance, and essentially χ²-divergence. The asymptotic expansion of the risk is given with respect to sample sizes up to order n^{? 2}. Each term in the expansion is expressed with the geometrical properties of the Riemannian manifold formed by the parametric probability model.     

The uniformly minimum variance unbiased estimator of odds ratio in case–control studies under inverse sampling

The stated goal of this paper is to propose the uniformly minimum variance unbiased estimator of odds ratio in case–control studies under inverse sampling design. The problem of estimating odds ratio plays a central role in case–control studies. However, the traditional sampling schemes appear inadequate when the expected frequencies of not exposed cases and exposed controls can be very low. In such a case, it is convenient to use the inverse sampling design, which requires that random drawings shall be continued until a given number of relevant events has emerged. In this paper we prove that a uniformly minimum variance unbiased estimator of odds ratio does not exist under usual binomial sampling, while the standard odds ratio estimator is uniformly minimum variance unbiased under inverse sampling. In addition, we compare these two sampling schemes by means of large-sample theory and small-sample simulation.