Reflected BSDE of Wiener-Poisson type in time-dependent domains

In this paper we study multidimensional reflected backward stochastic differential equations driven by Wiener-Poisson type processes. We prove existence and uniqueness of solutions, with reflection in the inward spatial normal direction, in the setting of certain time-dependent domains.     

Parameter estimation for generalized logistic distribution by estimating equations based on the order statistics

In this paper, we propose estimating equations estimators (EEE) based on the order statistics for the generalized Logistic distribution. Some asymptotic results are provided. Two simulation studies are undertaken to assess the performance of the proposed method and to compare them with other methods suggested in this paper. The simulation results indicate that EEE performs better than some other methods in terms of MSE. Finally, the proposed method is applied to two real data sets.     

Estimation of a two-equation panel model with mixed continuous and ordered categorical outcomes and missing data

Summary.  The paper proposes an estimation approach for panel models with mixed continuous and ordered categorical outcomes based on generalized estimating equations for the mean and pseudoscore equations for the covariance parameters. A numerical study suggests that efficiency can be gained in the mean parameter estimators by using individual covariance matrices in the estimating equations for the mean parameters. The approach is applied to estimate the returns to occupational qualification in terms of income and perceived job security in a 9-year period based on the German Socio-Economic Panel. To compensate for missing data, a combined multiple imputation–weighting approach is adopted.     

On tests of independence between stochastic regressors and disturbances

In this paper two tests for testing the hypothesis that the stochastic endogenous regressors in a structural equation, embedded in a system of simultaneous structural equations, are independent of the disturbances in that equation are presented.     

Partial Linear Models for Longitudinal Data Based on Quadratic Inference Functions

In this paper, we consider improved estimating equations for semiparametric partial linear models (PLM) for longitudinal data, or clustered data in general. We approximate the non‐parametric function in the PLM by a regression spline, and utilize quadratic inference functions (QIF) in the estimating equations to achieve a more efficient estimation of the parametric part in the model, even when the correlation structure is misspecified. Moreover, we construct a test which is an analogue to the likelihood ratio inference function for inferring the parametric component in the model. The proposed methods perform well in simulation studies and real data analysis conducted in this paper.     

A detection algorithm for the first jump time in sample trajectories of jump-diffusions driven by α-stable white noise

Abstract

The purpose of this paper is to develop a detection algorithm for the first jump point in sampling trajectories of jump-diffusions which are described as solutions of stochastic differential equations driven by α-stable white noise. This is done by a multivariate Lagrange interpolation approach. To this end, we utilize computer simulation algorithm in MATLAB to visualize the sampling trajectories of the jump-diffusions for various combinations of parameters arising in the modeling structure of stochastic differential equations.     

Jackknife empirical likelihood tests for distribution functions

It has been a long history for testing whether the underlying distribution belongs to a particular family. In this paper, we propose some jackknife empirical likelihood tests via estimating equations. The proposed new tests allow one to add more relevant constraints so as to improve the powers. A simulation study shows the effectiveness of the new tests.     

Two-Stage Welsh's Trimmed Mean for the Simultaneous Equations Model

This paper discusses the large sample theory of the two-stage Welsh's trimmed mean for the limited information simultaneous equations model. Besides having asymptotic normality, this trimmed mean, as the two-stage least squares estimator, is a generalized least squares estimator. It also acts as a robust Aitken estimator for the simultaneous equations model. Examples illustrate real data analysis and large sample inferences based on this trimmed mean.     

ESTIMATING PREDATOR-PREY SYSTEMS VIA ORDINARY DIFFERENTIAL EQUATIONS WITH CLOSED ORBITS

This paper considers periodic regression functions, which are solutions to a planar system of differential equations. In particular, it introduces a simple stochastic model which describes the interaction between predator and prey populations. The regression functions are solutions to the classical Lotka‐Volterra system of equations, which admits closed orbits. The proposed method of estimation can be applied whenever pairs of predator‐prey data are available, and the prey is the main source of food of the predator. Canadian mink‐muskrat data are analysed from this new viewpoint. The estimation method is based on the existence of closed trajectories that describe the relationship between the two population sizes, and the paper shows how it can be extended to other systems of differential equations which admit closed orbits (e.g. Hamiltonian systems).     

A Note on the Factorization of a Matrix Inverse

The decomposition of a matrix as a product of a lower triangular with ones on the diagonal and an upper triangular matrix is useful for solving systems of linear equations. For a given non singular matrix, this type of decomposition is unique and algorithms exist to obtain the two factors. However, in certain problems the factorization of the inverse matrix may be of interest. This note presents an algorithm for factoring the inverse matrix using simple operations of elements from the original matrix. As examples, we give factorizations for several well-known and widely used correlation matrices. The usefulness and practicality of these factorizations are provided in an application of statistical modeling using unbiased estimating equations.     

Comparative Study of Generalized Estimating Equations and Logistic Regressions on Different Sample Sizes and Correlation Levels

In this study, it was aimed to determine accuracy of generalized estimating equations versus logistic regressions on different correlation levels and sample sizes. For this aim, two methods were compared with different sample sizes 10, 25, 50 and 100 and correlation levels 0.0, 0.3, 0.5 and 0.8. Result of this study showed that using generalized estimating equations could be preferred versus logistic regression when the sample size is over than 25 and correlation level is higher than 0.3 on data taken from studies with repeated measurements, but logistic regression could be better when the autocorrelations do not exist.     

Stepanov-like almost automorphic solutions for stochastic differential equations with Lévy noise

In this paper, we introduce a new concept of Poisson Stepanov-like almost automorphy (or Poisson S²-almost automorphy). Under some suitable conditions on the coefficients, we establish the existence and uniqueness of Stepanov-like almost automorphic mild solution to a class of semilinear stochastic differential equations with infinite dimensional Lévy noise. We further discuss the global asymptotic stability of these solution. Finally, we give an example to illustrate the theoretical results obtained in this paper.     

Empirical likelihood test via estimating equations

It has been a long history for testing whether the underlying distribution belongs to a particular one or a parametric class of distributions. In this paper, we propose some empirical likelihood ratio tests via estimating equations. The proposed new tests allow one to add more relevant constraints so as to improve the powers. A simulation study shows the effectiveness of the new tests. The new method is then used to test employer size and market value distributions of US firms.     

HETEROGENEITY AMONG SUSCEPTIBLES IN EPIDEMIC MODELS

When the infection rate associated with an epidemic appears to decline over time, one explanation is a constant level of infectiousness combined with heterogeneity among the susceptible population. In this paper we consider random effects models for such heterogeneity, particularly in discrete time. Maximum likelihood techniques are discussed as well as a more convenient approach based on martingale estimating equations. An application to data on a smallpox outbreak is considered.     

GMM estimation of spatial autoregressive models in a system of simultaneous equations with heteroskedasticity

This paper proposes a GMM estimation framework for the SAR model in a system of simultaneous equations with heteroskedastic disturbances. Besides linear moment conditions, the proposed GMM estimator also utilizes quadratic moment conditions based on the covariance structure of model disturbances within and across equations. Compared with the QML approach, the GMM estimator is easier to implement and robust under heteroskedasticity of unknown form. We derive the heteroskedasticity-robust standard error for the GMM estimator. Monte Carlo experiments show that the proposed GMM estimator performs well in finite samples.     

Robust estimation of covariance parameters in partial linear model for longitudinal data

For longitudinal data, the within-subject dependence structure and covariance parameters may be of practical and theoretical interests. The estimation of covariance parameters has received much attention and been studied mainly in the framework of generalized estimating equations (GEEs). The GEEs method, however, is sensitive to outliers. In this paper, an alternative set of robust generalized estimating equations for both the mean and covariance parameters are proposed in the partial linear model for longitudinal data. The asymptotic properties of the proposed estimators of regression parameters, non-parametric function and covariance parameters are obtained. Simulation studies are conducted to evaluate the performance of the proposed estimators under different contaminations. The proposed method is illustrated with a real data analysis.     

Modelling heterogeneity in longitudinal binomial responses by generalized estimating equations

ABSTRACT

Extra-binomial variation in longitudinal/clustered binomial data is frequently observed in biomedical and observational studies. The usual generalized estimating equations method treats the extra-binomial parameter as a constant across all subjects. In this paper, a two-parameter variance function modelling the extraneous variance is proposed to account for heterogeneity among subjects. The new approach allows modelling the extra-binomial variation as a function of the mean and binomial size.     

Semi-parametric modelling and likelihood estimation with estimating equations

This paper proposes a semi-parametric modelling and estimating method for analysing censored survival data. The proposed method uses the empirical likelihood function to describe the information in data, and formulates estimating equations to incorporate knowledge of the underlying distribution and regression structure. The method is more flexible than the traditional methods such as the parametric maximum likelihood estimation (MLE), Cox's (1972) proportional hazards model, accelerated life test model, quasi-likelihood (Wedderburn, 1974) and generalized estimating equations (Liang & Zeger, 1986). This paper shows the existence and uniqueness of the proposed semi-parametric maximum likelihood estimates (SMLE) with estimating equations. The method is validated with known cases studied in the literature. Several finite sample simulation and large sample efficiency studies indicate that when the sample size is larger than 100 the SMLE is compatible with the parametric MLE; and in all case studies, the SMLE is about 15% better than the parametric MLE with a mis-specified underlying distribution.     

ASYMPTOTIC DISTRIBUTIONS OF SEMIPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATORS WITH ESTIMATING EQUATIONS FOR GROUP-CENSORED DATA

Semiparametric maximum likelihood estimation with estimating equations (SMLE) is more flexible than traditional methods; it has fewer restrictions on distributions and regression models. The required information about distribution and regression structures is incorporated in estimating equations of the SMLE to improve the estimation quality of non‐parametric methods. The likelihood of SMLE for censored data involves complicated implicit functions without closed‐form expressions, and the first derivatives of the log‐profile‐likelihood cannot be expressed as summations of independent and identically distributed random variables; it is challenging to derive asymptotic properties of the SMLE for censored data. For group‐censored data, the paper shows that all the implicit functions are well defined and obtains the asymptotic distributions of the SMLE for model parameters and lifetime distributions. With several examples the paper compares the SMLE, the regular non‐parametric likelihood estimation method and the parametric MLEs in terms of their asymptotic efficiencies, and illustrates application of SMLE. Various asymptotic distributions of the likelihood ratio statistics are derived for testing the adequacy of estimating equations and a partial set of parameters equal to some known values.     

Robust testing with generalized partial linear models for longitudinal data

By approximating the nonparametric component using a regression spline in generalized partial linear models (GPLM), robust generalized estimating equations (GEE), involving bounded score function and leverage-based weighting function, can be used to estimate the regression parameters in GPLM robustly for longitudinal data or clustered data. In this paper, score test statistics are proposed for testing the regression parameters with robustness, and their asymptotic distributions under the null hypothesis and a class of local alternative hypotheses are studied. The proposed score tests reply on the estimation of a smaller model without the testing parameters involved, and perform well in the simulation studies and real data analysis conducted in this paper.