Self-weighted quasi-maximum exponential likelihood estimator for ARFIMA-GARCH models

In this paper, a local self-weighted quasi-maximum exponential likelihood estimator for ARFIMA-GARCH models is proposed, asymptotic normality of this estimator is derived under the existence of second moment including stationary and non-stationary cases. A simulation study is given to evaluate the performance of the proposed self-weighted QMELE under the stationary case.     

Asymptotic Properties of the MLE in Nonlinear Reproductive Dispersion Models With Stochastic Regressors

Nonlinear reproductive dispersion models with stochastic regressors (NRDMWSR) includes generalized linear models with stochastic regressors (Fahrmer and Kaufmann, 1985 Fahrmer , L. , Kaufmann , H. ( 1985 ). Consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models . Ann. Statist. 13 : 342 – 368 . [Google Scholar]) as a special case. This article presents some mild regularity conditions. On the basis of those mild conditions, the existence, strong consistency, and asymptotic normality of maximum likelihood estimator (MLE) are obtained in NRDMWSR.     

Estimators for the Drift of Subfractional Brownian Motion

In this paper, we consider, using technique based on Girsanov theorem, the problem of efficient estimation for the drift of subfractional Brownian motion S^H ? (S^H_t)_{t ∈ [0, T]}. We also construct a class of biased estimators of James-Stein type which dominate, under the usual quadratic risk, the natural maximum likelihood estimator.     

EGARCH Model with Weighted Liquidity

We analyze a variant of the EGARCH model which captures the variation of the intra-day price. We study the asymptotic behavior of the estimators for the parameters of the model. We also illustrate our theoretical results by empirical studies.     

Asymptotic Properties of the Maximum Quasi-Likelihood Estimator in Quasi-Likelihood Nonlinear Models

Quasi-likelihood nonlinear models (QLNM) are a further extension of generalized linear models by only specifying the expectation and variance functions of the response variable. In this article, some mild regularity conditions are proposed. These regularity conditions, respectively, assure the existence, strong consistency, and the asymptotic normality of the maximum quasi-likelihood estimator (MQLE) in QLNM.     

Affiliation networks with an increasing degree sequence

Affiliation network is one kind of two-mode social network with two different sets of nodes (namely, a set of actors and a set of social events) and edges representing the affiliation of the actors with the social events. Although a number of statistical models are proposed to analyze affiliation networks, the asymptotic behaviors of the estimator are still unknown or have not been properly explored. In this article, we study an affiliation model with the degree sequence as the exclusively natural sufficient statistic in the exponential family distributions. We establish the uniform consistency and asymptotic normality of the maximum likelihood estimator when the numbers of actors and events both go to infinity. Simulation studies and a real data example demonstrate our theoretical results.     

Estimation of a multiple-threshold AR(p) model

The present paper deals with the multiple-threshold p-order autoregressive model which has been introduced by Tong and Lim [H. Tong, K.S. Lim, Threshold autoregression, limit cycles and cyclical data, J. R. Stat. Soc. Ser. B 42 (1980) 245–292] in nonlinear system modelling. Under some conditions on the coefficients of the model which ensure the stationarity, the existence of moments and the strong mixing property of this process and under other mild assumptions, we establish the asymptotic properties (consistency and asymptotic normality) of the minimum Hellinger distance estimates of the autoregressive coefficients of the model.     

Estimation and Asymptotic Properties in Periodic GARCH(1, 1) Models

This article studies the probabilistic structure and asymptotic inference of the first-order periodic generalized autoregressive conditional heteroscedasticity (PGARCH(1, 1)) models in which the parameters in volatility process are allowed to switch between different regimes. First, we establish necessary and sufficient conditions for a PGARCH(1, 1) process to have a unique stationary solution (in periodic sense) and for the existence of moments of any order. Second, using the representation of squared PGARCH(1, 1) model as a PARMA(1, 1) model, we then consider Yule-Walker type estimators for the parameters in PGARCH(1, 1) model and derives their consistency and asymptotic normality. The estimator can be surprisingly efficient for quite small numbers of autocorrelations and, in some cases can be more efficient than the least squares estimate (LSE). We use a residual bootstrap to define bootstrap estimators for the Yule-Walker estimates and prove the consistency of this bootstrap method. A set of numerical experiments illustrates the practical relevance of our theoretical results.     

Confidence intervals based on the deviance statistic for the hyperparameters in state space models

The main objective of this work is to evaluate the performance of confidence intervals, built using the deviance statistic, for the hyperparameters of state space models. The first procedure is a marginal approximation to confidence regions, based on the likelihood test, and the second one is based on the signed root deviance profile. Those methods are computationally efficient and are not affected by problems such as intervals with limits outside the parameter space, which can be the case when the focus is on the variances of the errors. The procedures are compared to the usual approaches existing in the literature, which includes the method based on the asymptotic distribution of the maximum likelihood estimator, as well as bootstrap confidence intervals. The comparison is performed via a Monte Carlo study, in order to establish empirically the advantages and disadvantages of each method. The results show that the methods based on the deviance statistic possess a better coverage rate than the asymptotic and bootstrap procedures.     

Affiliation discrete weighted networks with an increasing degree sequence

Affiliation network is one kind of two-mode social network with two different sets of nodes (namely, a set of actors and a set of social events) and edges representing the affiliation of the actors with the social events. The connections in many affiliation networks are only binary weighted between actors and social events that can not reveal the affiliation strength relationship. Although a number of statistical models are proposed to analyze affiliation binary weighted networks, the asymptotic behaviors of the maximum likelihood estimator (MLE) are still unknown or have not been properly explored in affiliation weighted networks. In this paper, we study an affiliation model with the degree sequence as the exclusively natural sufficient statistic in the exponential family distributions. We derive the consistency and asymptotic normality of the maximum likelihood estimator in affiliation finite discrete weighted networks when the numbers of actors and events both go to infinity. Simulation studies and a real data example demonstrate our theoretical results.     

Jackknife Estimation for Smooth Functions of the Parametric Component in Partially Linear Regression Models

Abstract

It is known that due to the existence of the nonparametric component, the usual estimators for the parametric component or its function in partially linear regression models are biased. Sometimes this bias is severe. To reduce the bias, we propose two jackknife estimators and compare them with the naive estimator. All three estimators are shown to be asymptotically equivalent and asymptotically normally distributed under some regularity conditions. However, through simulation we demonstrate that the jackknife estimators perform better than the naive estimator in terms of bias when the sample size is small to moderate. To make our results more useful, we also construct consistent estimators of the asymptotic variance, which are robust against heterogeneity of the error variances.     

Estimation of Measures in M/M/1 Queue

Maximum likelihood and uniform minimum variance unbiased estimators of steady-state probability distribution of system size, probability of at least ? customers in the system in steady state, and certain steady-state measures of effectiveness in the M/M/1 queue are obtained/derived based on observations on X, the number of customer arrivals during a service time. The estimators are compared using Asympotic Expected Deficiency (AED) criterion leading to recommendation of uniform minimum variance unbiased estimators over maximum likelihood estimators for some measures.     

A new test for high-dimensional regression coefficients in partially linear models

Partially linear regression models are semiparametric models that contain both linear and nonlinear components. They are extensively used in many scientific fields for their flexibility and convenient interpretability. In such analyses, testing the significance of the regression coefficients in the linear component is typically a key focus. Under the high-dimensional setting, i.e., “large p, small n,” the conventional F-test strategy does not apply because the coefficients need to be estimated through regularization techniques. In this article, we develop a new test using a U-statistic of order two, relying on a pseudo-estimate of the nonlinear component from the classical kernel method. Using the martingale central limit theorem, we prove the asymptotic normality of the proposed test statistic under some regularity conditions. We further demonstrate our proposed test's finite-sample performance by simulation studies and by analyzing some breast cancer gene expression data.     

Estimation of the linear EV model with censored data

In this paper, a censored linear errors-in-variables model is investigated. The asymptotic normality of the unknown parameter's estimator is obtained. Two empirical log-likelihood ratio statistics for the unknown parameter in the model are suggested. It is proved that the proposed statistics are asymptotically chi-squared under some mild conditions, and hence can be used to construct the confidence regions of the parameter of interest. Finite sample performance of the proposed method is illustrated in a simulation study.     

Estimation and hypothesis testing in multivariate linear regression models under non normality

This paper discusses the problem of statistical inference in multivariate linear regression models when the errors involved are non normally distributed. We consider multivariate t-distribution, a fat-tailed distribution, for the errors as alternative to normal distribution. Such non normality is commonly observed in working with many data sets, e.g., financial data that are usually having excess kurtosis. This distribution has a number of applications in many other areas of research as well. We use modified maximum likelihood estimation method that provides the estimator, called modified maximum likelihood estimator (MMLE), in closed form. These estimators are shown to be unbiased, efficient, and robust as compared to the widely used least square estimators (LSEs). Also, the tests based upon MMLEs are found to be more powerful than the similar tests based upon LSEs.     

The Two-Piece Normal-Laplace Distribution

This article considers the two-piece normal-Laplace (TPNL) distribution, a split skew distribution consisting of a normal part, and a Laplace part. The distribution is indexed by three parameters, representing location, scale, and shape. As illustrated with several examples, the TPNL family of distributions provides a useful alternative to other families of asymmetric distributions on the real line. However, because the likelihood function is not well behaved, standard theory of maximum-likelihood (ML) estimation does not apply to the TPNL family. In particular, the likelihood function can have multiple local maxima. We provide a procedure for computing ML estimators, and prove consistency and asymptotic normality of ML estimators, using non standard methods.     

Regression spline bivariate probit models: A practical approach to testing for exogeneity

Bivariate probit models can deal with a problem usually known as endogeneity. This issue is likely to arise in observational studies when confounders are unobserved. We are concerned with testing the hypothesis of exogeneity (or absence of endogeneity) when using regression spline recursive and sample selection bivariate probit models. Likelihood ratio and gradient tests are discussed in this context and their empirical properties investigated and compared with those of the Lagrange multiplier and Wald tests through a Monte Carlo study. The tests are illustrated using two datasets in which the hypothesis of exogeneity needs to be tested.     

Quasi-maximum exponential likelihood estimation for a non stationary GARCH(1,1) model

ABSTRACT

This article investigates a quasi-maximum exponential likelihood estimator(QMELE) for a non stationary generalized autoregressive conditional heteroscedastic (GARCH(1,1)) model. Asymptotic normality of this estimator is derived under a non stationary condition. A simulation study and a real example are given to evaluate the performance of QMELE for this model.     

Asymptotically Informative Prior for Bayesian Analysis

In classical Bayesian inference the prior is treated as fixed and its effects are ignored asymptotically, and useful information, if any, is wasted. However, in practice often an informative prior is summarized from previous similar or the same kind of studies, which contains useful cumulative information for the current study. We treat such prior to be non-fixed, i.e., we give the data sizes in the prior studies similar status as the that of the current dataset. Under this formulation, the prior is asymptotically non-negligible, and its original information is transferred to the new study. We explore some basic properties of Bayesian estimators under such prior formulation, and illustrate the method via simulation.     

Statistical Inference on a Stochastic Epidemic Model

In this work, we develop statistical inference for the parameters of a discrete-time stochastic SIR epidemic model. We use a Markov chain for describing the dynamic behavior of the epidemic. Specifically, we propose estimators for the contact and removal rates based on the maximum likelihood and martingale methods, and establish their asymptotic distributions. The obtained results are applied in the statistical analysis of the basic reproduction number, a quantity that is useful in establishing vaccination policies. In order to evaluate the population size for which the results are useful, a numerical study is carried out. Finally, a comparison of the maximum likelihood and martingale estimators is conducted by means of Monte Carlo simulations.