Orthogonal Locally Ancillary Estimating Functions with Application to Stratified Clustered Data

We propose an orthogonal locally ancillary estimating function that provides first-order bias correction of inferences. It requires the specification of merely the first two moments of the observations when applying to analysis of stratified clustered (continuous or binary) data with the parameters of interest in both the first and second joint moments of dependent data. Simulation results confirm that the estimators obtained using the proposed method are substantially improved over those using regular profile estimating functions.     

Bounded Estimation in the Presence of Nuisance Parameters

The aim of this paper is to extend in a natural fashion the results on the treatment of nuisance parameters from the profile likelihood theory to the field of robust statistics. Similarly to what happens when there are no nuisance parameters, the attempt is to derive a bounded estimating function for a parameter of interest in the presence of nuisance parameters. The proposed method is based on a classical truncation argument of the theory of robustness applied to a generalized profile score function. By means of comparative studies, we show that this robust procedure for inference in the presence of a nuisance parameter can be used successfully in a parametric setting.     

Second-Order Powers of a Class of Tests in the Presence of a Nuisance Parameter

The second-order local powers of a broad class of asymptotic chi-squared tests are considered in a composite case where both the parameter of interest and the nuisance parameter are possibly multidimensional for which no assumption has been made regarding global parametric orthogonality or curved exponentiality. The main result is that the second-order (point-by-point) local power identity holds if approximate third cumulants of a square-root version of the (modified) test statistic in the class vanish up to the second-order, which is an extension of Kakizawa (2010a Kakizawa , Y. ( 2010a ). Second-order power comparison of tests . Commun. Statist. Theor. Meth. 39 : 1424 – 1436 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) in the absence of the nuisance parameter. It is also shown that in the presence of the nuisance parameter, such a third cumulant condition does not always imply the second-order local unbiasedness of the resulting test. Then, the adjusted likelihood ratio test by Mukerjee (1993b Mukerjee , R. ( 1993b ). An extension of the conditional likelihood ratio test to the general multiparameter case . Ann. Inst. Statist. Math. 45 : 759 – 771 .[Crossref], [Web of Science ®] , [Google Scholar]) can be interpreted as the second-order local unbiased modification after applying the third cumulant condition.     

Polynomial Regression and Estimating Functions in the Presence of Multiplicative Measurement Error

We consider the polynomial regression model in the presence of multiplicative measurement error in the predictor. Two general methods are considered, with the methods differing in their assumptions about the distributions of the predictor and the measurement errors. Consistent parameter estimates and asymptotic standard errors are derived by using estimating equation theory. Diagnostics are presented for distinguishing additive and multiplicative measurement error. Data from a nutrition study are analysed by using the methods. The results from a simulation study are presented and the performances of the methods are compared.     

Estimating functions in the Bayesian paradigm

Received: August 3, 1998; revised version: June 9, 1999     

Maximizing a Family of Optimal Statistics over a Nuisance Parameter with Applications to Genetic Data Analysis

In this article, a simple algorithm is used to maximize a family of optimal statistics for hypothesis testing with a nuisance parameter not defined under the null hypothesis. This arises from genetic linkage and association studies and other hypothesis testing problems. The maximum of optimal statistics over the nuisance parameter space can be used as a robust test in this situation. Here, we use the maximum and minimum statistics to examine the sensitivity of testing results with respect to the unknown nuisance parameter. Examples from genetic linkage analysis using affected sub pairs and a candidate-gene association study in case-parents trio design are studied.     

Quadratic Artificial Likelihood Functions Using Estimating Functions

Abstract.  A vector-valued estimating function, such as the quasi-score, is typically not the gradient of any objective function. Consequently, an analogue of the likelihood function cannot be unambiguously defined by integrating the estimating function. This paper studies an analogue of the likelihood inference in the framework of optimal estimating functions. We propose a quadratic artificial likelihood function for an optimal estimating function. The objective function is uniquely identified as the potential function from the vector field decomposition by imposing some natural restriction on the divergence-free part. The artificial likelihood function is shown to resemble a genuine likelihood function in a number of respects. A bootstrap version of the artificial likelihood function is also studied, which may be used for selecting a root as an estimate from among multiple roots to an estimating equation.     

On the Mean Value Theorem for Estimating Functions

Abstract

Feng et al. revealed that the usual mean value theorem (MVT) should not be applied directly to a vector-valued function (e.g., the score function or a general estimating function under a multiparametric model). This note shows that the application of the Cramer–Wold’s device to a corrected version of the MVT is sufficient to obtain standard asymptotics for the estimators attained from vector-valued estimating functions.     

On Some Optimal Bayesian Nonparametric Rules for Estimating Distribution Functions

In this paper, we present a novel approach to estimating distribution functions, which combines ideas from Bayesian nonparametric inference, decision theory and robustness. Given a sample from a Dirichlet process on the space (𝒳, A), with parameter η in a class of measures, the sampling distribution function is estimated according to some optimality criteria (mainly minimax and regret), when a quadratic loss function is assumed. Estimates are then compared in two examples: one with simulated data and one with gas escapes data in a city network.     

Bayesian Analysis in Regression Models Using Pseudo-Likelihoods

This article deals with the issue of using a suitable pseudo-likelihood, instead of an integrated likelihood, when performing Bayesian inference about a scalar parameter of interest in the presence of nuisance parameters. The proposed approach has the advantages of avoiding the elicitation on the nuisance parameters and the computation of multidimensional integrals. Moreover, it is particularly useful when it is difficult, or even impractical, to write the full likelihood function.

We focus on Bayesian inference about a scalar regression coefficient in various regression models. First, in the context of non-normal regression-scale models, we give a theroetical result showing that there is no loss of information about the parameter of interest when using a posterior distribution derived from a pseudo-likelihood instead of the correct posterior distribution. Second, we present non trivial applications with high-dimensional, or even infinite-dimensional, nuisance parameters in the context of nonlinear normal heteroscedastic regression models, and of models for binary outcomes and count data, accounting also for possibile overdispersion. In all these situtations, we show that non Bayesian methods for eliminating nuisance parameters can be usefully incorporated into a one-parameter Bayesian analysis.     

Optimal estimating function for estimation and prediction in semi-parametric models

In a seminal paper, Godambe [1985. The foundations of finite sample estimation in stochastic processes. Biometrika 72, 419–428.] introduced the ‘estimating function’ approach to estimation of parameters in semi-parametric models under a filtering associated with a martingale structure. Later, Godambe [1987. The foundations of finite sample estimation in stochastic processes II. Bernoulli, Vol. 2. V.N.V. Science Press, 49–54.] and Godambe and Thompson [1989. An extension of quasi-likelihood Estimation. J. Statist. Plann. Inference 22, 137–172.] replaced this filtering by a more flexible conditioning. Abraham et al. [1997. On the prediction for some nonlinear time-series models using estimating functions. In: Basawa, I.V., et al. (Eds.), IMS Selected Proceedings of the Symposium on Estimating Functions, Vol. 32. pp. 259–268.] and Thavaneswaran and Heyde [1999. Prediction via estimating functions. J. Statist. Plann. Inference 77, 89–101.] invoked the theory of estimating functions for one-step ahead prediction in time-series models. This paper addresses the problem of simultaneous estimation of parameters and multi-step ahead prediction of a vector of future random variables in semi-parametric models by extending the inimitable approach of 13 and 14. The proposed technique is in conformity with the paradigm of the modern theory of estimating functions leading to finite sample optimality within a chosen class of estimating functions, which in turn are used to get the predictors. Particular applications of the technique give predictors that enjoy optimality properties with respect to other well-known criteria.     

On the Simultaneous Estimation of Scale Parameters

This note is an extension of Das Gupta's results (1986) on the estimation of multiparameter gamma distribution. Consider p (p ? 2) independent positive random variables with possibly different scale-parameter densities. For the estimation of the powers of the scale parameters it is shown that the “best multiple estimator” is inadmissible with respect to a large class of weighted quadratic loss functions.     

Parameter Orthogonality and Bias Adjustment for Estimating Functions

Abstract.  We consider an extended notion of parameter orthogonality for estimating functions, called nuisance parameter insensitivity, which allows a unified treatment of nuisance parameters for a wide range of methods, including Liang and Zeger's generalized estimating equations. Nuisance parameter insensitivity has several important properties in common with conventional parameter orthogonality, such as the nuisance parameter causing no loss of efficiency for estimating the interest parameter, and a simplified estimation algorithm. We also consider bias adjustment for profile estimating functions, and apply the results to restricted maximum likelihood estimation of dispersion parameters in generalized estimating equations.     

Optimal Weight Functions for Marginal Proportional Hazards Analysis of Clustered Failure Time Data

The choice of weights in estimating equations for multivariate survival data is considered. Specifically, we consider families of weight functions which are constant on fixed time intervals, including the special case of time-constant weights. For a fixed set of time intervals, the optimal weights are identified as the solution to a system of linear equations. The optimal weights are computed for several scenarios. It is found that for the scenarios examined, the gains in efficiency using the optimal weights are quite small relative to simpler approaches except under extreme dependence, and that a simple estimator of an exchangeable approximation to the weights also performs well.     

On expected volumes of multidimensional confidence sets associated with the usual and adjusted likelihoods

We consider a general multiparameter set-up, where both the interest and the nuisance parameters are possibly vector valued. We derive an explicit higher order asymptotic formula to compare the expected volumes of confidence sets given by likelihood ratio statistics arising from the usual profile likelihood and various adjustments thereof. Our general framework also allows us to include highest posterior density regions, with approximate frequentist validity, in the study. The fact that our interest parameter is possibly vector valued complicates the derivation and warrants the development of special tools and techniques.     

我国各地区常住人口总量推算方法探讨

 本文以年度人口变动调查为基础,通过调查指标之间的关系、人口变动自身特征与抽查的情况,对我国各地区常住人口的推算方法进行了分析研究,并对各种评估推算方法的优缺点和适用条件进行了分析比较,根据各地区人口流动特点与人口基数,选择合适的评估推算方法,准确推算出各地常住人口数据。如何加强全国与各地数据衔接问题是今后推算评估数据工作的重点。     

On the Computation of Gauss Hypergeometric Functions

The pioneering study undertaken by Liang et al. in 2008 (Journal of the American Statistical Association, 103, 410–423) and the hundreds of papers citing that work make use of certain hypergeometric functions. Liang et al. and many others claim that the computation of the hypergeometric functions is difficult. Here, we show that the hypergeometric functions can in fact be reduced to simpler functions that can often be computed using a pocket calculator.     

Estimating the First- and Second-Order Parameters of a Heavy-Tailed Distribution

This paper suggests censored maximum likelihood estimators for the first‐ and second‐order parameters of a heavy‐tailed distribution by incorporating the second‐order regular variation into the censored likelihood function. This approach is different from the bias‐reduced maximum likelihood method proposed by Feuerverger and Hall in 1999. The paper derives the joint asymptotic limit for the first‐ and second‐order parameters under a weaker assumption. The paper also demonstrates through a simulation study that the suggested estimator for the first‐order parameter is better than the estimator proposed by Feuerverger and Hall although these two estimators have the same asymptotic variances.     

Nuisance Parameters and the Use of Exploratory Graphical Methods in a Bayesian Analysis

Consider the problem of inference about a parameter θ in the presence of a nuisance parameter v. In a Bayesian framework, a number of posterior distributions may be of interest, including the joint posterior of (θ, ν), the marginal posterior of θ, and the posterior of θ conditional on different values of ν. The interpretation of these various posteriors is greatly simplified if a transformation (θ, h(θ, ν)) can be found so that θ and h(θ, v) are approximately independent. In this article, we consider a graphical method for finding this independence transformation, motivated by techniques from exploratory data analysis. Some simple examples of the use of this method are given and some of the implications of this approximate independence in a Bayesian analysis are discussed.