Parameter estimation of the generalized Pareto distribution—Part II

This is the second part of a paper which focuses on reviewing methods for estimating the parameters of the generalized Pareto distribution (GPD). The GPD is a very important distribution in the extreme value context. It is commonly used for modeling the observations that exceed very high thresholds. The ultimate success of the GPD in applications evidently depends on the parameter estimation process. Quite a few methods exist in the literature for estimating the GPD parameters. Estimation procedures, such as the maximum likelihood (ML), the method of moments (MOM) and the probability weighted moments (PWM) method were described in Part I of the paper. We shall continue to review methods for estimating the GPD parameters, in particular methods that are robust and procedures that use the Bayesian methodology. As in Part I, we shall focus on those that are relatively simple and straightforward to be applied to real world data.     

On estimation and influence diagnostics for log-Birnbaum–Saunders Student-t regression models: Full Bayesian analysis

The purpose of this paper is to develop a Bayesian approach for log-Birnbaum–Saunders Student-t regression models under right-censored survival data. Markov chain Monte Carlo (MCMC) methods are used to develop a Bayesian procedure for the considered model. In order to attenuate the influence of the outlying observations on the parameter estimates, we present in this paper Birnbaum–Saunders models in which a Student-t distribution is assumed to explain the cumulative damage. Also, some discussions on the model selection to compare the fitted models are given and case deletion influence diagnostics are developed for the joint posterior distribution based on the Kullback–Leibler divergence. The developed procedures are illustrated with a real data set.     

Analysis of a longitudinal ordinal response clinical trial using dynamic models

Summary.  In many areas of pharmaceutical research, there has been increasing use of categorical data and more specifically ordinal responses. In many cases, complex models are required to account for different types of dependences among the responses. The clinical trial that is considered here involved patients who were required to remain in a particular state to enable the doctors to examine their heart. The aim of this trial was to study the relationship between the dose of the drug administered and the time that was spent by the patient in the state permitting examination. The patient's state was measured every second by a continuous Doppler signal which was categorized by the doctors into one of four ordered categories. Hence, the response consisted of repeated ordinal series. These series were of different lengths because the drug effect wore off faster (or slower) on certain patients depending on the drug dose administered and the infusion rate, and therefore the length of drug administration. A general method for generating new ordinal distributions is presented which is sufficiently flexible to handle unbalanced ordinal repeated measurements. It consists of obtaining a cumulative mixture distribution from a Laplace transform and introducing into it the integrated intensity of a binary logistic, continuation ratio or proportional odds model. Then, a multivariate distribution is constructed by a procedure that is similar to the updating process of the Kalman filter. Several types of history dependences are proposed.     

A non linear wavelet based estimator for long memory processes

Two wavelet based estimators are considered in this paper for the two parameters that characterize long range dependence processes. The first one is linear and is based on the statistical properties of the coefficients of a discrete wavelet transform of long range dependence processes. The estimator consists in measuring the slope (related to the long memory parameter) and the intercept (related to the variance of the process) of a linear regression after a discrete wavelet transform is performed (Veitch and Abry, 1999). In this paper its properties are reviewed, and analytic evidence is produced that the linear estimator is applicable only when the second parameter is unknown. To overcome this limitation a non linear wavelet based estimator - that takes into account that the intercept depends on the long memory parameter - is proposed here for the cases in which the second parameter is known or the only parameter of interest is the long memory parameter. Under the same hypothesis assumed for the linear estimator, the non linear estimator is shown to be asymptotically more efficient for the long memory parameter. Numerical simulations show that, even for small data sets, the bias is very small and the variance close to optimal. An application to ATM based Internet traffic is presented.Financial support from the Italian Ministry of University and Scientific Research (MIUR), also in the context of the COFIN 2002 ALINWEB (Algorithms for the Internet and the Web) Project, is gratefully acknowledged.     

Small sample comparison of six bivariate survival curve estimators 1

We study the performance of six proposed bivariate survival curve estimators on simulated right censored data. The performance of the estimators is compared for data generated by three bivariate models with exponential marginal distributions. The estimators are compared in their ability to estimate correlations and survival functions probabilities. Simulated data results are presented so that the proposed estimators in this relatively new area of analysis can be explicitly compared to the known distribution of the data and the parameters of the underlying model. The results show clear differences in the performance of the estimators.     

N-stage splitting for maximum penalized likelihood estimation

An n-stage splitting algorithm for the solution of maximum penalized likelihood estimation (MPLE) problems is compared to the one-step-late (OSL) algorithm. General conditions under which the asymptotic rate of convergence of this splitting algorithm. exceeds that of the OSL algorithm are given. A one-dimensional positive data example, illustrates the comparison of the rates of convergence of these two algorithms.     

Indirect inference for fractional time series models

In this paper we present an indirect estimation procedure for (ARFIMA) fractional time series models.The estimation method is based on an ‘incorrect’criterion which does not directly provide a consistent estimator of the parameters of interest,but leads to correct inference by using simulations.

The main steps are the following. First,we consider an auxiliary model which can be easily estimated.Specifically,we choose the finite lag Autoregressive model.Then, this is estimated on the observations and simulated values drawn from the ARFIMA model associated with a given value of the parameters of interest.Finally,the latter is calibrated in order to obtain close values of the two estimators of the auxiliary parameters.

In this article,we describe the estimation procedure and compare the performance of the indirect estimator with some alternative estimators based on the likelihood function by a Monte Carlo study.     

Estimation of Causal Effects in Latent Strata with an Encouragement for Response

We consider a new approach to deal with non ignorable non response on an outcome variable, in a causal inference framework. Assuming that a binary instrumental variable for non response is available, we provide a likelihood-based approach to identify and estimate heterogeneous causal effects of a binary treatment on specific latent subgroups of units, named principal strata, defined by the non response behavior under each level of the treatment and of the instrument. We show that, within each stratum, non response is ignorable and respondents can be properly compared by treatment status. In order to assess our method and its robustness when the usually invoked assumptions are relaxed or misspecified, we simulate data to resemble a real experiment conducted on a panel survey which compares different methods of reducing panel attrition.     

“似然”是一种错误的说法

文章指出统计教学中关于"似然"这一名词的讲解存在一定的偏差,学生理解时也有一定的障碍.因此,作者通过追根溯源,发现"似然"这一统计名词在翻译过程中存在着明显地错误.     

Efficient sampling schemes for Bayesian MARS models with many predictors

Multivariate adaptive regression spline fitting or MARS (Friedman 1991) provides a useful methodology for flexible adaptive regression with many predictors. The MARS methodology produces an estimate of the mean response that is a linear combination of adaptively chosen basis functions. Recently, a Bayesian version of MARS has been proposed (Denison, Mallick and Smith 1998a, Holmes and Denison, 2002) combining the MARS methodology with the benefits of Bayesian methods for accounting for model uncertainty to achieve improvements in predictive performance. In implementation of the Bayesian MARS approach, Markov chain Monte Carlo methods are used for computations, in which at each iteration of the algorithm it is proposed to change the current model by either (a) Adding a basis function (birth step) (b) Deleting a basis function (death step) or (c) Altering an existing basis function (change step). In the algorithm of Denison, Mallick and Smith (1998a), when a birth step is proposed, the type of basis function is determined by simulation from the prior. This works well in problems with a small number of predictors, is simple to program, and leads to a simple form for Metropolis-Hastings acceptance probabilities. However, in problems with very large numbers of predictors where many of the predictors are useless it may be difficult to find interesting interactions with such an approach. In the original MARS algorithm of Friedman (1991) a heuristic is used of building up higher order interactions from lower order ones, which greatly reduces the complexity of the search for good basis functions to add to the model. While we do not exactly follow the intuition of the original MARS algorithm in this paper, we nevertheless suggest a similar idea in which the Metropolis-Hastings proposals of Denison, Mallick and Smith (1998a) are altered to allow dependence on the current model. Our modification allows more rapid identification and exploration of important interactions, especially in problems with very large numbers of predictor variables and many useless predictors. Performance of the algorithms is compared in simulation studies.