Robust M- and L-Estimators of Scale Parameter

We consider first the class of M-estimators of scale that are location-scale equivariant and Fisher consistent at the error distribution of the shrinking contamination neighborhood and derive an expression for the maximal asymptotic mean-squared-error, for a suitably regular score function, followed by a lower bound on it. We next show that the minimax asymptotic mean-squzred-error is attained at an M-estimator of scale with the truncated MLE score function which, when specialized to the Standard Normal error distribution has the form of Huber's Proposal 2. The latter minimax property is also shown to hold for α-trimmed variance as an L-estimator of scale.     

Truncated Poisson Regression for Time Series of Counts

We consider partial likelihood analysis of a truncated Poisson regression model for time series of counts. We focus our attention on the study of asymptotic theory for the maximum partial likelihood estimator of a vector of regression parameters. Simulations and data analysis integrate the presentation.     

Median regression model with doubly truncated data

We study the problem of fitting a heteroscedastic median regression model with doubly truncated data. A self-consistency equation is proposed to obtain an estimator. We set up a least absolute deviation estimating function. We establish the consistency and asymptotic normality for the case when covariates are discrete. The finite sample performance of the proposed estimators are investigated through simulation studies. The proposed method is illustrated using the AIDS Blood Transfusion Data.     

A Class of Non-parametric Tests in the Competing Risks Model for Comparing Two Samples

We consider a model when a process involving the production of elements is under inspection. The elements have possible failures due to competing risks. We assume the availability of a data set of failure times, D₁, obtained when the process is under control. Our main goal is to test if the failure rates in D₁ are equal to or less than the failure rates in another data set D₂, against "undesirable" neighbouring alternatives. A class of tests based on a two-dimensional vector statistic is obtained. Linear test statistics with weight functions giving optimal local asymptotic power are derived. Martingale techniques are used. An example illustrates the derivation of reasonable tests     

Local polynomial quasi-likelihood regression with truncated and dependent data

The local polynomial quasi-likelihood estimation has several good statistical properties such as high minimax efficiency and adaptation of edge effects. In this paper, we construct a local quasi-likelihood regression estimator for a left truncated model, and establish the asymptotic normality of the proposed estimator when the observations form a stationary and α-mixing sequence, such that the corresponding result of Fan et al. [Local polynomial kernel regression for generalized linear models and quasilikelihood functions, J. Amer. Statist. Assoc. 90 (1995), pp. 141–150] is extended from the independent and complete data to the dependent and truncated one. Finite sample behaviour of the estimator is investigated via simulations too.     

Expressing Estimators of Expected Quality Adjusted Survival as Functions of Nelson-Aalen Estimators

Quality adjusted survival has been increasingly advocated in clinical trials to be assessed as a synthesis of survival and quality of life. We investigate nonparametric estimation of its expectation for a general multistate process with incomplete follow-up data. Upon establishing a representation of expected quality adjusted survival through marginal distributions of a set of defined events, we propose two estimators for expected quality adjusted survival. Expressed as functions of Nelson-Aalen estimators, the two estimators are strongly consistent and asymptotically normal. We derive their asymptotic variances and propose sample-based variance estimates, along with evaluation of asymptotic relative efficiency. Monte Carlo studies show that these estimation procedures perform well for practical sample sizes. We illustrate the methods using data from a national, multicenter AIDS clinical trial.     

A nonparametric test for a constant correlation matrix

We propose a nonparametric procedure to test for changes in correlation matrices at an unknown point in time. The new test requires constant expectations and variances, but only mild assumptions on the serial dependence structure, and has considerable power in finite samples. We derive the asymptotic distribution under the null hypothesis of no change as well as local power results and apply the test to stock returns.     

A New Kind of Asymptotic Quasi-Score Estimating Function

A new definition of asymptotic quasi-score sequence of estimating functions is given and studied. The relationship between asymptotic quasi-likelihood and quasi-likelihood estimates is investigated. A new practical approach for obtaining a good estimate of θ in the model y _t = f_t (θ) + m_t without any prior knowledge on the nature of E ( m ² _t |F _{t −1}) is suggested, where f_t is a predictable process and m_t is a martingale difference process. Two examples are used to show that the approach is practicable.     

A test for abrupt change in hazard regression models with Weibull baselines

We develop a likelihood ratio test for an abrupt change point in Weibull hazard functions with covariates, including the two-piece constant hazard as a special case. We first define the log-likelihood ratio test statistic as the supremum of the profile log-likelihood ratio process over the interval which may contain an unknown change point. Using local asymptotic normality (LAN) and empirical measure, we show that the profile log-likelihood ratio process converges weakly to a quadratic form of Gaussian processes. We determine the critical values of the test and discuss how the test can be used for model selection. We also illustrate the method using the Chronic Granulomatous Disease (CGD) data.     

Estimation of parameters for the truncated exponential distribution

We consider the right truncated exponential distribution where the truncation point is unknown and show that the ML equation has a unique solution over an extended parameter space. In the case of the estimation of the truncation point T we show that the asymptotic distribution of the MLE is not centered at T. A modified MLE is introduced which outperforms all other considered estimators including the minimum variance unbiased estimator. Asymptotic as well as small sample properties of different estimators are investigated and compared. The truncated exponential distribution has an increasing failure rate, ideally suited for use as a survival distribution for biological and industrial data.     

Survivor Function Estimators Under Group Sequential Monitoring Based on the Logrank Statistic

In this paper we investigate a group sequential analysis of censored survival data with staggered entry, in which the trial is monitored using the logrank test while comparisons of treatment and control Kaplan-Meier curves at various time points are performed at the end of the trial. We concentrate on two-sample tests under local alternatives. We describe the relationship of the asymptotic bias of Kaplan-Meier curves between the two groups. We show that even if the asymptotic bias of the Kaplan-Meier curve is negligible relative to the true survival, this is not the case for the difference between the curves of the two arms of the trial. A corrected estimator for the difference between the survival curves is presented and by simulations we show that the corrected estimator reduced the bias dramatically and has a smaller variance. The methods of estimation are applied to the Beta-Blocker Heart Attack Trial (1982), a well-known group sequential trial.     

Asymptotic Normality for Regression Function Estimate Under Truncation and α-Mixing Conditions

In this article we establish pointwise asymptotic normality of nonparametric kernel estimator of regression function for a left truncation model. It is assumed that the lifetime observations with multivariate covariates form a stationary α-mixing sequence. Also, the asymptotic normality of the estimation of the covariable's density is considered. As a by-product, we obtain a uniform weak convergence rate for the product-limit estimator of the lifetime and truncated distributions under dependence, which is interesting independently. Finite sample behavior of the estimator of the regression function is investigated as well.     

Estimation of conditional mode with truncated,censored, and dependent data

In this paper, we studied the uniform convergence with rates for the kernel estimator of the conditional mode function for a left truncated and right censored model. It is assumed that the lifetime observations with multivariate covariates form a stationary α-mixing sequence. Also, the asymptotic normality of the estimator is established.     

On the Local Polynomial Estimators of the Counting Process Intensity Function and its Derivatives

Abstract. We consider the properties of the local polynomial estimators of a counting process intensity function and its derivatives. By expressing the local polynomial estimators in a kernel smoothing form via effective kernels, we show that the bias and variance of the estimators at boundary points are of the same magnitude as at interior points and therefore the local polynomial estimators in the context of intensity estimation also enjoy the automatic boundary correction property as they do in other contexts such as regression. The asymptotically optimal bandwidths and optimal kernel functions are obtained through the asymptotic expressions of the mean square error of the estimators. For practical purpose, we suggest an effective and easy‐to‐calculate data‐driven bandwidth selector. Simulation studies are carried out to assess the performance of the local polynomial estimators and the proposed bandwidth selector. The estimators and the bandwidth selector are applied to estimate the rate of aftershocks of the Sichuan earthquake and the rate of the Personal Emergency Link calls in Hong Kong.     

Local Linear GMM Estimation of Functional Coefficient IV Models With an Application to Estimating the Rate of Return to Schooling

We consider the local linear generalized method of moment (GMM) estimation of functional coefficient models with a mix of discrete and continuous data and in the presence of endogenous regressors. We establish the asymptotic normality of the estimator and derive the optimal instrumental variable that minimizes the asymptotic variance-covariance matrix among the class of all local linear GMM estimators. Data-dependent bandwidth sequences are also allowed for. We propose a nonparametric test for the constancy of the functional coefficients, study its asymptotic properties under the null hypothesis as well as a sequence of local alternatives and global alternatives, and propose a bootstrap version for it. Simulations are conducted to evaluate both the estimator and test. Applications to the 1985 Australian Longitudinal Survey data indicate a clear rejection of the null hypothesis of the constant rate of return to education, and that the returns to education obtained in earlier studies tend to be overestimated for all the work experience.     

Robust kernel estimators for additive models with dependent observations

Robust nonparametric estimators for additive regression or autoregression models under an α-mixing condition are proposed. They are based on local M-estimators or local medians with kernel weights, and their asymptotic behaviour is studied. Moreover, diese local M-estimators achieve the same univariate rate of convergence as their linear relatives.     

Asymptotic power of tests of normality under local alternatives

Seven tests of univariate normality are studied in view of their asymptotic power under local alternatives. The procedures under consideration are either based on the empirical skewness and/or kurtosis, including the popular Jarque-Bera statistic, as well as Cramér-von Mises, Anderson-Darling and Kolmogorov-Smirnov functionals of an empirical process with estimated parameters. The large-sample behavior of these test statistics under contiguous sequences is obtained; this allows for the computation of their associated local power curves and of their asymptotic relative efficiency in the light of a measure proposed by Berg and Quessy (2009). Comparisons are made under four classes of local alternatives, including those used by Thadewald and Büning (2007) in a recent Monte-Carlo power study. These theoretical results are related to empirical ones and many recommendations are formulated.     

Testing for residual correlation of any order in the autoregressive process

We are interested in the implications of a linearly autocorrelated driven noise on the asymptotic behavior of the usual least-squares estimator in a stable autoregressive process. We show that the least-squares estimator is not consistent and we suggest a sharp analysis of its almost sure limiting value as well as its asymptotic normality. We also establish the almost sure convergence and the asymptotic normality of the estimated serial correlation parameter of the driven noise. Then, we derive a statistical procedure enabling to test for correlation of any order in the residuals of an autoregressive modelling, giving clearly better results than the commonly used portmanteau tests of Ljung–Box and Box–Pierce, and appearing to outperform the Breusch–Godfrey procedure on small-sized samples.     

Hypotheses testing for a multidimensional parameter of inhomogeneous Poisson processes

We observe a realization of an inhomogeneous Poisson process whose intensity function depends on an unknown multidimensional parameter. We consider the asymptotic behaviour of the Rao score test for a simple null hypothesis against the multilateral alternative. By using the Edgeworth type expansion (under the null hypothesis) for a vector of stochastic integrals with respect to the Poisson process, we refine the (classic) threshold of the test (obtained by the central limit theorem), which improves the first type probability of error. The expansion allows us to describe the power of the test under the local alternative, i.e. a sequence of alternatives, which converge to the null hypothesis with a certain rate. The rates can be different for components of the parameter.