Low dimensional semiparametric estimation in a censored regression model

A new estimation procedure for a partial linear additive model with censored responses is proposed. To this aim, ideas of Lewbel and Linton [A. Lewbel, O. Linton, Nonparametric censored and truncated regression, Econometrica 70 (2002) 765-779] on censored model regression are combined with those of Kim et al. [W. Kim, O. Linton, N.W. Hengartner, A computationally efficient estimator for additive nonparametric regression with bootstrap confidence intervals, Journal of Computational and Graphical Statistics, 8 (1999) 278-297] on marginal integration and those on average derivatives. This allows for dimension reduction, interpretability and — depending on the context — for weights yielding computationally attractive estimates. Asymptotic behavior is provided for all proposed estimators.     

Addendum to “a third-order optimum property of the maximum likelihood estimator”

A third-order optimum property of the maximum likelihood estimator is extended to not necessarily symmetric loss functions under an appropriate restriction on the class of competing estimators.     

A third-order optimum property of the maximum likelihood estimator

Let θ⁽ⁿ⁾ denote the maximum likelihood estimator of a vector parameter, based on an i.i.d. sample of size n. The class of estimators θ⁽ⁿ⁾ + n^?1q(θ⁽ⁿ⁾), with q running through a class of sufficiently smooth functions, is essentially complete in the following sense: For any estimator T⁽ⁿ⁾ there exists q such that the risk of θ⁽ⁿ⁾ + n^?1q(θ⁽ⁿ⁾) exceeds the risk of T⁽ⁿ⁾ by an amount of order o(n^?1) at most, simultaneously for all loss functions which are bounded, symmetric, and neg-unimodal. If $\text{q}{}^1$ is chosen such that $\text{θ}{}^{\mathrm{(n)}}\text{+ n}{}^{\mathrm{?1}}\text{q}{}^1\text{(θ}{}^{\mathrm{(n)}}\text{)}$ is unbiased up to $\text{o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$, then this estimator minimizes the risk up to an amount of order o(n^?1) in the class of all estimators which are unbiased up to $\text{o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$.The results are obtained under the assumption that T⁽ⁿ⁾ admits a stochastic expansion, and that either the distributions have—roughly speaking—densities with respect to the lebesgue measure, or the loss functions are sufficiently smooth.     

Asymptotic expansions of maximum likelihood estimators for small diffusions via the theory of Malliavin-Watanabe

Summary The asymptotic expansions of the probability distributions of statistics for the small diffusion are derived by means of the Malliavin calculus. From this the second order efficiency of the maximum likelihood estimator is proved.The research was supported in part by Grant-in-Aid for Encouragement of Young Scientists from the Ministry of Education, Science and Culture     

Asymptotics for estimation and testing procedures under loss of identifiability

Statistical analyses commonly make use of models that suffer from loss of identifiability. In this paper, we address important issues related to the parameter estimation and hypothesis testing in models with loss of identifiability. That is, there are multiple parameter points corresponding to the same true model. We refer the set of these parameter points to as the set of true parameter values. We consider the case where the set of true parameter values is allowed to be very large or even infinite, some parameter values may lie on the boundary of the parameter space, and the data are not necessarily independently and identically distributed. Our results are applicable to a large class of estimators and their related testing statistics derived from optimizing an objective function such as a likelihood. We examine three specific examples: (i) a finite mixture logistic regression model; (ii) stationary ARMA processes; (iii) general quadratic approximation using Hellinger distance. The applications to these examples demonstrate the applicability of our results in a broad range of difficult statistical problems.     

Nonlinear least-squares estimation

The paper uses empirical process techniques to study the asymptotics of the least-squares estimator (LSE) for the fitting of a nonlinear regression function. By combining and extending ideas of Wu and Van de Geer, it establishes new consistency and central limit theorems that hold under only second moment assumptions on the errors. An application to a delicate example of Wu's illustrates the use of the new theorems, leading to a normal approximation to the LSE with unusual logarithmic rescalings.     

Robust estimation of periodic autoregressive processes in the presence of additive outliers

This paper suggests a robust estimation procedure for the parameters of the periodic AR (PAR) models when the data contains additive outliers. The proposed robust methodology is an extension of the robust scale and covariance functions given in, respectively, Rousseeuw and Croux (1993) [28], and Ma and Genton (2000) [23] to accommodate periodicity. These periodic robust functions are used in the Yule-Walker equations to obtain robust parameter estimates. The asymptotic central limit theorems of the estimators are established, and an extensive Monte Carlo experiment is conducted to evaluate the performance of the robust methodology for periodic time series with finite sample sizes. The quarterly Fraser River data was used as an example of application of the proposed robust methodology. All the results presented here give strong motivation to use the methodology in practical situations in which periodically correlated time series contain additive outliers.     

Comparison of Bartlett-type adjusted tests in the multiparameter case

Considering some Bartlett-type adjusted tests for a simple hypothesis about a multidimensional parameter, this paper clarifies similarities and dissimilarities with the one-parameter case developed in the 1990s, where a major emphasis is put on the issue posed by Rao and Mukerjee [C.R. Rao, R. Mukerjee, Comparison of Bartlett-type adjustments for the efficient score statistic, J. Statist. Plann. Inference 46 (1995) 137-146] on the power under a sequence of local alternatives. Not surprisingly, there is an infinite number of adjustments which extend Chandra-Mukerjee and Taniguchi approaches to the multiparameter case. Revisiting their ideas, this paper presents four specific cases (type K, K=0,1,2,3) and gives a sufficient condition under which our generalized adjustment for each case is uniquely determined, where type 0 is a counterpart of Chandra and Mukerjee’s original proposal for Rao’s test statistic, whereas the latter three types are introduced as double adjustments related to the Cordeiro and Ferrari approach. If the adjustment of type 1 is made instead of type K, K=0,2,3, it is shown that Chandra and Mukerjee’s approach is equivalent to Taniguchi’s approach in terms of the third-order local power. The same is partially true for type 0, depending on the model under consideration. However, the adjustments of type K, K=2,3, reveal, in general, the non-equivalence of these two approaches in terms of the third-order local power.     

Preliminary test estimation for spectra

For a general non-Gaussian stationary linear process, quasi-maximum likelihood estimation of a subset of the parameters of the spectral density is considered when the complementary subset is suspected to be superfluous. A preliminary test quasi-maximum likelihood estimator (q-MLE) of parameters is introduced and, in the light of its mean square error, is compared with the restricted and unrestricted q-MLE.     

Minimum Hellinger distance estimation in a two-sample semiparametric model

We investigate the estimation problem of parameters in a two-sample semiparametric model. Specifically, let X₁,…,X_n be a sample from a population with distribution function G and density function g. Independent of the X_i’s, let Z₁,…,Z_m be another random sample with distribution function H and density function h(x)=exp[α+r(x)β]g(x), where α and β are unknown parameters of interest and g is an unknown density. This model has wide applications in logistic discriminant analysis, case-control studies, and analysis of receiver operating characteristic curves. Furthermore, it can be considered as a biased sampling model with weight function depending on unknown parameters. In this paper, we construct minimum Hellinger distance estimators of α and β. The proposed estimators are chosen to minimize the Hellinger distance between a semiparametric model and a nonparametric density estimator. Theoretical properties such as the existence, strong consistency and asymptotic normality are investigated. Robustness of proposed estimators is also examined using a Monte Carlo study.     

Parametric estimation of a bivariate stable Lévy process

We propose a parametric model for a bivariate stable Lévy process based on a Lévy copula as a dependence model. We estimate the parameters of the full bivariate model by maximum likelihood estimation. As an observation scheme we assume that we observe all jumps larger than some ε>0 and base our statistical analysis on the resulting compound Poisson process. We derive the Fisher information matrix and prove asymptotic normality of all estimates when the truncation point ε→0. A simulation study investigates the loss of efficiency because of the truncation.     

Linear combination of concomitants of order statistics with application to testing and estimation

Summary Let (X ₁,Y ₁), (X ₂,Y ₂),…, (X _n,Y _n) be i.i.d. as (X, Y). TheY-variate paired with therth orderedX-variateX _rn is denoted byY _rn and terms the concomitant of therth order statistic. Statistics of the form are considered. The asymptotic normality ofT _n is established. The asymptotic results are used to test univariate and bivariate normality, to test independence and linearity ofX andY, and to estimate regression coefficient based on complete and censored samples.     

Asymptotic normality of test statistics under alternative hypotheses

The aim of this paper is to present a framework for asymptotic analysis of likelihood ratio and minimum discrepancy test statistics. First order asymptotics are presented in a general framework under minimal regularity conditions and for not necessarily nested models. In particular, these asymptotics give sufficient and in a sense necessary conditions for asymptotic normality of test statistics under alternative hypotheses. Second order asymptotics, and their implications for bias corrections, are also discussed in a somewhat informal manner. As an example, asymptotics of test statistics in the analysis of covariance structures are discussed in detail.     

Robust estimation of Cronbach's alpha

Cronbach's alpha is a popular method to measure reliability, e.g. in quantifying the reliability of a score to summarize the information of several items in questionnaires. The alpha coefficient is known to be non-robust. We study the behavior of this coefficient in different settings to identify situations where Cronbach's alpha is extremely sensitive to violations of the classical model assumptions. Furthermore, we construct a robust version of Cronbach's alpha which is insensitive to a small proportion of data that belong to a different source. The idea is that the robust Cronbach's alpha reflects the reliability of the bulk of the data. For example, it should not be possible that some small amount of outliers makes a score look reliable if it is not.     

Second order optimality for estimators in time series regression models

We consider the second order asymptotic properties of an efficient frequency domain regression coefficient estimator proposed by Hannan [Regression for time series, Proc. Sympos. Time Series Analysis (Brown Univ., 1962), Wiley, New York, 1963, pp. 17-37]. This estimator is a semiparametric estimator based on nonparametric spectral estimators. We derive the second order Edgeworth expansion of the distribution of . Then it is shown that the second order asymptotic properties are independent of the bandwidth choice for residual spectral estimator, which implies that has the same rate of convergence as in regular parametric estimation. This is a sharp contrast with the general semiparametric estimation theory. We also examine the second order Gaussian efficiency of . Numerical studies are given to confirm the theoretical results.     

Survival function estimation when lifetime and censoring time are dependent

In this paper we consider a model for dependent censoring and derive a consistent asymptotically normal estimator for the underlying survival distribution from a sample of censored data. The methodology is illustrated with an application to the analysis of cancer data. Some simulations to evaluate the performance of our estimator are also presented. The results indicate that our estimator performs reasonably well in comparison to the other dependent censoring survival curve estimators.     

Functional nonparametric estimation of conditional extreme quantiles

We address the estimation of quantiles from heavy-tailed distributions when functional covariate information is available and in the case where the order of the quantile converges to one as the sample size increases. Such “extreme” quantiles can be located in the range of the data or near and even beyond the boundary of the sample, depending on the convergence rate of their order to one. Nonparametric estimators of these functional extreme quantiles are introduced, their asymptotic distributions are established and their finite sample behavior is investigated.     

Nonparametric regression function estimation with surrogate data and validation sampling

This paper develops estimation approaches for nonparametric regression analysis with surrogate data and validation sampling when response variables are measured with errors. Without assuming any error model structure between the true responses and the surrogate variables, a regression calibration kernel regression estimate is defined with the help of validation data. The proposed estimator is proved to be asymptotically normal and the convergence rate is also derived. A simulation study is conducted to compare the proposed estimators with the standard Nadaraya-Watson estimators with the true observations in the validation data set and the complete observations, respectively. The Nadaraya-Watson estimator with the complete observations can serve as a gold standard, even though it is practically unachievable because of the measurement errors.