Nonparametric estimation of the conditional distribution function in a semiparametric censorship model

In this paper we propose a new nonparametric estimator of the conditional distribution function under a semiparametric censorship model. We establish an asymptotic representation of the estimator as a sum of iid random variables, balanced by some kernel weights. This representation is used for obtaining large sample results such as the rate of uniform convergence of the estimator, or its limit distributional law. We prove that the new estimator outperforms the conditional Kaplan–Meier estimator for censored data, in the sense that it exhibits lower asymptotic variance. Illustration through real data analysis is provided.     

Conditional inference in finite population sampling under a calibration model

Several estimators, including the classical and the regression estimators of finite population mean, are compared, both theoretically and empirically, under a calibration model, where the dependent variable(y), and not the independent variable(x), can be observed for all units of the finite population. It is shown asymptotically that when conditioned on x, the bias of the classical estimator may be much smaller than that of the regression estimators; whereas when conditioned on y, the regression estimator may have much smaller conditional bias than the classical estimator. Since all the y's(not x's) can be observed, it seems appropriate to make comparison under the conditional distribution of each estimator with y fixed. In this case, the regression estimator has smaller variance, smaller conditional bias, and the conditional coverage probability closer to its nominal level     

Berry–Esseen type bound of conditional mode estimation under truncation and strong mixing assumptions

ABSTRACT

In order to investigate the convergence rate of the asymptotic normality for the estimator of the conditional mode function for the left-truncation model, we derive a Berry–Esseen type bound of the estimator when the lifetime observations with multivariate covariates form a stationary α-mixing sequence. The finite sample performance of the estimator of the conditional mode function is explored through simulations.     

A conditionally-unbiased estimator of population size based on plant-capture in continuous time

This paper proposes an estimator of the unknown size of a target population to which has been added a planted population of known size. The augmented population is observed for a fixed time and individuals are sighted according to independent Poisson processes. These processes may be time-inhomogeneous, but, within each population, the intensity function is the same for all individuals. When the two populations have the same intensity function, known results on factorial series distributions suggest that the proposed estimator is approximately unbiased and provide a useful estimator of standard deviation. Except for short sampling times, computational results confirm that the proposed population-size estimator is nearly unbiased, and indicate that it gives a better overall performance than existing estimators in the literature. The robustness of this performance is investigated in situations in which it cannot be assumed that the behaviour of the plants matches that of individuals from the target population.     

Conditional estimation using prior information in 2‐stage group sequential designs assuming asymptotic normality when the trial terminated early

Two‐stage designs are widely used to determine whether a clinical trial should be terminated early. In such trials, a maximum likelihood estimate is often adopted to describe the difference in efficacy between the experimental and reference treatments; however, this method is known to display conditional bias. To reduce such bias, a conditional mean‐adjusted estimator (CMAE) has been proposed, although the remaining bias may be nonnegligible when a trial is stopped for efficacy at the interim analysis. We propose a new estimator for adjusting the conditional bias of the treatment effect by extending the idea of the CMAE. This estimator is calculated by weighting the maximum likelihood estimate obtained at the interim analysis and the effect size prespecified when calculating the sample size. We evaluate the performance of the proposed estimator through analytical and simulation studies in various settings in which a trial is stopped for efficacy or futility at the interim analysis. We find that the conditional bias of the proposed estimator is smaller than that of the CMAE when the information time at the interim analysis is small. In addition, the mean‐squared error of the proposed estimator is also smaller than that of the CMAE. In conclusion, we recommend the use of the proposed estimator for trials that are terminated early for efficacy or futility.     

Simple non-parametric estimators for unemployment duration analysis

Summary.  We consider an extension of conventional univariate Kaplan–Meier-type estimators for the hazard rate and the survivor function to multivariate censored data with a censored random regressor. It is an Akritas-type estimator which adapts the non-parametric conditional hazard rate estimator of Beran to more typical data situations in applied analysis. We show with simulations that the estimator has nice finite sample properties and our implementation appears to be fast. As an application we estimate non-parametric conditional quantile functions with German administrative unemployment duration data.     

Weak convergence for the conditional distribution function in a Koziol–Green model under dependent censoring

In this paper we consider the conditional Koziol–Green model of Braekers and Veraverbeke [2008. A conditional Koziol–Green model under dependent censoring. Statist. Probab. Lett., accepted for publication] in which they generalized the Koziol–Green model of Veraverbeke and Cadarso Suárez [2000. Estimation of the conditional distribution in a conditional Koziol–Green model. Test 9, 97–122] by assuming that the association between a censoring time and a time until an event is described by a known Archimedean copula function. They got in this way, an informative censoring model with two different types of informative censoring. Braekers and Veraverbeke [2008. A conditional Koziol–Green model under dependent censoring. Statist. Probab. Lett., accepted for publication] derived in this model a non-parametric Koziol–Green estimator for the conditional distribution function of the time until an event, for which they showed the uniform consistency and the asymptotic normality. In this paper we extend their results and prove the weak convergence of the process associated to this estimator. Furthermore we show that the conditional Koziol–Green estimator is asymptotically more efficient in this model than the general copula-graphic estimator of Braekers and Veraverbeke [2005. A copula-graphic estimator for the conditional survival function under dependent censoring. Canad. J. Statist. 33, 429–447]. As last result, we construct an asymptotic confidence band for the conditional Koziol–Green estimator. Through a simulation study, we investigate the small sample properties of this asymptotic confidence band. Afterwards we apply this estimator and its confidence band on a practical data set.     

Bernstein conditional density estimation with application to conditional distribution and regression functions

In this paper we propose a smooth nonparametric estimation for the conditional probability density function based on a Bernstein polynomial representation. Our estimator can be written as a finite mixture of beta densities with data-driven weights. Using the Bernstein estimator of the conditional density function, we derive new estimators for the distribution function and conditional mean. We establish the asymptotic properties of the proposed estimators, by proving their asymptotic normality and by providing their asymptotic bias and variance. Simulation results suggest that the proposed estimators can outperform the Nadaraya–Watson estimator and, in some specific setups, the local linear kernel estimators. Finally, we use our estimators for modeling the income in Italy, conditional on year from 1951 to 1998, and have another look at the well known Old Faithful Geyser data.     

Setting confidence intervals by ratio estimator

For the survey population total of a variable y when values of an auxiliary variable x are available a popular procedure is to employ the ratio estimator on drawing a simple random sample without replacement (SRSWOR) especially when the size of the sample is large. To set up a confidence interval for the total, various variance estimators are available to pair with the ratio estimator. We add a few more variance estimators studded with asymptotic design-cum-model properties. The ratio estimator is traditionally known to be appropriate when the regression of y on x is linear through the origin and the conditional variance of y given x is proportional to x. But through a numerical exercise by simulation we find the confidence intervals to fare better if the regression line deviates from the origin or if the conditional variance is disproportionate with x. Also, comparing the confidence intervals using alternative variance estimators we find our newly proposed variance estimators to yield favourably competitive results.     

Adaptive estimators for parameters of the autoregression function of a Markov chain

Suppose we observe an ergodic Markov chain on the real line, with a parametric model for the autoregression function, i.e. the conditional mean of the transition distribution. If one specifies, in addition, a parametric model for the conditional variance, one can define a simple estimator for the parameter, the maximum quasi-likelihood estimator. It is robust against misspecification of the conditional variance, but not efficient. We construct an estimator which is adaptive in the sense that it is efficient if the conditional variance is misspecified, and asymptotically as good as the maximum quasi-likelihood estimator if the conditional variance is correctly specified. The adaptive estimator is a weighted nonlinear least-squares estimator, with weights given by predictors for the conditional variance.     

Estimation Under Inequality Constraints: Semiparametric Estimation of Conditional Duration Models

This article proposes a semiparametric estimator of the parameter in a conditional duration model when there are inequality constraints on some parameters and the error distribution may be unknown. We propose to estimate the parameter by a constrained version of an unrestricted semiparametrically efficient estimator. The main requirement for applying this method is that the initial unrestricted estimator converges in distribution. Apart from this, additional regularity conditions on the data generating process or the likelihood function, are not required. Hence the method is applicable to a broad range of models where the parameter space is constrained by inequality constraints, such as the conditional duration models. In a simulation study involving conditional duration models, the overall performance of the constrained estimator was better than its competitors, in terms of mean squared error. A data example is used to illustrate the method.     

Evaluating dynamic treatment strategies: does it have to be more costly?

Dynamic treatment strategies are designed to change treatments over time in response to intermediate outcomes. They can be deployed for primary treatment as well as for the introduction of adjuvant treatment or other treatment‐enhancing interventions. When treatment interventions are delayed until needed, more cost‐efficient strategies will result. Sequential multiple assignment randomized (SMAR) trials allow for unbiased estimation of the marginal effects of different sequences of history‐dependent treatment decisions. Because a single SMAR trial enables evaluation of many different dynamic regimes at once, it is naturally thought to require larger sample sizes than the parallel randomized trial. In this paper, we compare power between SMAR trials studying a regime, where treatment boosting enters when triggered by an observed event, versus the parallel design, where a treatment boost is consistently prescribed over the entire study period. In some settings, we found that the dynamic design yields the more efficient trial for the detection of treatment activity. We develop one particular trial to compare a dynamic nursing intervention with telemonitoring for the enhancement of medication adherence in epilepsy patients. To this end, we derive from the SMAR trial data either an average of conditional treatment effects (‘conditional estimator’) or the population‐averaged (‘marginal’) estimator of the dynamic regimes. Analytical sample size calculations for the parallel design and the conditional estimator are compared with simulated results for the population‐averaged estimator. We conclude that in specific settings, well‐chosen SMAR designs may require fewer data for the development of more cost‐efficient treatment strategies than parallel designs. Copyright © 2012 John Wiley & Sons, Ltd.     

A Profile Conditional Likelihood Approach for the Semiparametric Transformation Regression Model with Missing Covariates

We propose a profile conditional likelihood approach to handle missing covariates in the general semiparametric transformation regression model. The method estimates the marginal survival function by the Kaplan-Meier estimator, and then estimates the parameters of the survival model and the covariate distribution from a conditional likelihood, substituting the Kaplan-Meier estimator for the marginal survival function in the conditional likelihood. This method is simpler than full maximum likelihood approaches, and yields consistent and asymptotically normally distributed estimator of the regression parameter when censoring is independent of the covariates. The estimator demonstrates very high relative efficiency in simulations. When compared with complete-case analysis, the proposed estimator can be more efficient when the missing data are missing completely at random and can correct bias when the missing data are missing at random. The potential application of the proposed method to the generalized probit model with missing continuous covariates is also outlined.     

Correcting for Population Stratification in Genomewide Association Studies

Genomewide association studies have become the primary tool for discovering the genetic basis of complex human diseases. Such studies are susceptible to the confounding effects of population stratification, in that the combination of allele-frequency heterogeneity with disease-risk heterogeneity among different ancestral subpopulations can induce spurious associations between genetic variants and disease. This article provides a statistically rigorous and computationally feasible solution to this challenging problem of unmeasured confounders. We show that the odds ratio of disease with a genetic variant is identifiable if and only if the genotype is independent of the unknown population substructure conditional on a set of observed ancestry-informative markers in the disease-free population. Under this condition, the odds ratio of interest can be estimated by fitting a semiparametric logistic regression model with an arbitrary function of a propensity score relating the genotype probability to ancestry-informative markers. Approximating the unknown function of the propensity score by B-splines, we derive a consistent and asymptotically normal estimator for the odds ratio of interest with a consistent variance estimator. Simulation studies demonstrate that the proposed inference procedures perform well in realistic settings. An application to the well-known Wellcome Trust Case-Control Study is presented. Supplemental materials are available online.     

Analysis of transformation models with right-truncated data

In many applications, statistical data are frequently observed subject to a retrospective sampling criterion resulting in right-truncated data. In this article, a general class of semiparametric transformation models that include proportional hazards model and proportional odds model as special cases is studied for the analysis of right-truncated data. We proposed two estimators for regression coefficients. The first estimator is based on martingale estimating equations. The second estimator is based on the conditional likelihood function given the truncation times. The asymptotic properties of both estimators are derived. The finite sample performance is examined through a simulation study.     

Asymptotic normality of a conditional hazard function estimate in the single index for quasi-associated data

Abstract

The main goal of this paper is to study the estimation of the conditional hazard function of a scalar response variable Y given a hilbertian random variable X in functional single-index model. We construct an estimator of this nonparametric function and we study its asymptotic properties, under quasi-associated structure. Precisely, we establish the asymptotic normality of the constructed estimator. We carried out simulation experiments to examine the behavior of this asymptotic property over finite sample data.     

Averaging estimation for conditional covariance models

Estimating conditional covariance matrices is important in statistics and finance. In this paper, we propose an averaging estimator for the conditional covariance, which combines the estimates of marginal conditional covariance matrices by Model Averaging MArginal Regression of Li, Linton, and Lu. This estimator avoids the “curse of dimensionality” problem that the local constant estimator of Yin et al. suffered from. We establish the asymptotic properties of the averaging weights and that of the proposed conditional covariance estimator. The finite sample performances are augmented by simulation. An application to portfolio allocation illustrates the practical superiority of the averaging estimator.     

Extreme Quantile Estimation for Autoregressive Models

ABSTRACT

A quantile autoregresive model is a useful extension of classical autoregresive models as it can capture the influences of conditioning variables on the location, scale, and shape of the response distribution. However, at the extreme tails, standard quantile autoregression estimator is often unstable due to data sparsity. In this article, assuming quantile autoregresive models, we develop a new estimator for extreme conditional quantiles of time series data based on extreme value theory. We build the connection between the second-order conditions for the autoregression coefficients and for the conditional quantile functions, and establish the asymptotic properties of the proposed estimator. The finite sample performance of the proposed method is illustrated through a simulation study and the analysis of U.S. retail gasoline price.     

Bias reduction of the maximum-likelihood estimator for a conditional Gaussian MA(1) model

In this paper, we consider an estimation for the unknown parameters of a conditional Gaussian MA(1) model. In the majority of cases, a maximum-likelihood estimator is chosen because the estimator is consistent. However, for small sample sizes the error is large, because the estimator has a bias of O(n^{? 1}). Therefore, we provide a bias of O(n^{? 1}) for the maximum-likelihood estimator for the conditional Gaussian MA(1) model. Moreover, we propose new estimators for the unknown parameters of the conditional Gaussian MA(1) model based on the bias of O(n^{? 1}). We investigate the properties of the bias, as well as the asymptotical variance of the maximum-likelihood estimators for the unknown parameters, by performing some simulations. Finally, we demonstrate the validity of the new estimators through this simulation study.     

A Generalization of Turnbull’s Estimator for Interval-Censored and Doubly Truncated Data

Nonparametric estimates of the conditional distribution of a response variable given a covariate are important for data exploration purposes. In this article, we propose a nonparametric estimator of the conditional distribution function in the case where the response variable is subject to interval censoring and double truncation. Using the approach of Dehghan and Duchesne (2011), the proposed method consists in adding weights that depend on the covariate value in the self-consistency equation of Turnbull (1976), which results in a nonparametric estimator. We demonstrate by simulation that the estimator, bootstrap variance estimation and bandwidth selection all perform well in finite samples.