Nonparametric estimators of the bivariate survival function under random censoring

A large number of proposals for estimating the bivariate survival function under random censoring have been made. In this paper we discuss the most prominent estimators, where prominent is meant in the sense that they are best for practical use; Dabrowska's estimator, the Prentice–Cai estimator, Pruitt's modified EM-estimator, and the reduced data NPMLE of van der Laan. We show how these estimators are computed and present their intuitive background. The asymptotic results are summarized. Furthermore, we give a summary of the practical performance of the estimators under different levels of dependence and censoring based on extensive simulation results. This leads also to a practical advise.     

Estimation of a survival curve with randomly censored data in the presence of a covariate

This paper deals with the estimation of a survival curve in models with random right censoring and dependent censoring mechanism. We consider a specific dependent censorship model in which conditional on a covariate, the survival and censoring times are assumed to be independent. We investigate asymptotic properties of a corrected version of a survival curve estimator introduced by Cheng (1989). In particular we show uniform strong consistency and weak convergence to a Gaussian process. Comparisons of this estimator with the well-known Kaplan-Meier-estimator are included. Finally, some examples illustrate how the estimator performs. Received: February 2000     

Strong representation of the presmoothed quantile function estimator for censored data

We consider lifetime data subject to right random censorship. In this context, this paper deals with the topic of estimating the distribution function of the lifetime and the corresponding quantile function. As it has been shown that the classical Kaplan–Meier estimator of the distribution function can be improved by means of presmoothing ideas, we introduce a quantile function estimator via the presmoothed distribution function estimator studied by Cao et al. [Journal of Nonparametric statistics, Vol. 17 (2005) pp. 31–56.] The main result of this paper is an almost sure representation of this presmoothed estimator. As a consequence, its strong consistency and asymptotic normality are established. The performance of this new quantile estimator is analyzed in a simulation study and applied to a real data example.     

A smoothed maximum score estimator for the binary choice panel data model with an application to labour force participation

In a binary choice panel data model with individual effects and two time periods, Manski proposed the maximum score estimator based on a discontinuous objective function and proved its consistency under weak distributional assumptions. The rate of convergence is low ( N ^1/3) and its limit distribution cannot easily be used for statistical inference. In this paper we apply the idea of Horowitz to smooth Manski's objective function. The resulting smoothed maximum score estimator is consistent and asymptotically normal with a rate of convergence that can be made arbitrarily close to N ^1/2, depending on the strength of the smoothness assumptions imposed. The estimator can be applied to panels with more than two time periods and to unbalanced panels. We apply the estimator to analyze labour force participation of married Dutch females.     

A continuous mapping theorem for the argmax-functional in the non-unique case

The Argmax-Continuous Mapping Theorem (Argmax-CMT) of K im and P ollard resp. van der Vaart and Wellner has been proved to be a very useful tool in statistics for deriving distributional convergence of M-estimators. However it only works as long as the limit process possesses an almost sure unique maximizing point. In this article we prove an extension of the Argmax-CMT where almost sure uniqueness is no longer needed. Moreover our Argmax-CMT is also valid in the function space D ( R ) equipped with L indvall's version of the Skorokhod-topology. As an example the result is applied to change-point estimators.     

Rates of almost sure convergence of plug-in estimates for distortion risk measures

In this article, we consider plug-in estimates for distortion risk measures as for instance the Value-at-Risk, the Expected Shortfall or the Wang transform. We allow for fairly general estimates of the underlying unknown distribution function (beyond the classical empirical distribution function) to be plugged in the risk measure. We establish strong consistency of the estimates, we investigate the rate of almost sure convergence, and we study the small sample behavior by means of simulations.     

Convergence problems in stochastic programming models with probabilistic constraints

We consider a stochastic programming model with probabilistic constraints and assume to have only empirical estimates of the true probability distribution of the random variables involved in the model. By using properties of semicontinuous and quasi-convex/quasi-concave limetions, we can weaken the traditional assumptions on the functions involved. We generalize both the functional form and the mathematical properties of the functions and then prove the almost sure uniform convergence of the sequence of approximate problems to the true one.     

Strong uniform convergence for the estimator of the regression function under ϕ-mixing conditions

Suppose the observations (X _i,Y _i), i=1,…, n, are ϕ-mixing. The strong uniform convergence and convergence rate for the estimator of the regression function was studied by serveral authors, e.g. G. Collomb (1984), L. Gy?rfi et al. (1989). But the optimal convergence rates are not reached unless the Y _i are bounded or the E exp (a|Y _i|) are bounded for some a>0. Compared with the i.i.d. case the convergence of the Nadaraya-Watson estimator under ϕ-mixing variables needs strong moment conditions. In this paper we study the strong uniform convergence and convergence rate for the improved kernel estimator of the regression function which has been suggested by Cheng P. (1983). Compared with Theorem A in Y. P. Mack and B. Silverman (1982) or Theorem 3.3.1 in L. Gy?rfi et al. (1989), we prove the convergence for this kind of estimators under weaker moment conditions. The optimal convergence rate for the improved kernel estimator is attained under almost the same conditions of Theorem 3.3.2 in L. Gy?rfi et al. (1989). Received: September 1999     

Characterizations and Modelling of Multivariate Lack of Memory Property

In this article we establish characterizations of multivariate lack of memory property in terms of the hazard gradient (whenever exists), the survival function and the cumulative hazard function. Based on one of these characterizations we establish a method of generating bivariate lifetime distributions possessing bivariate lack of memory property (BLMP) with specified marginals. It is observed that the marginal distributions have to satisfy certain conditions to be stated. The method generates absolutely continuous bivariate distributions as well as those containing a singular component. Bivariate exponential distributions due to Proschan and Sullo (Reliability and biometry, pp 423–440, 1974), Freund (in J Am Stat Assoc 56:971–977, 1961), Block and Basu (J Am Stat Assoc 89:1091–1097, 1974) and Marshall and Olkin (J Am Math Assoc 62:30–44, 1967) are generated as particular cases among others using the proposed method. Some other distributions generated using the method may be of practical importance. Shock models leading to bivariate distributions possessing BLMP are given. Some closure properties of a class of univariate failure rate functions that can generate distributions possessing BLMP and of the class of bivariate survival functions having BLMP are studied.     

Kernel spatial density estimation in infinite dimension space

In this paper, we propose a nonparametric method to estimate the spatial density of a functional stationary random field. This latter is with values in some infinite dimensional normed space and admitted a density with respect to some reference measure. We study both the weak and strong consistencies of the considered estimator and also give some rates of convergence. Special attention is paid to the links between the probabilities of small balls and the rates of convergence of the estimator. The practical use and the behavior of the estimator are illustrated through some simulations and a real data application.     

Semiparametric estimation of a bivariate Tobit model

The existing semiparametric estimation literature has mainly focused on univariate Tobit models and no semiparametric estimation has been considered for bivariate Tobit models. In this paper, we consider semiparametric estimation of the bivariate Tobit model proposed by Amemiya (1974), under the independence condition without imposing any parametric restriction on the error distribution. Our estimator is shown to be consistent and asymptotically normal, and simulation results show that our estimator performs well in finite samples. It is also worth noting that while Amemiya’s (1974) instrumental variables estimator (IV) requires the normality assumption, our semiparametric estimator actually outperforms his IV estimator even when normality holds. Our approach can be extended to higher dimensional multivariate Tobit models.     

Functional-coefficient models for nonstationary time series data

This paper studies functional coefficient regression models with nonstationary time series data, allowing also for stationary covariates. A local linear fitting scheme is developed to estimate the coefficient functions. The asymptotic distributions of the estimators are obtained, showing different convergence rates for the stationary and nonstationary covariates. A two-stage approach is proposed to achieve estimation optimality in the sense of minimizing the asymptotic mean squared error. When the coefficient function is a function of a nonstationary variable, the new findings are that the asymptotic bias of its nonparametric estimator is the same as the stationary covariate case but convergence rate differs, and further, the asymptotic distribution is a mixed normal, associated with the local time of a standard Brownian motion. The asymptotic behavior at boundaries is also investigated.     

Nonparametric identification and estimation of current status data in the presence of death

We present a nonparametric study of current status data in the presence of death. Such data arise from biomedical investigations in which patients are examined for the onset of a certain disease, for example, tumor progression, but may die before the examination. A key difference between such studies on human subjects and the survival–sacrifice model in animal carcinogenicity experiments is that, due to ethical and perhaps technical reasons, deceased human subjects are not examined, so that the information on their disease status is lost. We show that, for current status data with death, only the overall and disease‐free survival functions can be identified, whereas the cumulative incidence of the disease is not identifiable. We describe a fast and stable algorithm to estimate the disease‐free survival function by maximizing a pseudo‐likelihood with plug‐in estimates for the overall survival rates. It is then proved that the global rate of convergence for the nonparametric maximum pseudo‐likelihood estimator is equal to O_p(n^?1/3) or the convergence rate of the estimated overall survival function, whichever is slower. Simulation studies show that the nonparametric maximum pseudo‐likelihood estimators are fairly accurate in small‐ to medium‐sized samples. Real data from breast cancer studies are analyzed as an illustration.     

Uniform minimum variance unbiased estimators for bivariate families

Summary We consider uniform minimum variance unbiased (UMVU) estimation of an unbiased estimable function of distribution parameters for bivariate truncation (non-regular) parameter families. In particular, we derive the UMVU estimator of the probability thatY is less thanX.     

Estimating the conditional mode of a stationary stochastic process from noisy observations

Let {(X _i, Y _i,)}, i≥1, be a strictly stationary process from noisy observations. We examine the effect of the noise in the response Y and the covariates X on the nonparametric estimation of the conditional mode function. To estimate this function we are using deconvoluting kernel estimators. The asymptotic behavior of these estimators depends on the smoothness of the noise distribution, which is classified as either ordinary smooth or super smooth. Uniform convergence with almost sure convergence rates is established for strongly mixing stochastic processes, when the noise distribution is ordinary smooth. Received: April 1998     

Estimating a generalized correlation coefficient for a generalized bivariate probit model

In this paper we consider semiparametric estimation of a generalized correlation coefficient in a generalized bivariate probit model. The generalized correlation coefficient provides a simple summary statistic measuring the relationship between the two binary decision processes in a general framework. Our semiparametric estimation procedure consists of two steps, combining semiparametric estimators for univariate binary choice models with the method of maximum likelihood for the bivariate probit model with nonparametrically generated regressors. The estimator is shown to be consistent and asymptotically normal. The estimator performs well in our simulation study.     

Functional partially linear quantile regression model

This paper considers estimation of a functional partially quantile regression model whose parameters include the infinite dimensional function as well as the slope parameters. We show asymptotical normality of the estimator of the finite dimensional parameter, and derive the rate of convergence of the estimator of the infinite dimensional slope function. In addition, we show the rate of the mean squared prediction error for the proposed estimator. A simulation study is provided to illustrate the numerical performance of the resulting estimators.     

Concentration behavior of the penalized least squares estimator

Consider the standard nonparametric regression model and take as estimator the penalized least squares function. In this article, we study the trade‐off between closeness to the true function and complexity penalization of the estimator, where complexity is described by a seminorm on a class of functions. First, we present an exponential concentration inequality revealing the concentration behavior of the trade‐off of the penalized least squares estimator around a nonrandom quantity, where such quantity depends on the problem under consideration. Then, under some conditions and for the proper choice of the tuning parameter, we obtain bounds for this nonrandom quantity. We illustrate our results with some examples that include the smoothing splines estimator.     

Nonparametric fixed effects model for panel data with locally stationary regressors

We develop methods for inference in nonparametric time-varying fixed effects panel data models that allow for locally stationary regressors and for the time series length $T$ and cross-section size $N$ both being large. We first develop a pooled nonparametric profile least squares dummy variable approach to estimate the nonparametric function, and establish the optimal convergence rate and asymptotic normality of the resultant estimator. We then propose a test statistic to check whether the bivariate nonparametric function is time-varying or the time effect is separable, and derive the asymptotic distribution of the proposed test statistic. We present several simulated examples and two real data analyses to illustrate the finite sample performance of the proposed methods.     

Large sample estimation and testing procedures for dynamic equation systems

Joint two-step estimation procedures which have the same asymptotic properties as the maximum likelihood (ML) estimator are developed for the final equation, transfer function and structural form of a multivariate dynamic model with normally distributed vector-moving average errors. The ML estimator under fixed and known initial values is obtained by iterating the procedure until convergence. The asymptotic distribution of the two-step estimators is used to construct large sample testing procedures for the different forms of the model.