Asymptotic properties of conditional maximum likelihood estimator in a certain exponential model

The conditional maximum likelihood estimator is suggested as an alternative to the maximum likelihood estimator and is favorable for an estimator of a dispersion parameter in the normal distribution, the inverse-Gaussian distribution, and so on. However, it is not clear whether the conditional maximum likelihood estimator is asymptotically efficient in general. Consider the case where it is asymptotically efficient and its asymptotic covariance depends only on an objective parameter in an exponential model. This remand implies that the exponential model possesses a certain parallel foliation. In this situation, this paper investigates asymptotic properties of the conditional maximum likelihood estimator and compares the conditional maximum likelihood estimator with the maximum likelihood estimator. We see that the bias of the former is more robust than that of the latter and that two estimators are very close, especially in the sense of bias-corrected version. The mean Pythagorean relation is also discussed.     

Hazard Rate Estimation in Nonparametric Regression with Censored Data

Consider a regression model in which the responses are subject to random right censoring. In this model, Beran studied the nonparametric estimation of the conditional cumulative hazard function and the corresponding cumulative distribution function. The main idea is to use smoothing in the covariates. Here we study asymptotic properties of the corresponding hazard function estimator obtained by convolution smoothing of Beran's cumulative hazard estimator. We establish asymptotic expressions for the bias and the variance of the estimator, which together with an asymptotic representation lead to a weak convergence result. Also, the uniform strong consistency of the estimator is obtained.     

Estimating and Visualizing Conditional Densities

Abstract

We consider the kernel estimator of conditional density and derive its asymptotic bias, variance, and mean-square error. Optimal bandwidths (with respect to integrated mean-square error) are found and it is shown that the convergence rate of the density estimator is order n ^–2/3. We also note that the conditional mean function obtained from the estimator is equivalent to a kernel smoother. Given the undesirable bias properties of kernel smoothers, we seek a modified conditional density estimator that has mean equivalent to some other nonparametric regression smoother with better bias properties. It is also shown that our modified estimator has smaller mean square error than the standard estimator in some commonly occurring situations. Finally, three graphical methods for visualizing conditional density estimators are discussed and applied to a data set consisting of maximum daily temperatures in Melbourne, Australia.     

Data sharpening methods in multivariate local quadratic regression

This paper is concerned with the conditional bias and variance of local quadratic regression to the multivariate predictor variables. Data sharpening methods of nonparametric regression were first proposed by Choi, Hall, Roussion. Recently, a data sharpening estimator of local linear regression was discussed by Naito and Yoshizaki. In this paper, to improve mainly the fitting precision, we extend their results on the asymptotic bias and variance. Using the data sharpening estimator of multivariate local quadratic regression, we are able to derive higher fitting precision. In particular, our approach is simple to implement, since it has an explicit form, and is convenient when analyzing the asymptotic conditional bias and variance of the estimator at the interior and boundary points of the support of the density function.     

左截断相依数据下条件分位数的双核局部线性估计

本文在左截断相依数据下,利用局部线性估计的方法,先提出了条件分布函数的双核估计;然后利用该估计导出了条件分位数的双核局部线性估计,并建立了这些估计的渐近正态性结果;最后,通过模拟显示该估计在偏移和边界点调节上要比一般的核估计更好.     

Estimation and Bootstrap with Censored Data in Fixed Design Nonparametric Regression

We study Beran's extension of the Kaplan-Meier estimator for thesituation of right censored observations at fixed covariate values. Thisestimator for the conditional distribution function at a given value of thecovariate involves smoothing with Gasser-Müller weights. We establishan almost sure asymptotic representation which provides a key tool forobtaining central limit results. To avoid complicated estimation ofasymptotic bias and variance parameters, we propose a resampling methodwhich takes the covariate information into account. An asymptoticrepresentation for the bootstrapped estimator is proved and the strongconsistency of the bootstrap approximation to the conditional distributionfunction is obtained.     

Wavelet estimation of conditional density with truncated, censored and dependent data

In this paper we define a new nonlinear wavelet-based estimator of conditional density function for a random left truncation and right censoring model. We provide an asymptotic expression for the mean integrated squared error (MISE) of the estimator. It is assumed that the lifetime observations form a stationary α-mixing sequence. Unlike for kernel estimators, the MISE expression of the wavelet-based estimators is not affected by the presence of discontinuities in the curves. Also, asymptotic normality of the estimator is established.     

Cross-validated bagged learning

Many applications aim to learn a high dimensional parameter of a data generating distribution based on a sample of independent and identically distributed observations. For example, the goal might be to estimate the conditional mean of an outcome given a list of input variables. In this prediction context, bootstrap aggregating (bagging) has been introduced as a method to reduce the variance of a given estimator at little cost to bias. Bagging involves applying an estimator to multiple bootstrap samples and averaging the result across bootstrap samples. In order to address the curse of dimensionality, a common practice has been to apply bagging to estimators which themselves use cross-validation, thereby using cross-validation within a bootstrap sample to select fine-tuning parameters trading off bias and variance of the bootstrap sample-specific candidate estimators. In this article we point out that in order to achieve the correct bias variance trade-off for the parameter of interest, one should apply the cross-validation selector externally to candidate bagged estimators indexed by these fine-tuning parameters. We use three simulations to compare the new cross-validated bagging method with bagging of cross-validated estimators and bagging of non-cross-validated estimators.     

Frontier estimation with local polynomials and high power-transformed data

We present a new method for estimating the frontier of a sample. The estimator is based on a local polynomial regression on the power-transformed data. We assume that the exponent of the transformation goes to infinity while the bandwidth goes to zero. We give conditions on these two parameters for obtaining almost complete convergence. The asymptotic conditional bias and variance of the estimator are provided and its good performance is illustrated for some finite sample situations.     

Estimating function with asymptotic bias and its estimator

We introduce the estimating function with asymptotic bias and investigate the asymptotic behavior of the estimator based on it by using their relationship. The estimator based on the estimating function with asymptotic bias has the asymptotic normality with asymptotic bias. We show that this theory has several interesting applications in practical statistics.     

A weighted estimator of conditional hazard rate with left-truncated and dependent data

Based on empirical likelihood method, we construct new weighted estimators of conditional density and conditional survival functions when the interest random variable is subject to random left-truncation; further, we define a plug-in weighted estimator of the conditional hazard rate. Under strong mixing assumptions, we derive asymptotic normality of the proposed estimators which permit to built a confidence interval for the conditional hazard rate. The finite sample behavior of the estimators is investigated via simulations too.     

A bias corrected nonparametric regression estimator

In this article, we propose a new method of bias reduction in nonparametric regression estimation. The proposed new estimator has asymptotic bias order h⁴, where h is a smoothing parameter, in contrast to the usual bias order h² for the local linear regression. In addition, the proposed estimator has the same order of the asymptotic variance as the local linear regression. Our proposed method is closely related to the bias reduction method for kernel density estimation proposed by Chung and Lindsay (2011). However, our method is not a direct extension of their density estimate, but a totally new one based on the bias cancelation result of their proof.     

Bias correction in extreme value statistics with index around zero

Applying extreme value statistics in meteorology and environmental science requires accurate estimators on extreme value indices that can be around zero. Without having prior knowledge on the sign of the extreme value indices, the probability weighted moment (PWM) estimator is a favorable candidate. As most other estimators on the extreme value index, the PWM estimator bears an asymptotic bias. In this paper, we develop a bias correction procedure for the PWM estimator. Moreover, we provide bias-corrected PWM estimators for high quantiles and, when the extreme value index is negative, the endpoint of a distribution. The choice of k, the number of high order statistics used for estimation, is crucial in applications. The asymptotically unbiased PWM estimators allows the choice of higher level k, which results in a lower asymptotic variance. Moreover, since the bias-corrected PWM estimators can be applied for a wider range of k compared to the original PWM estimator, one gets more flexibility in choosing k for finite sample applications. All advantages become apparent in simulations and an environmental application on estimating “once per 10,000 years” still water level at Hoek van Holland, The Netherlands.     

Un test d'adéquation global pour la fonction de répartition conditionnelle

We propose a global test of goodness-of-fit to assess the validity of an entertained statistical model by testing simultaneously all the assumptions made about it. This test is based on a local polynomial estimator of the conditional distribution function and on the standard paradigm relating the distance between the nonparametric estimator and the theoretical parametric model. We derive the asymptotic distribution of the resulting test statistic under both the null hypothesis and local alternatives. To cite this article: S. Ferrigno, G.R. Ducharme, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Location estimation in nonparametric regression with censored data

Consider the heteroscedastic model Y=m(X)+σ(X)?, where ? and X are independent, Y is subject to right censoring, m(·) is an unknown but smooth location function (like e.g. conditional mean, median, trimmed mean…) and σ(·) an unknown but smooth scale function. In this paper we consider the estimation of m(·) under this model. The estimator we propose is a Nadaraya-Watson type estimator, for which the censored observations are replaced by ‘synthetic’ data points estimated under the above model. The estimator offers an alternative for the completely nonparametric estimator of m(·), which cannot be estimated consistently in a completely nonparametric way, whenever high quantiles of the conditional distribution of Y given X=x are involved.We obtain the asymptotic properties of the proposed estimator of m(x) and study its finite sample behaviour in a simulation study. The method is also applied to a study of quasars in astronomy.     

A new projection estimate for multivariate location with minimax bias

The maximum asymptotic bias of an estimator is a global robustness measure of its performance. The projection median estimator for multivariate location shows a remarkable behavior regarding asymptotic bias. In this paper we consider a modification of the projection median estimator which renders an estimate with better bias performance for point mass contaminations (the worst situation for the projection median estimator). Moreover, it achieves the lowest bound for an equivariant estimate for point mass contaminations.     

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In this paper, we study the asymptotic behavior of the B-spline estimator for semiparametric panel data model with fixed effects.We give explicit expression for the asymptotic bias of B-spline estimator for nonparametric function m. Our study shows that the asymptotic bias of the B-spline estimator does not depend on the working correlation matrix. Simulations are conducted to demonstrate our conclusion.     

Fixed Design Nonparametric Regression with Truncated and Censored Data

In this paper we consider a fixed design model in which the observations axe subject to left truncation and right censoring. A generalized product-limit estimator for the conditional distribution at a given covaxiate value is proposed, and an almost sure asymptotic representation of this estimator is established. We also obtain the rate of uniform consistency, weak convergence and a modulus of continuity for this estimator.Applications include trimmed mean and quantile function estimators.     

Functional time series prediction via conditional mode estimation

This Note focuses on an estimator of the conditional mode of a scalar response Y given a functional random variable X. We start by building a kernel estimator of the conditional density of Y given X; the conditional mode is defined as the value which maximizes this conditional density. We establish the almost complete convergence for this estimate under α-mixing assumption. To cite this article: F. Ferraty et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Penalized empirical likelihood estimation of semiparametric models

We propose an empirical likelihood-based estimation method for conditional estimating equations containing unknown functions, which can be applied for various semiparametric models. The proposed method is based on the methods of conditional empirical likelihood and penalization. Thus, our estimator is called the penalized empirical likelihood (PEL) estimator. For the whole parameter including infinite-dimensional unknown functions, we derive the consistency and a convergence rate of the PEL estimator. Furthermore, for the finite-dimensional parametric component, we show the asymptotic normality and efficiency of the PEL estimator. We illustrate the theory by three examples. Simulation results show reasonable finite sample properties of our estimator.