Local robust and asymptotically unbiased estimation of conditional Pareto-type tails

We introduce a non-parametric robust and asymptotically unbiased estimator for the tail index of a conditional Pareto-type response distribution in presence of random covariates. The estimator is obtained from local fits of the extended Pareto distribution to the relative excesses over a high threshold using an adjusted minimum density power divergence estimation technique. We derive the asymptotic properties of the proposed estimator under some mild regularity conditions, and also investigate its finite sample performance with a small simulation experiment. The practical applicability of the methodology is illustrated on a dataset of calcium content measurements of soil samples.     

Local polynomial estimation of a conditional mean function with dependent truncated data

By applying local polynomial regression, we propose an estimator of a conditional mean function and its derivatives under a left truncation model. The target function includes the regression function, the conditional moment as well as the conditional distribution function as special cases. It is assumed that the observations form a stationary α-mixing sequence. Asymptotic normality of the estimator is established. The finite sample behavior of the estimator is investigated via simulations, too.     

Bernstein estimation for a copula derivative with application to conditional distribution and regression functionals

Bernstein estimators attracted considerable attention as smooth nonparametric estimators for distribution functions, densities, copulas and copula densities. The present paper adds a parallel result for the first-order derivative of a copula function. This result then leads to Bernstein estimators for a conditional distribution function and its important functionals such as the regression and quantile functions. Results of independent interest have been derived such as an almost sure oscillation behavior of the empirical copula process and a Bahadur-type almost sure asymptotic representation for the Bernstein estimator of a regression quantile function. Simulations demonstrate the good performance of the proposed estimators.     

Asymptotic properties of conditional distribution estimator with truncated, censored and dependent data

In this paper we study the strong and weak convergence with rates for the estimators of the conditional distribution function as well as conditional cumulative hazard rate function for a left truncated and right censored model. It is assumed that the lifetime observations with multivariate covariates form a stationary ??-mixing sequence. Also, the almost sure representations and asymptotic normality of the estimators are established. The finite sample performance of the estimators is investigated via simulations.     

Weiss-Hill estimator

In this paper the asymptotic distributional behaviour is derived for a new estimator for the extreme value index γ of distribution, which is a combination of two estimators proposed by Weiss and Hill (Weiss, 1971, and Hill, 1975). For |γ|>1/2, the estimator outperforms the Moment estimator (Dekkers and al., 1989). in the sense that it has a smaller asymptotic variance than the latter; moreover, for γ>1/2 (γ<0, resp.) the estimator behaves asymptoticaly like the Hillresp. Weiss—estimator; for |γ|<1/2 the estimator does not achieve the same rate of convergence as the Moment estimator. Simulation results concerning the comparison of the mentioned estimators are also presented. This research project was partially supported by FCT/PRAXIS XXI/FEDER and POCTI.     

Bayes‐conditional Control Charts for Weibull Quantiles under Type II Censoring

In this paper, we develop a Bayesian approach for monitoring Weibull quantiles under Type II censoring when prior information is negligible relative to the data. The posterior median of quantiles is considered as the monitored statistic. A method based on the relationship between Bayesian and conditional limits under an appropriate prior distribution is proposed to obtain the posterior median of quantiles in closed form. A pivotal quantity based on the monitored statistic is proposed, and its distribution is conditionally derived. Then, the Bayes‐conditional control limits are proposed. For the proposed charts, the probability of out‐of‐control can be derived without use of simulation. The performance of the Bayes‐conditional charts is compared with the bootstrap charts through the simulation methods. The results show that to monitor the first quantiles, the lower‐sided Bayes‐conditional charts perform better than bootstrap charts in detecting a downward shift caused by decreasing in the shape parameter. Finally, an illustrative example is provided. Copyright © 2016 John Wiley & Sons, Ltd.     

Serial independence tests for innovations of conditional mean and variance models

In this paper, one studies the asymptotic behavior of empirical processes based on consecutive residuals of univariate conditional mean and variance models. These processes are then used to develop tests of serial independence of the innovations. Even if the limiting distributions of the empirical processes depend on unknown parameters, it is shown that a Monte Carlo method based on the so-called multipliers can be applied to estimate the P values of the proposed test statistics. A simulation study is carried out to demonstrate the effectiveness of the proposed tests and the behavior of the statistics is also studied under contiguous alternatives.     

Some Complete and Censored Sampling Results for the Weibull or Extreme-Value Distribution

A simple, unbiased estimator, based on a censored sample, has been proposed by Rain [1] for the scale parameter of the Extreme-value distribution. This estimator was shown to have high efficiency and to be approximately distributed as a chi-square variable if substantial censoring occurs. Further small sample and asymptotic properties of this estimator are considered in this paper. The estimator is modified so that it is more applicable to the complete sample case and a close chi-square approximation is established for all cases. The estimator is also shown to be related to the maximum likelihood estimator.     

Process capability indices for Weibull distributions and upper specification limits

We consider a previously proposed class of capability indices that are useful when the quality characteristic of interest has a skewed, zero‐bound distribution with a long tail towards large values and there is an upper specification with a pre‐specified target value, T=0. We investigate this class of process capability indices when the underlying distribution is a Weibull distribution and focus on the situation when the Weibull distribution is highly skewed. We propose an estimator of the index in the studied class, based on the maximum likelihood estimators of the parameters in the Weibull distribution, and derive an asymptotic distribution for this estimator. Furthermore, we suggest a decision rule based on the estimated index and its asymptotic distribution and present a power comparison between the proposed estimator and a previously studied estimator. A simulation study is also performed to investigate the true significance level when the sample size is small or moderate. An example from a Swedish industry is presented. Copyright © 2008 John Wiley & Sons, Ltd.     

Weighted conditional least square estimators for bisexual branching processes with immigration

The aim of this paper is to study some inferential problems arising from the class of bisexual Galton–Watson branching processes with immigration of females and males (BGWPI). Immigrants are assumed to be unobservable, and it is only possible to sample the number of females, males, and couples (mating units) in each generation. Under these conditions, weighted conditional least square estimators are proposed for the offspring and immigration mean vectors. Asymptotic properties of such estimators are investigated when the process is subcritical and supercritical, paying especial attention to their strong consistency and limit distributions. Weighted conditional least square estimators are also developed for the offspring and immigration variance vectors, and their asymptotic properties are studied. Some comments on the critical case are also given to possibly provide a unified estimation theory for BGWPI.     

Reweighted least trimmed squares: an alternative to one-step estimators

A new class of robust regression estimators is proposed that forms an alternative to traditional robust one-step estimators and that achieves the $\sqrt{n}$ rate of convergence irrespective of the initial estimator under a wide range of distributional assumptions. The proposed reweighted least trimmed squares (RLTS) estimator employs data-dependent weights determined from an initial robust fit. Just like many existing one- and two-step robust methods, the RLTS estimator preserves robust properties of the initial robust estimate. However contrary to existing methods, the first-order asymptotic behavior of RLTS is independent of the initial estimate even if errors exhibit heteroscedasticity, asymmetry, or serial correlation. Moreover, we derive the asymptotic distribution of RLTS and show that it is asymptotically efficient for normally distributed errors. A simulation study documents benefits of these theoretical properties in finite samples.     

A method of evaluating zero conditional Lyapunov exponent from time series

A new method of evaluating zero conditional Lyapunov exponent is proposed that is valid for dynamic systems exhibiting periodic dynamics, in the presence of noise, and for coupled chaotic systems. The proposed method is based on analysis of the time series (phase difference versus time) of a system under consideration and allows a zero conditional Lyapunov exponent in a supercritical region of the control parameter to be estimated with sufficiently high accuracy. The main results are illustrated by the example of systems exhibiting periodic dynamics in the presence of noise.     

Reliability properties of systems with exchangeable components and exponential conditional distributions

We study basic properties for bivariate systems with exchangeable components and exponential conditional distributions which represent bi-component biological or engineering systems with structural dependency. This is equivalent to suppose that we have similar components with the bivariate exponential conditional joint distribution defined by Arnold and Strauss (1988). Specifically, we study the reliability functions, the moments, some aging measures, ordering and classification properties for series and parallel systems. Supported by Ministerio de Ciencia y Tecnología under grant BFM2003-02947     

An informative subset-based estimator for censored quantile regression

Quantile regression in the presence of fixed censoring has been studied extensively in the literature. However, existing methods either suffer from computational instability or require complex procedures involving trimming and smoothing, which complicates the asymptotic theory of the resulting estimators. In this paper, we propose a simple estimator that is obtained by applying standard quantile regression to observations in an informative subset. The proposed method is computationally convenient and conceptually transparent. We demonstrate that the proposed estimator achieves the same asymptotical efficiency as the Powell??s estimator, as long as the conditional censoring probability can be estimated consistently at a nonparametric rate and the estimated function satisfies some smoothness conditions. A simulation study suggests that the proposed estimator has stable and competitive performance relative to more elaborate competitors.     

A dimension reduction approach for conditional Kaplan–Meier estimators

Many quantities of interest in survival analysis are smooth, closed-form functionals of the law of the observations. For instance, the conditional law of a lifetime of interest under random right censoring, and the conditional probability of being cured. In such cases, one can easily derive nonparametric estimators for the quantities of interest by plugging-into the functional the nonparametric estimators of the law of the observations. However, with multivariate covariates, the nonparametric estimation suffers from the curse of dimensionality. Here, a new dimension reduction approach for survival analysis is proposed and investigated in the right-censored lifetime case. First, we consider a single-index hypothesis on the conditional law of the observations and propose a \(\sqrt{n}-\)asymptotically normal semiparametric estimator. Next, we apply the smooth functionals to this estimator. This results in semiparametric estimators of the quantities of interest that avoid the curse of dimensionality. Confidence regions for the index and the functional of interest are built by bootstrap. The new methodology allows to test the dimension reduction assumption, can be extended to other dimension reduction methods and can be applied to closed-form functionals of more general censoring mechanisms.     

Estimating an endpoint with high-order moments

We present a new method for estimating the endpoint of a unidimensional sample when the distribution function decreases at a polynomial rate to zero in the neighborhood of the endpoint. The estimator is based on the use of high-order moments of the variable of interest. It is assumed that the order of the moments goes to infinity, and we give conditions on its rate of divergence to get the asymptotic normality of the estimator. The good performance of the estimator is illustrated on some finite sample situations.     

Kernel estimators of extreme level curves

We address the estimation of extreme level curves of heavy-tailed distributions. This problem is equivalent to estimating quantiles when covariate information is available and when their order converges to one as the sample size increases. We show that, under some conditions, these so-called “extreme conditional quantiles” can still be estimated through a kernel estimator of the conditional survival function. Sufficient conditions on the rate of convergence of their order to one are provided to obtain asymptotically Gaussian distributed estimators. Making use of this result, some kernel estimators of the conditional tail-index are introduced and a Weissman type estimator is derived, permitting to estimate extreme conditional quantiles of arbitrary large order. These results are illustrated through simulated and real datasets.     

Moments,Variances, and Covariances of Sines and Cosines of Arguments which are Subject to Random Error

A simple, unbiased estimator, based on a censored sample, is proposed for the scale parameter of the extreme-value distribution. The exact distribution of the estimator is determined for the cases in which only the first two or only the first three ordered observations are available. The asymptotic distribution is derived, and an approximate distribution for small sample size is also provided. Interval estimation for the scale parameter is developed and a conservative interval estimate for reliability is also obtained.     

Orthogonality of two-dimensional separations based on conditional entropy

A new approach to assess the orthogonality of two-dimensional (2-D) separation systems based on conditional entropy is developed. It considers the quantitative distribution of peaks in the entire separation space such that the orthogonality obtained is independent of the number of peaks observed for each separation technique. Therefore, it can be used to compare the orthogonality of different 2-D separation protocols for a given sample. Herein, the developed method has been employed to estimate the orthogonality of peptide separation by off-gel electrophoresis (OGE) hyphenated to capillary zone electrophoresis (CZE).     

Relative hazard rate estimation for right censored and left truncated data

In medical research differences among treatment groups are a common focus of study. The concept of relative hazard rate is a tool for comparing two groups in terms of their difference in risk rates. A kernel estimator is proposed in the case where both samples are subject to left truncation and right censoring and an iid representation is obtained in this setup. The asymptotic distribution and the asymptotic mean squared error of the estimator are obtained. An application to the famous Channing House data set illustrates the theory. Research supported in part by MCyT Grants PB98-0182-C02-01 and BFM2002-00265 (ERDF support included) and XUGA Grant PGIDT00PX110501 PN for the first author and by the Ministry of the Flemish Community (Project BIL00/28, International Scientific and Technological Cooperation) and by the IAP research network nr P5/24 of the Belgian State (Federal Office for Scientific, Technical and Cultural Affairs) for the last two authors.