On the Nonparametric Mean Residual Life Estimator in Length-biased Sampling

In this article, we discuss nonparametric estimation of a mean residual life function from length-biased data. Precisely, we prove strong uniform consistency and weak converge of the nonparametric mean residual life estimator in length-biased setting.     

Nonparametric estimation of mean residual functions

Situations frequently arise in practice in which mean residual life (mrl) functions must be ordered. For example, in a clinical trial of three experiments, let e (1), e (2) and e (3) be the mrl functions, respectively, for the disease groups under the standard and experimental treatments, and for the disease-free group. The well-documented mrl functions e (1) and e (3) can be used to generate a better estimate for e (2) under the mrl restriction e (1) < or = e (2) < or = e (3). In this paper we propose nonparametric estimators of the mean residual life function where both upper and lower bounds are given. Small and large sample properties of the estimators are explored. Simulation study shows that the proposed estimators have uniformly smaller mean squared error compared to the unrestricted empirical mrl functions. The proposed estimators are illustrated using a real data set from a cancer clinical trial study.     

Nonparametric recursive estimation in stationary markov processes

This paper deals with a class of recursive kernel estimators of the transition probability density function t(y|x) of a stationary Markov process. A sufficient condition for such estimators to be weakly and strongly 2 consistent for almost all (x,y)∈R² is given. Further an L, convergence result is obtained. No continuity conditions are imposed on t(y|x).     

Estimation in inverse Gaussian regression: comparison of asymptotic and bootstrap distributions

This paper presents a brief review of the asymptotic properties of the pseudo-maximum likelihood estimator in the regression model where the reciprocal of the mean of the dependent variable is considered to be a linear function of the regressor variables, and the observations on the dependent variable are assumed to have an inverse Gaussian distribution. The large sample theory for the pseudo-maximum likelihood estimator presented in Babu and Chaubey (1996) is highlighted and a simulation study is carried out to compare the approximation yielded by the bootstrap distribution to that of the asymptotic distribution.     

Uniform strong consistency of histogram density estimation for dependent process

Histogram density estimator is very intuitive and easy to compute and has been widely adopted. Especially in today's big data environment, people pay more attention to the computational cost and are more willing to choose estimators with less to compute. And so, many scholars have been interested in the various estimates based on the histogram technique. Under strong mixing process, this article studies the uniform strong consistency of histogram density estimator and the convergence rate. Our conditions on the mixing coefficient and the bin width are very mild.     

On Robustness of Model-Based Bootstrap Schemes in Nonparametric Time Series Analysis

Theory in time series analysis is often developed under the assumption of finite-dimensional models for the data generating process. Whereas corresponding estimators such as those of a conditional mean function are reasonable even if the true dependence mechanism is more complex, it is usually necessary to capture the whole dependence structure asymptotically for the bootstrap to be valid. In contrast, we show that certain simplified bootstrap schemes which imitate only some aspects of the time series are consistent for quantities arising in nonparametric statistics. To this end, we generalize the well-known "whitening by windowing" principle to joint distributions of nonparametric estimators of the autoregression function. Consequently, we obtain that model-based nonparametric bootstrap schemes remain valid for supremum-type functionals as long as they mimic those finite-dimensional joint distributions consistently which determine the quantity of interest. As an application, we show that simple regression-type bootstrap schemes can be applied for the determination of critical values for nonparametric tests of parametric or semiparametric hypotheses on the autoregression function in the context of a general process.     

Nonparametric linear bayes estimation of survival curves from incomplete observations

A linear Bayes estimator of a survival curve is derived.The estimator has a relatively simple interpretation as a Kaplan-Meier estimator based on an augemented data base - prior information plus sampling information.It is Bayes if the prior is a Dirichlet process, and otherwise an approximation to the Bayes rule against any prior.     

Relative Errors of Difference-Based Variance Estimators in Nonparametric Regression

Difference-based estimators for the error variance are popular since they do not require the estimation of the mean function. Unlike most existing difference-based estimators, new estimators proposed by Müller et al. (2003 Müller , U. , Schick , A. , Wefelmeyer , W. ( 2003 ). Estimating the error variance in nonparametric regression by a covariate-matched U-statistic . Statistics 37 : 179 – 188 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Tong and Wang (2005 Tong , T. , Wang , Y. ( 2005 ). Estimating residual variance in nonparametric regression using least squares . Biometrika 92 : 821 – 830 .[Crossref], [Web of Science ®] , [Google Scholar]) achieved the asymptotic optimal rate as residual-based estimators. In this article, we study the relative errors of these difference-based estimators which lead to better understanding of the differences between them and residual-based estimators. To compute the relative error of the covariate-matched U-statistic estimator proposed by Müller et al. (2003 Müller , U. , Schick , A. , Wefelmeyer , W. ( 2003 ). Estimating the error variance in nonparametric regression by a covariate-matched U-statistic . Statistics 37 : 179 – 188 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), we develop a modified version by using simpler weights. We further investigate its asymptotic property for both equidistant and random designs and show that our modified estimator is asymptotically efficient.     

Nonparametric methodology for the time-dependent partial area under the ROC curve

To assess the classification accuracy of a continuous diagnostic result, the receiver operating characteristic (ROC) curve is commonly used in applications. The partial area under the ROC curve (pAUC) is one of the widely accepted summary measures due to its generality and ease of probability interpretation. In the field of life science, a direct extension of the pAUC into the time-to-event setting can be used to measure the usefulness of a biomarker for disease detection over time. Without using a trapezoidal rule, we propose nonparametric estimators, which are easily computed and have closed-form expressions, for the time-dependent pAUC. The asymptotic Gaussian processes of the estimators are established and the estimated variance-covariance functions are provided, which are essential in the construction of confidence intervals. The finite sample performance of the proposed inference procedures are investigated through a series of simulations. Our method is further applied to evaluate the classification ability of CD4 cell counts on patient's survival time in the AIDS Clinical Trials Group (ACTG) 175 study. In addition, the inferences can be generalized to compare the time-dependent pAUCs between patients received the prior antiretroviral therapy and those without it.     

Time varying long memory parameter estimation for locally stationary long memory processes

The semiparametric estimators of time varying long memory parameter are investigated for locally stationary long memory processes. The GPH estimator and the local Whittle estimator are considered. Under some mild regularity assumptions, the weak consistency and the asymptotic normality of the estimators are obtained. The finite sample performance of the estimators is discussed through a small simulation study.     

Nonparametric estimation of quality adjusted lifetime distribution in a simple illness-death model

In this work, we consider the nonparametric estimation of quality adjusted lifetime (QAL) distribution in a simple illness-death model. We first derive the expression of QAL distribution in terms of the distribution of sojourn time in each health state. Next we substitute the estimate of sojourn time distributions in the expression of QAL distribution to obtain its estimate. Consistency and asymptotic normality of the proposed nonparametric estimator are established. Estimation in the presence of some missing data on the transition time to illness is also discussed. We conduct a simulation study to investigate the performance of the proposed estimator. For illustration, we analyse a data set of the Stanford Heart Transplant program. Extension to multistate progressive model is discussed along with an analysis of International Breast Cancer Study Group (IBCSG) Trial V data.     

Copula-based reliability analysis for a parallel system with a cold standby

The traditional reliability models cannot well reflect the effect of performance dependence of subsystems on the reliability of system, and neglect the problems of initial reliability and standby redundancy. In this paper, the reliability of a parallel system with active multicomponents and a single cold-standby unit has been investigated. The simultaneously working components are dependent and the dependence is expressed by a copula function. Based on the theories of conditional probability, the explicit expressions for the reliability and the MTTF of the system, in terms of the copula function and marginal lifetime distributions, are obtained. Let the copula function be the FGM copula and the marginal lifetime distribution be exponential distribution, a system with two parallel dependent units and a single cold-standby unit is taken as an example. The effect of different degrees of dependence among components on system reliability is analyzed, and the system reliability can be expressed as the linear combination of exponential reliability functions with different failure rates. For investigating how the degree of dependence affects the mean lifetime, furthermore, the parallel system with a single cold standby, comprising different number of active components, is also presented. The effectiveness of the modeling method is verified, and the method presented provides a theoretical basis for reliability design of engineering systems and physics of failure.     

Nonparametric estimation of a regression function from backward recurrence times in a cross-sectional sampling

This study considers the nonparametric estimation of a regression function when the response variable is the waiting time between two consecutive events of a stationary renewal process, and where this variable is not completely observed. In these circumstances, our data are the recurrence times from the occurrence of the last event up to a pre-established time, along with the corresponding values of a certain set of covariates. Estimation of the error density function and some of its characteristics are also considered. For the proposed estimators, we first analyze their asymptotic behavior and, thereafter, carry out a simulation study to highlight their behavior in finite samples. Finally, we apply this methodology to an illustrative example with biomedical data.     

On a multidimensional general bootstrap for empirical estimator of continuous-time semi-Markov kernels with applications

The present paper introduces a general notion and presents results of bootstrapped empirical estimators of the semi-Markov kernels and of the conditional transition distributions for semi-Markov processes with countable state space, constructed by exchangeably weighting the sample. Our proposal provides a unification of bootstrap methods in the semi-Markov setting including, in particular, Efron's bootstrap. Asymptotic properties of these generalised bootstrapped empirical distributions are obtained, under mild conditions by a martingale approach. We also obtain some new results on the weak convergence of the empirical semi-Markov processes. We apply these general results in several statistical problems such as the construction of confidence bands and the goodness-of-fit tests where the limiting distributions are derived under the null hypothesis. Finally, we introduce the quantile estimators and their bootstrapped versions in the semi-Markov framework and we establish their limiting laws by using the functional delta methods. Our theoretical results and numerical examples by simulations demonstrate the merits of the proposed techniques.     

Improved estimation for the autocovariances of a Gaussian stationary process

For a Gaussian stationary process with mean μ and autocovariance function γ(·), we consider to improve the usual sample autocovariances with respect to the mean squares error (MSE) loss. For the cases μ=0 and μ≠0, we propose sort of empirical Bayes type estimators Γ? and Γ?, respectively. Then their MSE improvements upon the usual sample autocovariances are evaluated in terms of the spectral density of the process. Concrete examples for them are provided. We observe that if the process is near to a unit root process the improvement becomes quite large. Thus, consideration for estimators of this type seems important in many fields, e.g., econometrics.     

Asymptotic Results of a Nonparametric Conditional Quantile Estimator for Functional Time Series

We consider the estimation of the conditional quantile function when the covariates take values in some abstract function space. The main goal of this article is to establish the almost complete convergence and the asymptotic normality of the kernel estimator of the conditional quantile under the α-mixing assumption and on the concentration properties on small balls of the probability measure of the functional regressors. Some applications and particular cases are studied. This approach can be applied in time series analysis to the prediction and building of confidence bands. We illustrate our methodology with El Niño data.     

Statistical Analysis of Parameter Estimation for 2-D Harmonics in Multiplicative and Additive Noise

This article considers the problem of parameter estimation for two dimensional (2-D) multi-component harmonics in non zero-mean multiplicative and additive noise. The least squares estimators (LSEs) are proposed to estimate the coherent model parameters, and some statistical results of the LSEs are obtained, including strong consistency, strong convergence rate, and asymptotic normality. Furthermore, the LSEs-based estimators are proposed to estimate the noncoherent model parameters, and the strong consistency and the asymptotic normality are also proved. Finally, some numerical experiments are performed to see how the asymptotic results work for finite sample sizes.     

Detection of Change Point in Nonparametric Function with Unit-Root Noise by Wavelet

A wavelet method is proposed to detect jumps in a function which is observed with unit-root noise. We obtain critical values at any scale and prove the consistency of wavelet detection when the nonparametric function is smooth. It shows that the estimation of the number and locations of change points are consistent when there are change points in the nonparametric function. Simulation study supports our method.