《Journal of nonparametric statistics》2012,24(2):149-166

In this paper, we focus on a fundamental reliability measure, the discrete-time intensity of the hitting time (DTIHT), which is the discrete analogue of the rate of occurrence of failures. The problem of evaluating and estimating the DTIHT is addressed for the first time for semi-Markov chains. First, a simple formula for the evaluation of the DTIHT is derived. Following the previous result, a statistical estimator of this plug-in type function is proposed. The main results given here are the asymptotic properties of this estimator, including the strong consistency and the asymptotic normality. Second, the DTIHT is investigated for hidden Markov renewal chains. Following its evaluation, a statistical estimator is suggested whose asymptotic properties are studied. Finally, we give some numerical examples for illustration purposes. The derived models and results can be used to typical reliability problems encountered in different scientific disciplines.  相似文献   

    

《Journal of nonparametric statistics》2012,24(1):41-63

Unlike symmetric kernels, so far exploring asymptotics on asymmetric kernels has relied on diversified approaches. This paper proposes a family of the generalised gamma (GG) kernels that is built on the probability density function of the GG distribution [Stacy, E.W. (1962), ‘A Generalization of the Gamma Distribution’, Annals of Mathematical Statistics, 33, 1187–1192] and a few common conditions. The family can generate asymmetric kernels that share appealing properties with the modified gamma kernel [Chen, S.X. (2000), ‘Probability Density Function Estimation Using Gamma Kernels’, Annals of the Institute of Statistical Mathematics, 52, 471–480]. Asymptotics on the kernels generated from the family can be delivered by manipulating the conditions directly, as with symmetric kernels.  相似文献   

    

《Journal of nonparametric statistics》2012,24(3):561-571

We show that maximum likelihood weighted kernel density estimation offers a unified approach to density estimation and nonparametric inferences. For density estimation, the approach is a generalisation of the standard kernel density estimator that allows the weights attached to each kernel to be chosen by maximum likelihood, instead of being set to n ⁻¹ from the outset (see also Jones, M.C., and Henderson, D.A. (2005), ‘Maximum Likelihood Kernel Density Estimation’, Technical Report 01/05, Department of Statistics, The Open University, UK). For nonparametric inferences, the approach offers a natural, smoothed analogue to empirical likelihood (Owen, A.B. (2001), Empirical Likelihood, Boca Raton, FL: Chapman and Hall/CRC) for inferences on functionals of the underlying distribution, such as its mean or median. Numerical results demonstrate that the proposed method is comparable to the standard kernel density estimator (of the same bandwidth) for density estimation, but can offer noticeable small-sample improvements over empirical likelihood for inferences when the underlying distribution is continuous.  相似文献   

    

《Journal of nonparametric statistics》2012,24(2):77-88

We establish the asymptotic expressions for the bias and the variance of the kernel estimator of Radon–Nikodym derivatives. Under mixing conditions, we show that the kernel estimator has exactly the same asymptotic quadratic error as the i.i.d. case.  相似文献   

    

《Journal of nonparametric statistics》2012,24(3):315-331

In this article, we investigate how to apply the empirical likelihood method for the inference of average derivatives in nonparametric multiple regression models. Empirical likelihood ratios for the vectors of the average derivatives and the density-weighted average derivatives are defined and it is shown that their limiting distributions are weighted sums of independent chi-squared random variables with one degree of freedom. Monte Carlo simulation studies are presented to compare the empirical likelihood method with the normal-approximation-based method. It is found that the empirical likelihood method performs better than the normal-approximation-based method.  相似文献   

    

《Journal of nonparametric statistics》2012,24(2):215-226

Ever since the pioneering work of Parzen [Parzen, E., 1962, On estimation of a probability density function and mode. Annales of Mathematics and Statistics, 33, 1065–1076.], the mean-square error (MSE) and its integrated form (MISE) have been used as the criteria of error in choosing the window size in kernel density estimation. More recently, however, other criteria have been advocated as competitors to the MISE, such as the mean absolute deviation or the Kullback–Leibler loss. In this note, we define a weighted version of the Hellinger distance and show that it has an asymptotic form, which is one-fourth the asymptotic MISE under a slightly more stringent smoothness conditions on the density f. In addition, the proposed criteria give rise to a new way for data-dependent bandwidth selection, which is more stable in the sense of having smaller MSE than the usual least-squares cross-validation, biased cross-validation or the plug-in methodologies when estimating f. Analogous results for the kernel distribution function estimate are also presented.  相似文献   

    

《Journal of nonparametric statistics》2012,24(2):181-197

One popular application of kernel density estimation is in kernel discriminant analysis, where kernel estimates of population densities are plugged in the Bayes rule to develop a nonparametric classifier. Performance of these kernel density estimates and that of the corresponding classifier depends on the values of associated smoothing parameters commonly known as the bandwidths. Bandwidths that minimize mean integrated square errors of kernel density estimates often lead to poor misclassification rates in classification problems. In discriminant analysis, usually a cross-validated estimate of misclassification probability is minimized to find the optimal bandwidth, and that bandwidth is used for classifying all observations. However, in addition to depending on the training data set, a good choice of bandwidth should also depend on the specific observation to be classified. Therefore, instead of fixing the value of the bandwidth parameter, in practice it may be more useful to choose it adaptively. This article presents one such adaptive classification technique, where the bandwidth is chosen on the basis of the training sample and the data point to be classified. The performance of the proposed method has been illustrated using some benchmark data sets.  相似文献   

    

《Journal of nonparametric statistics》2012,24(8):721-738

A new estimate of the hazard rate function is proposed, based on nonparametric transformations of the data and motivated by the bias expression of conventional kernel hazard estimates. The squared error of this estimate is considered, and it is shown to be considerably smaller than that of ordinary kernel estimates. With the use of a practical bandwidth choice rule, the estimate is illustrated graphically on distributional and real-world data.  相似文献   

    

《Journal of nonparametric statistics》2012,24(4):505-518

This paper proposes a profile likelihood algorithm to compute the nonparametric maximum likelihood estimator of a convex hazard function. The maximisation is performed in two steps: First the support reduction algorithm is used to maximise the likelihood over all hazard functions with a given point of minimum (or antimode). Then it is shown that the profile (or partially maximised) likelihood is quasi-concave as a function of the antimode, so that a bisection algorithm can be applied to find the maximum of the profile likelihood, and hence also the global maximum. The new algorithm is illustrated using both artificial and real data, including lifetime data for Canadian males and females.  相似文献   

    

《Journal of nonparametric statistics》2012,24(2):241-259

One of the most frequently used methods to model the autocovariance function of a second-order stationary time series is to use the parametric framework of autoregressive and moving average models developed by Box and Jenkins. However, such parametric models, though very flexible, may not always be adequate to model autocovariance functions with sharp changes. Furthermore, if the data do not follow the parametric model and are censored at a certain value, the estimation results may not be reliable. We develop a Gaussian imputation method to estimate an autocovariance structure via nonparametric estimation of the autocovariance function in order to address both censoring and incorrect model specification. We demonstrate the effectiveness of the technique in terms of bias and efficiency with simulations under various rates of censoring and underlying models. We describe its application to a time series of silicon concentrations in the Arctic.  相似文献