《Journal of nonparametric statistics》2012,24(3):171-195

Let X₁…, X _n be a random sample of R ^d -valued random variables from the probability density function f, and let f _n ^*(x) be a recursive kernel estimate of f(x) based on this random sample. Conditions are given under which the exact rates of almost sure convergence of f _n ^*(x) to f(x) are determined. This is done both for d≦1 and also for the special case d = 1. These results are suitably used to establish exact rates for almost sure convergence of a recursive estimate m _n ^*(x) of the regression function m(x). A further application yields exact rates of almost sure convergence of a recursive estimate r _n (x) of the hazard rate r(x).  相似文献   

    

《Journal of nonparametric statistics》2012,24(1-2):127-141

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《Journal of nonparametric statistics》2012,24(4):409-426

We present a discrete kernel estimator appropriate for estimating probability mass functions (p.m.f's) for integer data. Discrete kernel functions analogous to the Beta functions used as kernels in the continuous case are derived for the interior and for the boundary of the domain. An integer bandwidth is considered. Cross validation is used for bandwidth selection. The estimator was motivated by the need to characterize processes (e.g., mixtures of geometric distributions) with long tailed distributions with high mass near the origin, and integer arguments of the random variable. Monte Carlo comparisons with the Hall and Titterington [8](HT) estimator are offered. An application for estimating the p.m.f.'s of wet and dry spell lengths for a nonparamet-ric renewal model of daily rainfall is also presented. Other possible methods for obtaining discrete weight sequences are also presented.  相似文献   

    

《Journal of nonparametric statistics》2012,24(2):203-209

Ratio functions for which nonparametric estimators have been considered include the hazard rate and density under random censoring. One estimation method involves individual estimates of the numerator and denominator. An alternative targets the entire function not the separate pieces. The two estimators are not comparable in terms of Mean Integrated Squared Error. However, the second type is seen to be more natural because it admits an elegant and useful martingale representation, and also is pictorially more attractive.  相似文献   

    

《Journal of nonparametric statistics》2012,24(1):45-64

A family of kernel-based estimators parametrised by a bandwidth is used to estimate the location of the discontinuity in discretely observed data consisting of a discontinuous regression function with added noise. Asymptotic bias and variance calculations for the estimated location are given as a function of the bandwidth and the sample size. For functions Which are Smooth apart from a single discontinuity, bias is of order h ², and is not in general negligible compared with variance, which is of order h/n. The asymptotic mean square error rate MSE = O(n ^−4/3), obtained by balancing squared bias and variance, can usually be improved to O(n ^−2+δ ) with δ > 0 arbitrarily small by choosing a smaller bandwidth h = O(n ^−1+δ ). Special cases are treated and bias and variance are calculated explicitly. Only bias depends on the type of discontinuous function considered: For a step function with added noise and for any function which is antisymmetric about the discontinuity point, squared bias becomes negligible compared to variance, and MSE can therefore converge at the faster rate O(n ^−2+δ )₁ For a smooth function with one added step and added white noise, bias is of order h ³ showing that bias decreases more rapidly as the function becomes simpler.  相似文献   

    

《Journal of nonparametric statistics》2012,24(3):279-293

This investigation proposes consistent nonparametric tests for the hypothesis that a distribution is symmetric about a known median, taken without loss of generality to be zero. These tests are based on the L L₂ no₂ norm and the celebrated kernel method of density estimation. The test statistics when appropriately centered and scaled are asymptotically normal under the null hypothesis. Generalization of the procedure to the case of testing symmetry of residuals in the linear regression model is given offering a more advantageous procedure to that of Fan and Gencay (1995). The procedure is also shown to apply to the case of testing “bivariate symmetry”, cf. Hollander (1971).  相似文献   

    

《Journal of nonparametric statistics》2012,24(1):57-66

The problem of bandwidth selection for kernel density estimation at a point is considered. Asymptotic lower bounds are established for the relative rate of convergence of data-driven bandwidth selectors to their optimal values. It is noted that some existing methods of local bandwidth selection, using high order kernel functions, attain these rates. Nevertheless, a simulation study indicates that improved performance predicted by asymptotic theory may not occur in practice for sample sizes as large as 10⁴ or 10⁵. The paper finishes with a comparison of local and global bandwidth selection, observing that in some sense the local problem is the more difficult.  相似文献   

    

《Journal of nonparametric statistics》2012,24(2-3):223-251

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《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