A numerical comparison of the normal and some saddlepoint approximations to a distribution-free test for stochastic ordering in the competing risks model

Calculating the exact critical value of the test statistic is important in nonparametric statistics. However, to evaluate the exact critical value is difficult when the sample sizes are moderate to large. Under these circumstances, to consider more accurate approximation for the distribution function of a test statistic is extremely important. A distribution-free test for stochastic ordering in the competing risks model has been proposed by Bagai et al. (1989). Herein, we performed a saddlepoint approximation in the upper tails for the Bagai statistic under finite sample sizes. We then compared the saddlepoint approximations with the Bagai approximation and investigate the accuracy of the approximations. Additionally, the orders of errors of the saddlepoint approximations were derived.     

截断样本下的强率函数和密度函数核估计的渐进正态性

本文用[1]发展的计数过程去研究截断样本下强率函数核估计的渐进正态性．在弱于[7]和[10]的条件下，得到了更一般的结果．接着我们将这种方法运用到密度函数核估计，在较弱的条件下，得到了截断样本下密度函数核估计的渐进正态性．     

Saddlepoint Approximations to Tail Probabilities and Quantiles of Inhomogeneous Discounted Compound Poisson Processes with Periodic Intensity Functions

This article provides saddlepoint approximations to tail probabilities and quantiles of the insurer discounted total claim amount, where the individual claim amounts are independent with a linear combination of exponential distributions and the number of claims is given by an inhomogeneous Poisson process with a periodic intensity function. It extends some previous results by Gatto (Methodol Comput Appl Probab 12:533?C551, 2010), which are given for tail probabilities only and for non-periodic intensities only. Both extensions proposed in this article are important in the actuarial practice, where phenomena generating claims are subject to seasonal variations and where the quantiles or the values-at-risk of the total claim amount are desired. Some numerical comparisons of the new methods with Monte Carlo simulation are shown. The methods proposed are numerically very accurate, computationally efficient and hence relevant for the actuarial practice.     

An approximation method using approximate approximations

The aim of this article is to extend the method of approximate approximations to boundary value problems. This method was introduced by V. Maz'ya in 1991 and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present article we develop an approximation procedure for the solution of the interior Dirichlet problem for the Laplace equation in two dimensions using approximate approximations. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In the first step, the unknown source density in the potential representation of the solution is replaced by approximate approximations. In the second, the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in the third, Nyström's method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.     

Finite-sample density and its small sample asymptotic approximation

To derive the exact density of a statistic, which can be intractable, is sometimes a difficult problem. The exact densities of estimates of the shift or regression parameters can be derived with the aid of score functions. Moreover, extremely accurate approximations can be obtained by the small sample asymptotics, based on the saddlepoint method. It is of interest to compare these two approaches, at least for small samples. We numerically compare the exact densities of estimates of the shift parameter with their small sample approximations for various parent distributions of the data. For some distributions both methods are in surprising concordance even under very small samples.     

Necessary conditions of the isolated zero-points with nonzero degree and choice of initial approximations for the secant method in the n-dimensional case

Necessary conditions of isolated zero-points with nonzero degree (in particular, point of local minimum or maximum either saddlepoint) in R ⁿ which is an extension of the rule that the first derivative of a function changes a sign in every neighborhood of an isolated point of local minimum or maximum in JR1 and connected with this choice of initial approximations for the secant method are given and proved. Most of the statements are given through using the basic topological facts such as degree, the fiberings of a finite dimensional sphere, homotopy, quotient topology, etc.     

Saddlepoint approximations to the distribution of the total distance of the multivariate isotropic and von Mises–Fisher random walks

This article considers the random walk over R^p, with p ≥ 2, where the directions taken by the individual steps follow either the isotropic or the vonMises–Fisher distributions. Saddlepoint approximations to the density and to upper tail probabilities of the total distance covered by the random walk, i.e., of the length of the resultant, are derived. The saddlepoint approximations are onedimensional and simple to compute, even though the initial problem is p-dimensional. Numerical illustrations of the high accuracy of the proposed approximations are provided.     

Malliavin calculus for difference approximations of multidimensional diffusions: Truncated local limit theorem

The truncated local limit theorem is proved for difference approximations of multidimensional diffusions. Under very mild conditions on the distributions of difference terms, this theorem states that the transition probabilities of these approximations, after truncation of some asymptotically negligible terms, possess densities uniformly convergent to the transition probability density for the limiting diffusion and satisfy certain uniform diffusion-type estimates. The proof is based on a new version of the Malliavin calculus for the product of a finite family of measures that may contain nontrivial singular components. Applications to the uniform estimation of mixing and convergence rates for difference approximations of stochastic differential equations and to the convergence of difference approximations of local times for multidimensional diffusions are presented. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 3, pp. 340–381, March, 2008.     

An efficient third-moment saddlepoint approximation for probabilistic uncertainty analysis and reliability evaluation of structures

This paper presents an efficient third-moment saddlepoint approximation approach for probabilistic uncertainty analysis and reliability evaluation of random structures. By constructing a concise cumulant generating function (CGF) for the state variable according to its first three statistical moments, approximate probability density function and cumulative distribution function of the state variable, which may possess any types of distribution, are obtained analytically by using saddlepoint approximation technique. A convenient generalized procedure for structural reliability analysis is then presented. In the procedure, the simplicity of general moment matching method and the accuracy of saddlepoint approximation technique are integrated effectively. The main difference of the presented method from existing moment methods is that the presented method may provide more detailed information about the distribution of the state variable. The main difference of the presented method from existing saddlepoint approximation techniques is that it does not strictly require the existence of the CGFs of input random variables. With the advantages, the presented method is more convenient and can be used for reliability evaluation of uncertain structures where the concrete probability distributions of input random variables are known or unknown. It is illustrated and examined by five representative examples that the presented method is effective and feasible.     

Robust tests based on dual divergence estimators and saddlepoint approximations

This paper is devoted to robust hypothesis testing based on saddlepoint approximations in the framework of general parametric models. As is known, two main problems can arise when using classical tests. First, the models are approximations of reality and slight deviations from them can lead to unreliable results when using classical tests based on these models. Then, even if a model is correctly chosen, the classical tests are based on first order asymptotic theory. This can lead to inaccurate p-values when the sample size is moderate or small. To overcome these problems, robust tests based on dual divergence estimators and saddlepoint approximations, with good performances in small samples, are proposed.     

Large Deviations Approximations to Distributions of the Total Distance of Compound Random Walks with von Mises Directions

This article considers the planar random walk where the direction taken by each consecutive step follows the von Mises distribution and where the number of steps of the random walk is determined by the class of inhomogeneous birth processes. Saddlepoint approximations to the distribution of the total distance covered by the random walk, i.e. of the length of the resultant vector of the individual steps, are proposed. Specific formulae are derived for the inhomogeneous Poisson process and for processes with linear contagion, which are the binomial and the negative binomial processes. A numerical example confirms the high accuracy of the proposed saddlepoint approximations.     

概率密度函数核估计的Bootstrap逼近

主要研究了密度函数核估计逼近的速度,用Bootstrap方法对核密度进行估计,在适当的条件下,进一步提高了密度核估计Bootstrap逼近的速度,所得到的结果使得密度核估计Bootstrap逼近的速度与密度函数及其导数之间的关系更加的明确.     

A filtered polynomial approach to density estimation

In this paper, a little known computational approach to density estimation based on filtered polynomial approximation is investigated. It is accompanied by the first online available density estimation computer program based on a filtered polynomial approach. The approximation yields the unknown distribution and density as the product of a monotonic increasing polynomial and a filter. The filter may be considered as a target distribution which gets fixed prior to the estimation. The filtered polynomial approach then provides coefficient estimates for (close) algebraic approximations to (a) the unknown density function and (b) the unknown cumulative distribution function as well as (c) a transformation (e.g., normalization) from the unknown data distribution to the filter. This approach provides a high degree of smoothness in its estimates for univariate as well as for multivariate settings. The nice properties as the high degree of smoothness and the ability to select from different target distributions are suited especially in MCMC simulations. Two applications in Sects. 1 and 7 will show the advantages of the filtered polynomial approach over the commonly used kernel estimation method.      

Locally adaptive hazard smoothing

Summary We consider nonparametric estimation of hazard functions and their derivatives under random censorship, based on kernel smoothing of the Nelson (1972) estimator. One critically important ingredient for smoothing methods is the choice of an appropriate bandwidth. Since local variance of these estimates depends on the point where the hazard function is estimated and the bandwidth determines the trade-off between local variance and local bias, data-based local bandwidth choice is proposed. A general principle for obtaining asymptotically efficient data-based local bandwiths, is obtained by means of weak convergence of a local bandwidth process to a Gaussian limit process. Several specific asymptotically efficient bandwidth estimators are discussed. We propose in particular an, asymptotically efficient method derived from direct pilot estimators of the hazard function and of the local mean squared error. This bandwidth choice method has practical advantages and is also of interest in the uncensored case as well as for density estimation.Research supported by UC Davis Faculty Research Grant and by Air Force grant AFOSR-89-0386Research supported by Air Force grant AFOSR-89-0386     

Boundary layer approximate approximations and cubature of potentials in domains

In this article we present a new approach to the computation of volume potentials over bounded domains, which is based on the quasi‐interpolation of the density by almost locally supported basis functions for which the corresponding volume potentials are known. The quasi‐interpolant is a linear combination of the basis function with shifted and scaled arguments and with coefficients explicitly given by the point values of the density. Thus, the approach results in semi‐analytic cubature formulae for volume potentials, which prove to be high order approximations of the integrals. It is based on multi‐resolution schemes for accurate approximations up to the boundary by applying approximate refinement equations of the basis functions and iterative approximations on finer grids. We obtain asymptotic error estimates for the quasi‐interpolation and corresponding cubature formulae and provide some numerical examples. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Saddlepoint Approximation to the Distribution of Inhomogeneous Discounted Compound Poisson Processes

In this article we propose an accurate approximation to the distribution of the discounted total claim amount, where the individual claim amounts are independent and identically distributed and the number of claims over a specified period is governed by an inhomogeneous Poisson process. More precisely, we compute cumulant generating functions of such discounted total claim amounts under various intensity functions and individual claim amount distributions, and invert them by the saddlepoint approximation. We provide precise conditions under which the saddlepoint approximation holds. The resulting approximation is numerically accurate, computationally fast and hence more efficient than Monte Carlo simulation.     

A note on testing hypotheses for stationary processes in the frequency domain

In a recent paper, Eichler (2008) [11] considered a class of non- and semiparametric hypotheses in multivariate stationary processes, which are characterized by a functional of the spectral density matrix. The corresponding statistics are obtained using kernel estimates for the spectral distribution and are asymptotically normally distributed under the null hypothesis and local alternatives. In this paper, we derive the asymptotic properties of these test statistics under fixed alternatives. In particular, we also show weak convergence but with a different rate compared to the null hypothesis. We also discuss potential statistical applications of the asymptotic theory by means of a small simulation study.     

Faster Adaptive Importance Sampling in Low Dimensions

Abstract

Adaptive importance sampling using kernel density estimation techniques was introduced by West. This technique adapts the importance sampling function to the underlying integrand, thus yielding small-variance estimates. One drawback of this approach is that evaluation of the kernel mixture density is slow. We present a linear tensor spline representation of the adaptive importance function using variable bandwidth kernels that retains the small variance properties of West's approach but executes more quickly.     

Saddlepoint Approximations to the Probability of Ruin in Finite Time for the Compound Poisson Risk Process Perturbed by Diffusion

A large deviations type approximation to the probability of ruin within a finite time for the compound Poisson risk process perturbed by diffusion is derived. This approximation is based on the saddlepoint method and generalizes the approximation for the non-perturbed risk process by Barndorff-Nielsen and Schmidli (Scand Actuar J 1995(2):169–186, 1995). An importance sampling approximation to this probability of ruin is also provided. Numerical illustrations assess the accuracy of the saddlepoint approximation using importance sampling as a benchmark. The relative deviations between saddlepoint approximation and importance sampling are very small, even for extremely small probabilities of ruin. The saddlepoint approximation is however substantially faster to compute.     

Transformation Kernel Density Estimation With Applications

One of the main objectives of this article is to derive efficient nonparametric estimators for an unknown density fX. It is well known that the ordinary kernel density estimator has, despite several good properties, some serious drawbacks. For example, it suffers from boundary bias and it also exhibits spurious bumps in the tails. We propose a semiparametric transformation kernel density estimator to overcome these defects. It is based on a new semiparametric transformation function that transforms data to normality. A generalized bandwidth adaptation procedure is also developed. It is found that the newly proposed semiparametric transformation kernel density estimator performs well for unimodal, low, and high kurtosis densities. Moreover, it detects and estimates densities with excessive curvature (e.g., modes and valleys) more effectively than existing procedures. In conclusion, practical examples based on real-life data are presented.