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.     

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.     

Improved inference in dispersion models

We derive a general matrix Bartlett–type correction factor to the gradient statistic in the class of dispersion models. The correction improves the large–sample χ² approximation to the null distribution of the gradient statistic when the sample size is finite. We conduct Monte Carlo simulation experiments to evaluate and compare the performance of various different tests, namely the usual Wald, likelihood ratio, score, and gradient tests, the Bartlett–corrected versions of the likelihood ratio, score, and gradient tests, and bootstrap–based tests. The simulation results suggest that the analytical and computational corrections are effective in removing size distortions of the type I error probability with no power loss. The impact of the corrections in two real data applications is considered for illustrative purposes.     

Similar tests for covariance structures in multivariate linear models

Nyblom (J. Multivariate Anal. 76 (2001) 294) has derived locally best invariant test for the covariance structure in a multivariate linear model. The class of invariant tests obtained by Nyblom [9] does not coincide with the class of similar tests for this testing set-up. This paper extends some of the results of Nyblom [9] by deriving the locally best similar tests for the covariance structure. Moreover, it develops a saddlepoint approximation to optimal weighted average power similar tests (i.e. tests which maximize a weighted average power).     

Asymptotic minimax properties of M-estimators of scale

We ask whether or not the saddlepoint property holds, for robust M-estimation of scale, in gross-errors and Kolmogorov neighbourhoods of certain distributions. This is of interest since the saddlepoint property implies the minimax property — that the supremum of the asymptotic variance of an M-estimator is minimized by the maximum likelihood estimator for that member of the distributional class with minimum Fisher information. Our findings are exclusively negative — the saddlepoint property fails in all cases investigated.     

Nonparametric evaluation of the first passage time of degradation processes

This paper discusses a nonparametric method to approximate the first passage time (FPT) distribution of the degradation processes incorporating random effects if the process type is unknown. The FPT of a degradation process is unnecessarily observed since its density function can be approximated by inverting the empirical Laplace transform using the empirical saddlepoint method. The empirical Laplace transform is composed of the measured increments of the degradation processes. To evaluate the performance of the proposed method, the approximated FPT is compared with the theoretical FPT assuming a true underlying process. The nonparametric method discussed in this paper is shown to possess the comparatively small relative errors in the simulation study and performs well to capture the heterogeneity in the practical data analysis. To justify the fitting results, the goodness‐of‐fit tests including Kolmogorov‐Smirnov test and Cramér‐von Mises test are conducted, and subsequently, a bootstrap confidence interval is constructed in terms of the 90th percentile of the FPT distribution.     

积分波动率二阶变差估计量分析:鞍点算法

本文利用鞍点逼近方法对Black-Scholes模型的积分波动率的二阶变差估计量的估计误差进行分析,得到了相对于中心极限定理更为精细的结果,并且给出了逼近的鞍点算法。结果表明鞍点逼近是中心极限定理的纠正。模拟结果表明鞍点算法给出的估计误差分布相对于正态逼近更合理。该结果在对积分波动率进行统计假设检验时是有意义的。     

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.     

Logistic响应分布分位数的鞍点近似置信区间

论文基于响应数据，应用鞍点近似方法，给出构造Logistic响应分布分位数的近似置信区间的方法. 论文还对这种置信区间进行了模拟，并将该方法应用于QD8电雷管. 模拟和实例结果表明，当样本量较小时，该方法能够较好地推断Logistic响应分布的分位数     

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.     

基于指数分布不同定时截尾数据的可靠度的置信下限

本文基于指数分布不同定时截尾数据,利用鞍点逼近法给出参数估计的概率分布的近似公式,进而给出可靠度的近似置信下限,并通过数值模拟及实例计算说明本文方法的可行性。     

A 3D non-linear k–ε turbulent model for prediction of flow and mass transport in channel with vegetation

The results from a 3D non-linear k–ε turbulence model with vegetation are presented to investigate the flow structure, the velocity distribution and mass transport process in a straight compound open channel and a curved open channel. The 3D numerical model for calculating flow is set up in non-orthogonal curvilinear coordinates in order to calculate the complex boundary channel. The finite volume method is used to disperse the governing equations and the SIMPLEC algorithm is applied to acquire the coupling of velocity and pressure. The non-linear k–ε turbulent model has good useful value because of taking into account the anisotropy and not increasing the computational time. The water level of this model is determined from 2D Poisson equation derived from 2D depth-averaged momentum equations. For concentration simulation, an expression for dispersion through vegetation is derived in the present work for the mixing due to flow over vegetation. The simulated results are in good agreement with available experimental data, which indicates that the developed 3D model can predict the flow structure and mass transport in the open channel with vegetation.     

Tensile tests on glass-reinforced plastics

The mechanism of failure of glass-reinforced plastics in short-time tensile tests is considered from the statistical viewpoint. It is thus possible to calculate for various materials the damage suffered by the test piece up to failure. This depends on the dispersion of the static properties but not on the absolute dimensions.Mekhanika Polimerov, Vol. 1, No. 2, pp. 93–103, 1965     

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.     

Longitudinal dispersion in flows that are homogeneous in the streamwise direction

The longitudinal dispersion of a pulse of passive scalar contaminant within a flow that is unchanging in the downstream direction is established to be a function of the mean-velocity-history of molecules from a uniform distribution on the flow cross-section. The specific case of laminar flow between parallel plates is investigated in detail, wherein it is found that $$\frac{1}{2}\frac{{\overline {d\eta (t)^2 } }}{{dt}} = \frac{9}{{2\pi ^6 }}\frac{{\bar U^2 d^2 }}{k}\sum\limits_{n = 1}^\infty {\frac{1}{{n^6 }}} \left( {1 - \exp \left( {\frac{{ - (2n\pi )^2 tK}}{{d^2 }}} \right)} \right),$$ where η(t) is the displacement relative to an axis located byūt whenū is the discharge velocity,K is the molecular diffusivity andd is the plate separation. This flow is used to show that the molecule mean velocity-histories can be approximated by a function that does not directly depend on the release position of the molecule on the flow cross-section. The results of a numerical simulation of the dispersion in this flow are presented for comparison.     

Saddlepoint approximations to P-values for comparison of density estimates

Summary  This article is concerned with computing approximate p-values for the maximum of the absolute difference between kernel density estimates. The approximations are based on treating the process of local extrema of the differences as a nonhomogeneous Poisson Process and estimating the corresponding local intensity function. The process of local extrema is characterized by the intensity function, which determines the rate of local extrema above a given threshold. A key idea of this article is to provide methods for more accurate estimation of the intensity function by using saddlepoint approximations for the joint density of the difference between kernel density estimates and using the first and second derivative of the difference. In this article, saddlepoint approximations are compared to gaussian approximations. Simulation results from saddlepoint approximations show consistently better agreement between empirical p-value and predetermined value with various bandwidths of kernel density estimates.     

Dual divergence estimators and tests: Robustness results

The class of dual ?-divergence estimators (introduced in Broniatowski and Keziou (2009) [5]) is explored with respect to robustness through the influence function approach. For scale and location models, this class is investigated in terms of robustness and asymptotic relative efficiency. Some hypothesis tests based on dual divergence criteria are proposed and their robustness properties are studied. The empirical performances of these estimators and tests are illustrated by Monte Carlo simulation for both non-contaminated and contaminated data.     

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.     

On high-dimensional two sample mean testing statistics: a comparative study with a data adaptive choice of coefficient vector

The key issues involved in two sample tests in high dimensional problems arise due to large dimension of the mean vector for a relatively small sample size. Recently, Wang et al. (Stat Sin 23:667–690, 2013) proposed a jackknife empirical likelihood test that works under weak assumptions on the dimension of variables (p), and showed that the test statistic has a chi-square limit regardless of whether p is finite or diverges. The sufficient condition required for this statistic is still restrictive. In this paper we significantly relax the sufficient condition for the asymptotic chi-square limit with models allowing flexible dependence structures and derive simpler alternative statistics for testing the equality of two high dimensional means. The proposed statistics have a chi-squared distribution or the maximum of two independent chi-square statistics as their limiting distributions, and the asymptotic results hold for either finite or divergent p. We also propose a data-adaptive method to select the coefficient vector, and compare the various methods in simulation studies. The proposed choice of coefficient vector substantially increases power in the simulation.     

LOgistic响应分布中刻度参数的中、小样本推断

在火炸药产品的敏感性推断中，对响应分布的标准差给出较精确的推断，是基础性工作之一．为此，本文基于Logistic响应分布，在二元响应数据下，应用鞍点近似方法构造了刻度参数的近似置信区间，并进行了模拟研究．最后，本文将该方法应用于QD-8电雷管．模拟结果和实例分析表明，在中、小样本情形，本文方法对刻度参数的推断结果较为精确，显著改进了现行的基于渐近正态性的方法．