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.     

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.     

Saddlepoint approximation at the edges of a conditional sample space

Saddlepoint methods present a convenient way to approximate probabilities associated with canonical sufficient statistic vectors in generalized linear models. Implementing saddlepoint approximations requires calculating maximum likelihood estimators for the associated parameters. When the sufficient statistic vector lies at the edge of the sample space, maximum likelihood estimators may not exist. This paper describes how to modify saddlepoint approximation to work in these cases.     

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.     

Tail probability approximations for Student's t-statistics

In this paper, we derive saddlepoint approximations for Student's t-statistics for strongly nonlattice random variables without moment conditions. Under very mild conditions, we show that saddlepoint equations always have solutions. Supported in part by the grants R-155-050-055-133/101 and R-155-000-035-112 at the National University of Singapore     

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.     

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.     

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.     

Boundary Crossing Probabilities for Scan Statistics and Their Applications to Change-Point Detection

This paper gives a brief review of recent developments in change-point detection and in the associated boundary crossing problems. Making use of saddlepoint approximations for Markov random walks, we give further extensions of a basic result on boundary crossing probabilities, leading to detection procedures that are not too de-manding in computational and memory requirements and yet are nearly optimal under several performance criteria.     

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.     

On the Lévy Measure of the Lognormal and the LogCauchy Distributions

The densities of the Lévy measure and the Thorin measure of the standard lognormal distribution are approximated and presented in graphs. Moreover, the behavior of these densities at 0 and is studied. Laplace and saddlepoint approximations are used. The infinite divisibility of the standard logCauchy distribution is given some attention in passing.     

Second-order accurate inference on eigenvalues of covariance and correlation matrices

Edgeworth expansions and saddlepoint approximations for the distributions of estimators of certain eigenfunctions of covariance and correlation matrices are developed. These expansions depend on second-, third-, and fourth-order moments of the sample covariance matrix. Expressions for and estimators of these moments are obtained. The expansions and moment expressions are used to construct second-order accurate confidence intervals for the eigenfunctions. The expansions are illustrated and the results of a small simulation study that evaluates the finite-sample performance of the confidence intervals are reported.     

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.     

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。     

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.     

Some Properties and Improvements of the Saddlepoint Approximation in Nonlinear Regression

We summarize properties of the saddlepoint approximation of the density of the maximum likelihood estimator in nonlinear regression with normal errors: accuracy, range of validity, equivariance. We give a geometric insight into the accuracy of the saddlepoint density for finite samples. The role of the Riemannian curvature tensor in the whole investigation of the properties is demonstrated. By adding terms containing this tensor we improve the saddlepoint approximation. When this tensor is zero, or when the number of observations is large, we have pivotal, independent, and ² distributed variables, like in a linear model. Consequences for experimental design or for constructions of confidence regions are discussed.     

Saddlepoint approximations for tests of dispersion

The nonparametric class of tests for dispersion is widely used for testing the equality of the scale parameters of two populations. This class includes Mood, Siegel–Tukey, Klotz and Moses tests. This paper uses the double saddlepoint approximation to calculate analytical mid-p-values for these tests that are almost exact as the permutation simulation method. The performance of the saddlepoint method is assessed using an extensive simulation study. The speed and accuracy of the saddlepoint method enable us to invert the dispersion tests to calculate (1 ? α)100% confidence intervals for the dispersion parameter.     

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.     

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 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.