Saddlepoint approximations for nonlinear statistics

It is well known that saddlepoint expansions lead to accurate approximations to the cumulative distributions and densities of a sample mean and other simple linear statistics. The use of such expansions is explored in a broader situation. The saddlepoint formula for the tail probability of a certain type of nonlinear statistic is derived. The relative error of O(n^–1), as in the linear case, is retained. A simple example is considered, to illustrate the great accuracy of the approximation.     

Saddlepoint Approximation for Sample and Bootstrap Quantiles

The paper gives the saddlepoint approximation for the distribution function of the sample quantile. A comparison of the saddlepoint approximations for the distribution functions of the sample quantile and the bootstrap quantile shows that the error of the bootstrap approximation to the distribution of the sample quantile obtained by Singh (1981) as an absolute error is actually a relative error.     

Saddlepoint p-values for a class of tests for comparing competing risks with censored data

One of the general problems in clinical trials and mortality rates is the comparison of competing risks. Most of the test statistics used for independent and dependent risks with censored data belong to the class of weighted linear rank tests in its multivariate version. In this paper, we introduce the saddlepoint approximations as accurate and fast approximations for the exact p-values of this class of tests instead of the asymptotic and permutation simulated calculations. Real data examples and extensive simulation studies showed the accuracy and stability performance of the saddlepoint approximations over different scenarios of lifetime distributions, sample sizes and censoring.     

Tail Exactness of Multivariate Saddlepoint Approximations

We consider a log-concave density f in R ^m satisfying certain weak conditions, particularly on the Hessian matrix of log f . For such a density, we prove tail exactness of the multivariate saddlepoint approximation. The proof is based on a local limit theorem for the exponential family generated by f . However, the result refers not to asymptotic behaviour under repeated sampling, but to a limiting property at the boundary of the domain of f . Our approach does not apply any complex analysis, but relies totally on convex analysis and exponential models arguments.     

Multivariate Saddlepoint test for the wrapped normal model

In this article we show the effectiveness and the accuracy of the test statistic based on the expnnent of the saddlepoint approximation for the density of M-estimators, proposed by Robinson, Ronchetti and Young (1999), for testing simultaneous hypotheses on the mean and on the variance of a wrapped normal distribution. We base this test statistic on the trigonometric method of moments estimator proposed by Gatto and Jammalamadaka (l999b), which admits the M-estimator representation necessary for this test. This test statistic has an approximate chi-squared distribution, asympiotically up to the second order, and the high accuracy of this approximation is shown by numerical simulations.     

Correlation properties of continuous-time autoregressive processes delayed by the inverse of the stable subordinator

Abstract

We define the delayed Lévy-driven continuous-time autoregressive process via the inverse of the stable subordinator. We derive correlation structure for the observed non-stationary delayed Lévy-driven continuous-time autoregressive processes of order p, emphasizing low orders, and we show they exhibit long-range dependence property. Distributional properties are discussed as well.     

Saddlepoint approximations for some models of circular data

In this article we provide saddlepoint approximations for some important models of circular data. The particularity of these saddlepoint approximations is that they do not require solving the saddlepoint equation iteratively, so their evaluation is immediate. We first give very accurate approximations to P-values, critical values and power functions for some optimal tests regarding the concentration parameter under wrapped symmetric α-stable and circular normal models. Then, we consider an approximation to the distribution of a projection of the two-dimensional Pearson random walk with exponential step sizes.     

Saddlepoint approximations to tail expectations under non-Gaussian base distributions: option pricing applications

The saddlepoint approximation formulas provide versatile tools for analytic approximation of the tail expectation of a random variable by approximating the complex Laplace integral of the tail expectation expressed in terms of the cumulant generating function of the random variable. We generalize the saddlepoint approximation formulas for calculating tail expectations from the usual Gaussian base distribution to an arbitrary base distribution. Specific discussion is presented on the criteria of choosing the base distribution that fits better the underlying distribution. Numerical performance and comparison of accuracy are made among different saddlepoint approximation formulas. Improved accuracy of the saddlepoint approximations to tail expectations is revealed when proper base distributions are chosen. We also demonstrate enhanced accuracy of the generalized saddlepoint approximation formulas under non-Gaussian base distributions in pricing European options on continuous integrated variance under the Heston stochastic volatility model.     

Saddlepoint Approximations to the Density and the Distribution Functions of Linear Combinations of Ratios of Partial Sums

A class of ratios of partial sums, including Normal, Weibull, Gamma, and Exponential distributions, is considered. The distribution of a linear combination of ratios of partial sums from this class is characterized by the distribution of a linear combination of Dirichlet components. This article presents two saddlepoint approaches to calculate the density and the distribution function for such a class of linear combinations. A simulation study is conducted to assess the performance of the saddlepoint methods and shows the great accuracy of the approximations over the usual asymptotic approximation. Applications of the presented approximations in statistical inferences are discussed.     

General Saddlepoint Approximation for Testing Separate Location-Scale Families of Hypotheses

For testing separate families of hypotheses, the likelihood ratio test does not have the usual asymptotic properties. This paper considers the asymptotic distribution of the ratio of maximized likelihoods (RML) statistic in the special case of testing separate scale or location-scale families of distributions. We derive saddlepoint approximations to the density and tail probabilities of the log of the RML statistic. These approximations are based on the expansion of the log of the RML statistic up to the second order, which is shown not to depend on the location and scale parameters. The resulting approximations are applied in several cases, including normal versus Laplace, normal versus Cauchy, and Weibull versus log-normal. Our results show that the saddlepoint approximations are satisfactory, even for fairly small sample sizes, and are more accurate than normal approximations and Edgeworth approximations, especially for tail probabilities that are the values of main interest in hypothesis testing problems.     

Implementation of saddlepoint approximations in resampling problems

In many situations saddlepoint approximations can replace the Monte Carlo simulation typically used to find the bootstrap distribution of a statistic. We explain how bootstrap and permutation distributions can be expressed as conditional distributions and how methods for linear programming and for fitting generalized linear models can be used to find saddlepoint approximations to these distributions. The ideas are illustrated using an example from insurance.     

Partial Saddlepoint Approximations for Transformed Means

The full saddlepoint approximation for real valued smooth functions of means requires the existence of the joint cumulant generating function for the entire vector of random variables which are being transformed. We propose a mixed saddlepoint-Edgeworth approximation requiring the existence of a cumulant generating function for only part of the random vector considered, while retaining partially the relative nature of the errors. Tail probability approximations are obtained under conditions which enable the approximations to be used in resampling situations and hence to obtain a result on the relative error of coverage in the case of the bootstrap approximation to the confidence interval for the Studentized mean.     

Saddlepoint p-values for group of linear rank tests

Many nonparametric tests in one sample problem, matched pairs, and competingrisks under censoring have the same underlying permutation distribution. This article proposes a saddlepoint approximation to the exact p-values of these tests instead of the asymptotic approximations. The performance of the saddlepoint approximation is assessed by using simulation studies that show the superiority of the saddlepoint methods over the asymptotic approximations in several settings. The use of the saddlepoint to approximate the p-values of class of two sample tests under complete randomized design is also discussed.     

Saddlepoint Approximation for Sample Quantiles with Some Applications

In this article, we use the integral form of the binomial distribution to derive saddlepoint approximations for sample quantiles. As an application, we present the calculation of the tail probability of the empirical log-likelihood ratio statistic for quantiles. Simulation results are also given to show that our approximations are extremely accurate.     

Saddlepoint Approximations for a Product and Its Application

A technique of the saddlepoint approximation with double exponential base, SPA_D is developed to evaluate the probability of a product of two random variables, which may be independent or dependent, normal or contaminated normal random variables. The SPA_D shows a slightly better approximation as compared to the saddlepoint approximation with Lagannani–Rice formula. However, both methods get remarkable results when applied to evaluate the tail probabilities of the Reynolds stress for soil erosion prediction.     

Densities and distribution functions of the ratio of two linear functions and the product of Gamma variates: A saddlepoint approach

Saddlepoint approximations for the densities and the distribution functions of the ratio of two linear functions of gamma random variables and the product of gamma random variables are derived. Ratios of linear functions with positive and negative weights and non identical gamma variables are considered. The saddlepoint approximations are very accurate in the tails as in the center of the distribution. Extensive simulation studies are used to evaluate the accuracy of the proposed methods.     

Enhancement for two commonly-used approximations for the inverse cumulative function of the normal distribution

Two commonly used approximations for the inverse distribution function of the normal distribution are Schmeiser's and Shore's. Both approximations are based on a power transformation of either the cumulative density function (CDF) or a simple function of it. In this note we demonstrate, that if these approximations are presented in the form of the classical one-parameter Box-Cox transformation, and the exponent of the transformation is expressed as a simple function of the CDF, then the accuracy of both approximations may be considerably enhanced, without losing much in algebraic simplicity. Since both approximations are special cases of more general four-parameter systems of distributions, the results presented here indicate that the accuracy of the latter, when used to represent non-normal density functions, may also be considerably enhanced.     

Analytical approximations for iterated bootstrap confidence intervals

Standard algorithms for the construction of iterated bootstrap confidence intervals are computationally very demanding, requiring nested levels of bootstrap resampling. We propose an alternative approach to constructing double bootstrap confidence intervals that involves replacing the inner level of resampling by an analytical approximation. This approximation is based on saddlepoint methods and a tail probability approximation of DiCiccio and Martin (1991). Our technique significantly reduces the computational expense of iterated bootstrap calculations. A formal algorithm for the construction of our approximate iterated bootstrap confidence intervals is presented, and some crucial practical issues arising in its implementation are discussed. Our procedure is illustrated in the case of constructing confidence intervals for ratios of means using both real and simulated data. We repeat an experiment of Schenker (1985) involving the construction of bootstrap confidence intervals for a variance and demonstrate that our technique makes feasible the construction of accurate bootstrap confidence intervals in that context. Finally, we investigate the use of our technique in a more complex setting, that of constructing confidence intervals for a correlation coefficient.