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 the Difference of Order Statistics and Studentized Sample Quantiles

For a sample from a given distribution the difference of two order statistics and the Studentized quantile are statistics whose distribution is needed to obtain tests and confidence intervals for quantiles and quantile differences. This paper gives saddlepoint approximations for densities and saddlepoint approximations of the Lugannani–Rice form for tail probabilities of these statistics. The relative errors of the approximations are n ⁻¹ uniformly in a neighbourhood of the parameters and this uniformity is global if the densities are log-concave.     

Saddlepoint approximations for subordinator processes

ABSTRACT

We develop the saddlepoint approximations in obtaining the transition functions for general subordinator processes. We derive explicit expressions of the first- and second-order approximations. Specifically, we consider some particular classes of subordinators including the Poisson processes, the Gamma processes, the α-stable subordinators, and the Poisson random integrals. We test this technique on the Poisson and Gamma processes, which have closed-form transition functions. Outcomes show that the approximate expressions are consistent with the true transition functions. We then use this method to predict transition density functions for the α-stable subordinator processes. Finally, we calculate approximated transition densities for some Poisson random integrations. Numerical analysis shows the perfect ability of the saddlepoint approximations to predict the transition densities of the α-stable processes and the Poisson random integrations.     

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.     

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 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.     

Saddlepoint-Based Bootstrap Inference for Quadratic Estimating Equations

Abstract.  We propose an easy to implement method for making small sample parametric inference about the root of an estimating equation expressible as a quadratic form in normal random variables. It is based on saddlepoint approximations to the distribution of the estimating equation whose unique root is a parameter's maximum likelihood estimator (MLE), while substituting conditional MLEs for the remaining (nuisance) parameters. Monotoncity of the estimating equation in its parameter argument enables us to relate these approximations to those for the estimator of interest. The proposed method is equivalent to a parametric bootstrap percentile approach where Monte Carlo simulation is replaced by saddlepoint approximation. It finds applications in many areas of statistics including, nonlinear regression, time series analysis, inference on ratios of regression parameters in linear models and calibration. We demonstrate the method in the context of some classical examples from nonlinear regression models and ratios of regression parameter problems. Simulation results for these show that the proposed method, apart from being generally easier to implement, yields confidence intervals with lengths and coverage probabilities that compare favourably with those obtained from several competing methods proposed in the literature over the past half-century.     

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.     

Asymptotic approximations to the distribution of Kendall's sample τ

This paper provides a saddlepoint approximation to the distribution of the sample version of Kendall's τ, which is a measure of association between two samples. The saddlepoint approximation is compared with the Edgeworth and the normal approximations, and with the bootstrap resampling distribution. A numerical study shows that with small sample sizes the saddlepoint approximation outperforms both the normal and the Edgeworth approximations. This paper gives also an analytical comparison between approximated and exact cumulants of the sample Kendall's τ when the two samples are independent.     

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.     

Monte Carlo Approximations of the Quantiles of a Sample Statistic

We consider in this article the problem of numerically approximating the quantiles of a sample statistic for a given population, a problem of interest in many applications, such as bootstrap confidence intervals. The proposed Monte Carlo method can be routinely applied to handle complex problems that lack analytical results. Furthermore, the method yields estimates of the quantiles of a sample statistic of any sample size though Monte Carlo simulations for only two optimally selected sample sizes are needed. An analysis of the Monte Carlo design is performed to obtain the optimal choices of these two sample sizes and the number of simulated samples required for each sample size. Theoretical results are presented for the bias and variance of the numerical method proposed. The results developed are illustrated via simulation studies for the classical problem of estimating a bivariate linear structural relationship. It is seen that the size of the simulated samples used in the Monte Carlo method does not have to be very large and the method provides a better approximation to quantiles than those based on an asymptotic normal theory for skewed sampling distributions.     

The Wild Bootstrap for Multilevel Models

In this paper, we study the performance of the most popular bootstrap schemes for multilevel data. Also, we propose a modified version of the wild bootstrap procedure for hierarchical data structures. The wild bootstrap does not require homoscedasticity or assumptions on the distribution of the error processes. Hence, it is a valuable tool for robust inference in a multilevel framework. We assess the finite size performances of the schemes through a Monte Carlo study. The results show that for big sample sizes it always pays off to adopt an agnostic approach as the wild bootstrap outperforms other techniques.     

Testing Normality Against the Logistic Distribution Using Saddlepoint Approximation

We consider the problem of testing normality against the logistic distribution, based on a random sample of observations. Since the two families are separate (non nested), the ratio of maximized likelihoods (RML) statistic does not have the usual asymptotic chi-square distribution. We derive the saddlepoint approximation to the distribution of the RML statistic and show that this approximation is more accurate than the normal and Edgeworth approximations, especially for tail probabilities that are the main values of interest in hypothesis testing. It is also shown that this test is almost identical to the most powerful invariant test.     

The size accuracy of combination tests

Adaptive clinical trials typically involve several independent stages. The P‐values from each stage are synthesized through a so‐called combination function which ensures that the overall test will be valid if the stagewise tests are valid. In practice however, approximate and possibly invalid stagewise tests are used. This paper studies how imperfections of the stagewise tests feed through into the combination test. Several general results are proven including some for discrete models. An approximation formula which directly links the combined size accuracy to the component size accuracy is given. In the wider context of adaptive clinical trials, the main conclusion is that the basic tests used should be size accurate at nominal sizes both much larger and also much smaller than nominal desired size. For binary outcomes, the implication is that the parametric bootstrap should be used.     

Optimal Bootstrap Sample Size in Construction of Percentile Confidence Bounds

In traditional bootstrap applications the size of a bootstrap sample equals the parent sample size, n say. Recent studies have shown that using a bootstrap sample size different from n may sometimes provide a more satisfactory solution. In this paper we apply the latter approach to correct for coverage error in construction of bootstrap confidence bounds. We show that the coverage error of a bootstrap percentile method confidence bound, which is of order O ( n ^−2/2) typically, can be reduced to O ( n ⁻¹) by use of an optimal bootstrap sample size. A simulation study is conducted to illustrate our findings, which also suggest that the new method yields intervals of shorter length and greater stability compared to competitors of similar coverage accuracy.     

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.     

Approximation of power in multivariate analysis

We consider the calculation of power functions in classical multivariate analysis. In this context, power can be expressed in terms of tail probabilities of certain noncentral distributions. The necessary noncentral distribution theory was developed between the 1940s and 1970s by a number of authors. However, tractable methods for calculating the relevant probabilities have been lacking. In this paper we present simple yet extremely accurate saddlepoint approximations to power functions associated with the following classical test statistics: the likelihood ratio statistic for testing the general linear hypothesis in MANOVA; the likelihood ratio statistic for testing block independence; and Bartlett's modified likelihood ratio statistic for testing equality of covariance matrices.