Saddlepoint density and distribution functions for the ratio of two linear functions and the product of generalized gamma variates

The generalized gamma distribution is a flexible and attractive distribution because it incorporates several well-known distributions, i.e., gamma, Weibull, Rayleigh, and Maxwell. This article derives saddlepoint density and distribution functions for the ratio of two linear functions of generalized gamma variables and the product of n independent generalized gamma variables. Simulation studies are used to evaluate the accuracy of the saddlepoint approximations. The saddlepoint approximations are fast, easy, and very accurate.     

Saddlepoint Approximations for Stopped-Sum Distributions

One of the common used classes of distributions is the stopped-sum class. This class includes Hermite distribution, Polya–Aeppli distribution, Poisson-Gamma distribution, and Neyman type A. This article introduces the saddlepoint approximations to the stopped-sum class in continuous and discrete settings. We discuss approximations for mass/density and cumulative distribution functions of stopped-sum distributions. Examples of continuous and discrete distributions from the Poisson stopped-sum class are presented. Comparisons between saddlepoint approximations and the exact calculations show the great accuracy of the saddlepoint methods.     

On the bootstrap and smoothed bootstrap

The standard bootstrap and two commonly used types of smoothed bootstrap are investigated. The saddlepoint approximations are used to evaluate the accuracy of the three bootstrap estimates of the density of a sample mean. The optimal choice for the smoothing parameter is obtained when smoothing is useful in reducing the mean squared 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.     

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 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.     

On the relative accuracy of certain bootstrap procedures

We show the second-order relative accuracy, on bounded sets, of the Studentized bootstrap, exponentially tilted bootstrap and nonparametric likelihood tilted bootstrap, for means and smooth functions of means. We also consider the relative errors for larger deviations. Our method exploits certain connections between Edgeworth and saddlepoint approximations to simplify the computations.     

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.     

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 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-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.     

Bootstrap estimates of the sample bivariate autocorrelation and partial autocorrelation distributions

This paper shows how the bootstrap method can be used to estimate the joint distribution of sample autocorrelations and partial autocorrelations. The exact joint distribution of sample autocorrelations is mathematically intractable and attempts at workable approximations are difficult and rely on special assumptions. The bootstrap offers an accurate solution to this problem without requiring special assumptions and in a way that avoids theoretical difficulties. The bootstrap-estimated joint distributions of the autocorrelations and partial autocorrelations of time series are shown to lead to better ARMA model identification. This is demonstrated using simulated series.     

Bayesian prediction of waiting times in stochastic models

The authors show how saddlepoint techniques lead to highly accurate approximations for Bayesian predictive densities and cumulative distribution functions in stochastic model settings where the prior is tractable, but not necessarily the likelihood or the predictand distribution. They consider more specifically models involving predictions associated with waiting times for semi‐Markov processes whose distributions are indexed by an unknown parameter θ. Bayesian prediction for such processes when they are not stationary is also addressed and the inverse‐Gaussian based saddlepoint approximation of Wood, Booth & Butler (1993) is shown to accurately deal with the nonstationarity whereas the normal‐based Lugannani & Rice (1980) approximation cannot, Their methods are illustrated by predicting various waiting times associated with M/M/q and M/G/1 queues. They also discuss modifications to the matrix renewal theory needed for computing the moment generating functions that are used in the saddlepoint methods.     

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.     

PROBABILITY APPROXIMATIONS FOR DIVISIBLE DISCRETE DISTRIBUTIONS

If an integer-valued random variable can be represented as a sum of independent random variables, then powerful tools exist to derive approximations to its distribution. We apply this idea to examples in some of which it is not clear how to give a physical interpretation to the independent sum-mands. We consider bounds on the accuracy of single term approximations, Edgeworth expansions and saddlepoint approximations for both individual probabilities and cumulative probabilities.     

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.     

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.     

Control variates and importance sampling for efficient bootstrap simulations

Importance sampling and control variates have been used as variance reduction techniques for estimating bootstrap tail quantiles and moments, respectively. We adapt each method to apply to both quantiles and moments, and combine the methods to obtain variance reductions by factors from 4 to 30 in simulation examples.We use two innovations in control variates—interpreting control variates as a re-weighting method, and the implementation of control variates using the saddlepoint; the combination requires only the linear saddlepoint but applies to general statistics, and produces estimates with accuracy of order n ^-1/2 B ^-1, where n is the sample size and B is the bootstrap sample size.We discuss two modifications to classical importance sampling—a weighted average estimate and a mixture design distribution. These modifications make importance sampling robust and allow moments to be estimated from the same bootstrap simulation used to estimate quantiles.     

Practical small sample inference for single lag subset autoregressive models

We propose a method for saddlepoint approximating the distribution of estimators in single lag subset autoregressive models of order one. By viewing the estimator as the root of an appropriate estimating equation, the approach circumvents the difficulty inherent in more standard methods that require an explicit expression for the estimator to be available. Plots of the densities reveal that the distributions of the Burg and maximum likelihood estimators are nearly identical. We show that one possible reason for this is the fact that Burg enjoys the property of estimation equation optimality among a class of estimators expressible as a ratio of quadratic forms in normal random variables, which includes Yule–Walker and least squares. By inverting a two-sided hypothesis test, we show how small sample confidence intervals for the parameters can be constructed from the saddlepoint approximations. Simulation studies reveal that the resulting intervals generally outperform traditional ones based on asymptotics and have good robustness properties with respect to heavy-tailed and skewed innovations. The applicability of the models is illustrated by analyzing a longitudinal data set in a novel manner.     

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.