Empirical Saddlepoint Approximation Research Articles

Semiparametric evaluation of first‐passage distribution for step‐stress accelerated degradation tests

AbstractIn reliability engineering, different types of accelerated degradation tests have been used to obtain reliability information for evaluating highly reliable or expensive products. The step‐stress accelerated degradation test (SSADT) is one of the useful experimental schemes that can be used to save the resources of an experiment. Motivated by the SSADT data for operational amplifiers collected in Xi'an Microelectronic Technology Institute, in which the underlying degradation mechanism of the operational amplifiers is unknown, we propose a semiparametric approach for SSADT data analysis that does not require strict distributional assumptions. Specifically, the empirical saddlepoint approximation method is proposed to estimate the items' lifetime (first‐passage time) distribution at both stress levels included and not included in the SSADT experiment. Monte Carlo simulation studies are used to evaluate the performance and illustrate the advantages of the proposed approach. Finally, the proposed semiparametric approach is applied to analyze the motivating data set.

Incorporating family disease history and controlling case-control imbalance for population-based genetic association studies.

ABSTRACTMotivationIn the genome-wide association analysis of population-based biobanks, most diseases have low prevalence, which results in low detection power. One approach to tackle the problem is using family disease history, yet existing methods are unable to address type I error inflation induced by increased correlation of phenotypes among closely related samples, as well as unbalanced phenotypic distribution.ResultsWe propose a new method for genetic association test with family disease history, mixed-model-based Test with Adjusted Phenotype and Empirical saddlepoint approximation, which controls for increased phenotype correlation by adopting a two-variance-component mixed model, accounts for case–control imbalance by using empirical saddlepoint approximation, and is flexible to incorporate any existing adjusted phenotypes, such as phenotypes from the LT-FH method. We show through simulation studies and analysis of UK Biobank data of white British samples and the Korean Genome and Epidemiology Study of Korean samples that the proposed method is robust and yields better calibration compared to existing methods while gaining power for detection of variant–phenotype associations.Availability and implementationThe summary statistics and code generated in this study are available at https://github.com/styvon/TAPE.Supplementary information Supplementary data are available at Bioinformatics online.

An empirical saddlepoint approximation method for producing smooth survival and hazard functions under interval-censoring.

We devise a new method to produce smooth estimates of baseline survival and hazard functions for incomplete data observed subject to interval-censoring, that can in principle be viewed as being nonparametric. The key idea is to start from the nonparametric maximum likelihood estimate, and to then construct an empirical moment generating function for the underlying data generating mechanism, which is subsequently inverted via a saddlepoint approximation in order to obtain smooth distributional estimates. Unlike the typical spline-based and other semiparametric methods that have thus far been devised for the same purpose, the proposed approach is unencumbered by the choice of tuning parameters. Simulation studies show that in terms of integrated squared error, the method is very close in performance to the parametric gold standard, and should generally be preferred over the well-established spline-based approach implemented in R package logspline. The methodology is illustrated on some publicly available real datasets, and its implications and limitations are discussed.

Beta approximation and its applications

ABSTRACTDuring the past 30 years, many statistics have been proposed for tackling change-point problems. However, the null distributions of these statistics are not easy to obtain in both theory and practice. In this paper, we consider to use the beta distribution to approximate a standard distribution by matching their first four moments. Sufficient conditions are derived for the beta approximation to any distribution. The distance is used to evaluate the beta approximation to a distribution. To demonstrate the advantage of this approximation over the Rayleigh approximation, the Gumbel approximation and the empirical saddlepoint approximation in change-point analysis, three applications and one real data example are provided for illustration.

Calibrated bootstrap and saddlepoint approximations of finite population L-statistics

We propose two methods to approximate the distribution function of a Studentized linear combination of order statistics for a simple random sample drawn without replacement from a finite population. Using auxiliary data available for the population units, the first method modifies a nonparametric bootstrap approximation, and the second one corrects an empirical saddlepoint approximation based on the bootstrap. We conclude from simulations that, on the tails of distribution of interest, both approximations improve their initial versions and alternative Edgeworth approximations.

An empirical saddlepoint approximation based method for smoothing survival functions under right censoring

AbstractThe Kaplan–Meier (KM) estimator is ubiquitously used for estimating survival functions, but it provides only a discrete approximation at the observation times and does not deliver a proper distribution if the largest observation is censored. Using KM as a starting point, we devise an empirical saddlepoint approximation‐based method for producing a smooth survival function that is unencumbered by choice of tuning parameters. The procedure inverts the moment generating function (MGF) defined through a Riemann–Stieltjes integral with respect to an underlying mixed probability measure consisting of the discrete KM mass function weights and an absolutely continuous exponential right‐tail completion. Uniform consistency, and weak and strong convergence results are established for the resulting MGF and its derivatives, thus validating their usage as inputs into the saddlepoint routines. Relevant asymptotic results are also derived for the density and distribution function estimates. The performance of the resulting survival approximations is examined in simulation studies, which demonstrate a favourable comparison with the log spline method (Kooperberg & Stone, 1992) in small sample settings. For smoothing survival functions we argue that the methodology has no immediate competitors in its class, and we illustrate its application on several real data sets. The Canadian Journal of Statistics 47: 238–261; 2019 © 2019 Statistical Society of Canada

An extended empirical saddlepoint approximation for intractable likelihoods

The challenges posed by complex stochastic models used in computational ecology, biology and genetics have stimulated the development of approximate approaches to statistical inference. Here we focus on Synthetic Likelihood (SL), a procedure that reduces the observed and simulated data to a set of summary statistics, and quantifies the discrepancy between them through a synthetic likelihood function. SL requires little tuning, but it relies on the approximate normality of the summary statistics. We relax this assumption by proposing a novel, more flexible, density estimator: the Extended Empirical Saddlepoint approximation. In addition to proving the consistency of SL, under either the new or the Gaussian density estimator, we illustrate the method using three examples. One of these is a complex individual-based forest model for which SL offers one of the few practical possibilities for statistical inference. The examples show that the new density estimator is able to capture large departures from normality, while being scalable to high dimensions, and this in turn leads to more accurate parameter estimates, relative to the Gaussian alternative. The new density estimator is implemented by the esaddle R package, which is freely available on the Comprehensive R Archive Network (CRAN).

A Classical Moment-Based Approach with Bayesian Properties: Econometric Theory and Empirical Evidence from Asset Pricing

Moment-based estimation often yields instable estimates, such as the RRA (relative risk aversion) estimate in consumption-based asset pricing. This paper establishes novel theoretical results for the ESP (empirical saddlepoint) approximation, and then use them to investigate this instability. We prove that there exists an intensity distribution of the solutions to empirical moment conditions, and approximate it with the integral of the ESP approximation, calling the result the ESP intensity. Global consistency and asymptotic normality of the ESP intensity are proved. The application provides an explanation for the instability of the RRA estimates reported in the literature (fat and long right tail of the ESP approximation), and it suggests that consumption-based asset-pricing theory is more consistent with data than standard inference approaches indicate.

Saddle-Point Approximations, Integrodifference Equations, and Invasions

Invasion, the growth in numbers and spatial spread of a population over time, is a fundamental process in ecology. Governments and businesses expend vast sums to prevent and control invasions of pests and pestilences and to promote invasions of endangered species and biological control agents. Many mathematical models of biological invasions use nonlinear integrodifference equations to describe the growth and dispersal processes and to predict the speed of invasion fronts. Linear models have received less attention, perhaps because they are difficult to simulate for large times. In this paper, we use the saddle-point method, alias the method of steepest descent, to derive asymptotic approximations for the solutions of linear integrodifference equations. We work through five examples, for Gaussian, Laplace, and uniform dispersal kernels in one dimension and for asymmetric Gaussian and radially symmetric Laplace kernels in two dimensions. Our approximations are extremely close to the exact solutions, even for intermediate times. We also employ an empirical saddle-point approximation to predict densities using dispersal data. We use our approximations to examine the effects of censored dispersal data on estimates of invasion speed and population density.

Empirical saddlepoint approximations of the Studentized mean under stratified random sampling

We obtain a saddlepoint approximation for the Studentized mean under stratified random sampling. This is of theoretical interest only, since its calculation requires knowledge of the entire population. We also obtain an empirical saddlepoint approximation based on the stratified sample alone. This empirical approximation can be used for tests of significance and confidence intervals for the population mean and can be regarded as a saddlepoint approximation to a bootstrap approximation. We compare the empirical approximation to the true saddlepoint approximation, both theoretically, to obtain results on the relative error of the bootstrap, and numerically, using Monte Carlo methods to compare the true and bootstrap distributions with the saddlepoint approximations.

Empirical saddlepoint approximations of the Studentized ratio and regression estimates for finite populations

We obtain saddlepoint approximations for tail probabilities of the Studentized ratio and regression estimates of the population mean for a simple random sample taken without replacement from a finite population. This is only possible if we know the entire population, so we also obtain empirical saddlepoint approximations based on the sample alone. These empirical approximations can be used for tests of significance and confidence intervals for the population mean. We compare the empirical approximation to the true saddlepoint approximation, both theoretically and numerically. The empirical saddlepoint is related to a bootstrap method for finite populations and we give numerical comparisons of these. We show that for data which contains extreme outliers, poor approximations can be obtained in the case of regression estimates, both for the saddlepoint and empirical saddlepoint, but for less extreme data the saddlepoint and empirical saddlepoint approximations are extremely close to the corresponding Monte Carlo and bootstrap approximations.

Empirical saddlepoint approximations of the Studentized mean under simple random sampling

We obtain a saddlepoint approximation for the Studentized mean of a simple random sample taken without replacement from a finite population. This is only possible if we know the entire population, so we also obtain an empirical saddlepoint approximation based on the sample alone. This empirical approximation can be used for tests of significance and confidence intervals for the population mean. We compare the empirical approximation to the true saddlepoint approximation, both theoretically and numerically. We also compare both approximations to values obtained in a large Monte Carlo simulation for a population of survival times. The comparisons show that good accuracy can be obtained from the empirical saddlepoint approximation. In addition, the approximations are compared numerically to the Edgeworth approximation of Sugden and Smith (Statist. Probab. Lett. 34 (3) (1997) 293–299; Statist. Probab. Lett. 37 (3) (1998) 317), with a correction in Sugden et al. (J. Roy. Statist. Soc. B 62 (2000) 787–794).

Nonparametric statistical methods for confidence sets.Part2: The empirical likelihood and empirical saddlepoint approximation

The paper deals with some nonparametric methods for the construction of confidence sets; the empirical likelihood, the empirical saddlepoint and the nonparametric tilting. After introducing the basic principles of each method, some refined or alternative versions are considered. The paper gives an account of different aspects, advantages and drawbacks of these methods. Furthermore, examples show their applications and provide computational details. Finally, comparisons are carried out and analogous approaches and analytical links are remarked.

Bounds for probabilities of small relative errors for empirical saddlepoint and bootstrap tail approximations

To obtain test probabilities based on empirical approximations to the distribution of a Studentized function of a mean, we need the approximations to be accurate with sufficiently high probability. In particular, when these test probabilities are small it is best to consider relative errors. Here we show that in the case of univariate standardized means and in the general case of tests based on smooth functions of means, the empirical approximations have asymptotically small relative errors on sets with probability differing from 1 by an exponentially small quantity and that these error rates hold for moderately large deviations. In particular, for standardized deviations of order $n^{1/6}$, the probabilities approximated are exponentially small with exponents of order $n^{1/3}$ and the corresponding relative errors tend to zero on sets whose complements have probabilities of the order of the probabilities being approximated.

Empirical Saddlepoint Approximations for Multivariate M-Estimators

SUMMARY In this paper, we investigate the use of the empirical distribution function in place of the underlying distribution function F to construct an empirical saddlepoint approximation to the density fn of a general multivariate M-estimator. We obtain an explicit form for the error term in the approximation, investigate the effect of renormalizing the estimator, carry out some numerical comparisons and discuss the regression problem.

On the bootstrap saddlepoint approximations

We compare saddlepoint approximations to the exact distributions of a studentized mean and to its bootstrap approximation. We show that, on bounded sets, these empirical saddlepoint approximations achieve second order relative errors uniformly. We also consider the relative errors for larger deviations. It follows that the studentized-t bootstrap p-value and the coverage of the bootstrap confidence interval have second order relative errors

On the Bootstrap Saddlepoint Approximations

We compare saddlepoint approximations to the exact distributions of a studentized mean and to its bootstrap approximation. We show that, on bounded sets, these empirical saddlepoint approximations achieve second order relative errors uniformly. We also consider the relative errors for larger deviations. It follows that the studentized-t bootstrap p-value and the coverage of the bootstrap confidence interval have second order relative errors.

On the Relationship Between Empirical Likelihood and Empirical Saddlepoint Approximation for Multivariate M-Estimators

On the Relationship Between Empirical Likelihood and Empirical Saddlepoint Approximation for Multivariate M-Estimators

On the relationship between empirical likelihood and empirical saddlepoint approximation for multivariate M-estimators

By comparing the expansions of the empirical log-likelihood ratio and the empirical cumulant generating function calculated at the saddlepoint, we investigate the relationship between empirical likelihood and empirical saddlepoint approximations. This leads to a nonparametric approximation of the density of a multivariate M-estimator based on the empirical likelihood and, on the other hand, it provides nonparametric confidence regions based on the empirical cumulant generating function. Some examples illustrate the use of the empirical likelihood in saddlepoint approximations and vice versa.

On The Empirical Saddlepoint Approximation

On The Empirical Saddlepoint Approximation