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.     

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.     

General Saddlepoint Approximations: Application to the Anderson-Darling Test Statistic

We consider the relative merits of various saddlepoint approximations for the cumulative distribution function (cdf) of a statistic with a possibly non normal limit distribution. In addition to the usual Lugannani-Rice approximation, we also consider approximations based on higher-order expansions, including the case where the base distribution for the approximation is taken to be non normal. This extends earlier work by Wood et al. (1993 Wood , A. T. A. , Booth , J. G. , Butler , R. W. ( 1993 ). Saddlepoint approximations to the CDF of some statistics with nonnormal limit distributions . Journal of the American Statistical Association 88 : 680 – 686 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). These approximations are applied to the distribution of the Anderson-Darling test statistic. While these generalizations perform well in the middle of the distribution's support, a conventional normal-based Lugannani-Rice approximation (Giles, 2001 Giles , D. E. A. ( 2001 ). A Saddlepoint approximation to the distribution function of the Anderson-Darling test statistic . Communications in Statistics B 30 : 899 – 905 .[Taylor & Francis Online] , [Google Scholar]) is superior for conventional critical regions.     

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 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 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.     

A Saddlepoint Approximation to the Distribution of the Half-Life Estimator in a Stationary Autoregressive Model

We derive saddlepoint approximations for the distribution and density functions of the half-life estimated by OLS from autoregressive time-series models. Our results are used to prove that none of the integer-order moments of these half-life estimators exist. This provides an explanation for the very large estimates of persistency, and the extremely wide confidence intervals, that have been reported by various authors, i.e., in the empirical economics literature relating to purchasing power parity.     

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.     

Posterior Distributions for the Gini Coefficient Using Grouped Data

When available data comprise a number of sampled households in each of a number of income classes, the likelihood function is obtained from a multinomial distribution with the income class population proportions as the unknown parameters. Two methods for going from this likelihood function to a posterior distribution on the Gini coefficient are investigated. In the first method, two alternative assumptions about the underlying income distribution are considered, namely a lognormal distribution and the Singh–Maddala (1976) income distribution. In these cases the likelihood function is reparameterized and the Gini coefficient is a nonlinear function of the income distribution parameters. The Metropolis algorithm is used to find the corresponding posterior distributions of the Gini coefficient from a sample of Bangkok households. The second method does not require an assumption about the nature of the income distribution, but uses (a) triangular prior distributions, and (b) beta prior distributions, on the location of mean income within each income class. By sampling from these distributions, and the Dirichlet posterior distribution of the income class proportions, alternative posterior distributions of the Gini coefficient are calculated.     

Testing the Difference of Two Binomial Proportions: Comparison of Continuity Corrections for Saddlepoint Approximation

We carried out a simulation study based on the methodology of Newcombe (1998 Newcombe , R. G. ( 1998 ). Interval estimation for the difference between independent proportions: comparison of eleven methods . Statist. Med. 17 : 873 – 890 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) to compare tests for the difference of two binomial proportions by applying different continuity corrections on saddlepoint approximation to tail probabilities. In this article, we proposed a new continuity correction based on the least common multiple of two sample sizes. We evaluated that the best test should have the actual Type I error rates that are, on the whole, closest to α, but not exceeding α, where α is nominal level of significance.     

Saddlepoint Approximation for the Distribution of the Modified Signed Root of Likelihood Ratio Statistics Near the Mean

The Tracy-Singh product, which can be viewed as generalized Kronecher product, is studied and used widely in matrix theory and statistics. Therefore, some well-known Kantorovich inequalities are generalized to Tracy-Singh product in this article. Furthermore, some applications to statistics are presented.     

Maximum Entropy Approximations for Asymptotic Distributions of Smooth Functions of Sample Means

Abstract. We propose an information‐theoretic approach to approximate asymptotic distributions of statistics using the maximum entropy (ME) densities. Conventional ME densities are typically defined on a bounded support. For distributions defined on unbounded supports, we use an asymptotically negligible dampening function for the ME approximation such that it is well defined on the real line. We establish order n^?1 asymptotic equivalence between the proposed method and the classical Edgeworth approximation for general statistics that are smooth functions of sample means. Numerical examples are provided to demonstrate the efficacy of the proposed method.     

Improved Tolerance Factors for Multivariate Normal Distributions

The Q_os and Qm are two leading estimators of the probability of misclassification which are based on the asymptotic expansion of the the expected value of the Error Rate, P_i. The estimators are, however, not suitable for estimating the Error rates for certain ranges of the parameters p , n₁, n₂ and ß.We investigate the regions in which they produce unacceptable estimates , and show that the Q_os is, in general, better than the Q_m in producing acceptable estimates     

Inference for Income Distributions Using Grouped Data

We develop a general approach to estimation and inference for income distributions using grouped or aggregate data that are typically available in the form of population shares and class mean incomes, with unknown group bounds. We derive generic moment conditions and an optimal weight matrix that can be used for generalized method-of-moments (GMM) estimation of any parametric income distribution. Our derivation of the weight matrix and its inverse allows us to express the seemingly complex GMM objective function in a relatively simple form that facilitates estimation. We show that our proposed approach, which incorporates information on class means as well as population proportions, is more efficient than maximum likelihood estimation of the multinomial distribution, which uses only population proportions. In contrast to the earlier work of Chotikapanich, Griffiths, and Rao, and Chotikapanich, Griffiths, Rao, and Valencia, which did not specify a formal GMM framework, did not provide methodology for obtaining standard errors, and restricted the analysis to the beta-2 distribution, we provide standard errors for estimated parameters and relevant functions of them, such as inequality and poverty measures, and we provide methodology for all distributions. A test statistic for testing the adequacy of a distribution is proposed. Using eight countries/regions for the year 2005, we show how the methodology can be applied to estimate the parameters of the generalized beta distribution of the second kind (GB2), and its special-case distributions, the beta-2, Singh–Maddala, Dagum, generalized gamma, and lognormal distributions. We test the adequacy of each distribution and compare predicted and actual income shares, where the number of groups used for prediction can differ from the number used in estimation. Estimates and standard errors for inequality and poverty measures are provided. Supplementary materials for this article are available online.     

Case-deletion Influence Measures for the Data from Multivariate t Distributions

For the data from multivariate t distributions, it is very hard to make an influence analysis based on the probability density function since its expression is intractable. In this paper, we present a technique for influence analysis based on the mixture distribution and EM algorithm. In fact, the multivariate t distribution can be considered as a particular Gaussian mixture by introducing the weights from the Gamma distribution. We treat the weights as the missing data and develop the influence analysis for the data from multivariate t distributions based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm. Several case-deletion measures are proposed for detecting influential observations from multivariate t distributions. Two numerical examples are given to illustrate our methodology.     

Moments for Some Kumaraswamy Generalized Distributions

Explicit expansions for the moments of some Kumaraswamy generalized (Kw-G) distributions (Cordeiro and de Castro, 2011 Cordeiro, G.M., de Castro, M. (2011). A new family of generalized distributions. J. Statist. Computat. Simul. 81:883–898.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) are derived using special functions. We explore the Kw-normal, Kw-gamma, Kw-beta, Kw-t, and Kw-F distributions. These expressions are given as infinite weighted linear combinations of well-known special functions for which numerical routines are readily available.     

Improved approximations for the inverse moments of the positive binomial distribution

Several alternatives to the most common approximation to the inverse moments of the positive binomial distribution are obtained. The method is based on equating moments and gives considerably better approximations for some values of the parameters.     

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.     

A Bootstrap Test for Circular Data

In this article, variance stabilizing filters are discussed. A new filter with nice properties is proposed which makes use of moving averages and moving standard deviations, the latter smoothed with the Hodrick-Prescott filter. This filter is compared to a GARCH-type filter. An ARIMA model is estimated for the filtered GDP series, and the parameter estimates are used in forecasting the unfiltered series. These forecasts compare well with those of ARIMA, ARFIMA, and GARCH models based on the unfiltered data. The filter does not color white noise.