A comparison of some approximations to the Wilcoxon-Mann-Whitney distribution

The following approximations to the exact distribution of the Wilcoxon rank sum test (Mann-Whitney U-test) are compared at or near significance levels .05, .025, .01 and .005: the normal approximation with continuity correction, the simple and complete version of the Edgeworth series approximation proposed by Fix and Hodges (1955), Buckle, Kraft and van Eeden’s (1969a) uniform approximation and Iman’s (1976) recently proposed approximation. The comparison takes into account simplicity of application as well as closeness, and some preference rules are suggested for different user objectives.     

Discrete approximations to the Csiszár,Renyi, and Fisher measures of information

The problem of loss of information due to the discretization of data and its estimate is studied for various measures of information. The results of Ghurye and Johnson (1981) are generalized and supplemented for the Csiszár and Renyi measures of information as well as for Fisher's information matrix.     

Two normal approximations to the binomial distribution

Two new normal approximations are proposed for the cumulative binomial distribution when the mean-is reasonably large. Their adequacy is compared with that of certain well-known approximations- The first is recommended for its simplicity and accuracy relative to the standard and Gram-Charlier approximations. The second Is shown to be more accurate than all known approximations for a certain range of the probability of success.     

Discrete approximations to continuous univariate distributions—an alternative to simulation

A method to replace a continuous univariate distribution with a discrete distribution that takes MN different values is analysed. Both distributions share the same r th moments for r =0, . . ., 2 N −1 and their corresonding distribution functions coincide at least at M +1 points. Several statistical and engineering examples are considered in which the discrete approximation may be used to avoid a simulation study that would be much more demanding computationally.     

Patched approximations and their convergence

Abstract

Patched approximations of copulas unify ordinal sums, shuffles of Min, checkerboard, and checkmin approximations. We give a characterization of patched approximations and an error bound of the approximations in Sobolev norm. Patched approximations with uniform marginal conditional distributions are shown to arise naturally. We prove that these uniform patched approximations converge uniformly and in the Sobolev norm. The latter convergence is settled by showing the convergence almost everywhere of the first partial derivatives. We also show that the independence copula can be approximated by conditional mutual complete copulas in the Sobolev norm.     

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.     

Spectral and circulant approximations to the likelihood for stationary Gaussian random fields

Consider a Gaussian random field model on , observed on a rectangular region. Suppose it is desired to estimate a set of parameters in the covariance function. Spectral and circulant approximations to the likelihood are often used to facilitate estimation of the parameters. The purpose of the paper is to give a careful treatment of the quality of these approximations. A spectral approximation for the likelihood was given by Guyon (Biometrika 69 (1982) 95–105) but without proof. The results given here generalize those of Guyon, and fill in the details of the proof. In addition some matrix results are derived which may be of independent interest. Applications are made to Fisher information and bias calculations for maximum likelihood estimates.     

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.     

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.     

Additional approximations to the distribution of a structural coefficient estimator for the case of two included endogenous variables

Four new approximations t o the exact distribution of the two-stage l e a s t squares estimator of astructuralcoefficient for

the case of two included endogeneous variables are introduced and compared with the others in the literatur e . Two of the new approximations are based on the Pearson distribution and are found to be adequate throughout the parameter space. A normal approximation using exact moments and an approximation based on the saddlepoint method (Holly and Phillips,1979) are found to be

poor for a wide range of parameter values.     

Coupling methods in approximations

Let X and Y be two arbitrary k-dimensional discrete random vectors, for k ≥ 1. We prove that there exists a coupling method which minimizes P( X ≠ Y ). This result is used to find the least upper bound for the metric d( X, Y ) = sup_A|P( X ∈ A ) ? P( Y ∈ A )| and to derive the inequality d(Σ X _i, Σ Y _i) ≤ Σd( X _i, Y _i). We thus obtain a unified method to measure the disparity between the distributions of sums of independent random vectors. Several examples are given.     

Laplace approximations for censored linear regression models

We consider approximate Bayesian inference about scalar parameters of linear regression models with possible censoring. A second-order expansion of their Laplace posterior is seen to have a simple and intuitive form for logconcave error densities with nondecreasing hazard functions. The accuracy of the approximations is assessed for normal and Gumbel errors when the number of regressors increases with sample size. Perturbations of the prior and the likelihood are seen to be easily accommodated within our framework. Links with the work of DiCiccio et al. (1990) and Viveros and Sprott (1987) extend the applicability of our results to conditional frequentist inference based on likelihood-ratio statistics.     

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.     

Bootstrapping survival times in stochastic systems by using saddlepoint approximations

The single bootstrap is implemented by using a saddlepoint approximation to determine estimates for the survival and hazard functions of first-passage times in complicated semi-Markov processes. The double bootstrap is also implemented by resampling saddlepoint inversions and provides BC_a confidence bands for these functions. Confidence intervals for the mean and variance of first-passage times are easily computed. A new characterization of the asymptotic hazard rate for survival times is presented and leads to an indirect method for constructing its bootstrap confidence interval.     

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.     

Simple computation of a bivariate normal integral arising from a problem of misclassification with applications to the diagnosis of hypertension

The joint distribution of the true and observed values of a variable that is subject to measurement error is bivariate normal.An important special case occurs when we want the joint probability of the true value being below a cutoff point and the observed value above it.In that case the required integral can be simply evaluated using a Gaussian quadrature formula, which can easily be evaluated using a calculator.This formula is used to estimate the probabilities of misclassification of participants in screening programs for hypertension.It shows that basing a diagnosis on a single visit, at which a single measurement was made leads to a very high risk of misclassification.The probability of a subject having a blood pressure below the cutoff point, given that the observed pressure is above it, would be 0.45.Increasing the number of visits to three, and measuring the blood pressure twice at each visit, as advocated by Rosner and Polk (1979), would bring the probability down to 0.29.     

Laplace approximations and Bayesian information criteria in possibly misspecified models

We provide general conditions to ensure the valid Laplace approximations to the marginal likelihoods under model misspecification, and derive the Bayesian information criteria including all terms of order O_p(1). Under conditions in theorem 1 of Lv and Liu [J. R. Statist. Soc. B, 76, (2014), 141–167] and a continuity condition for prior densities, asymptotic expansions with error terms of order o_p(1) are derived for the log-marginal likelihoods of possibly misspecified generalized linear models. We present some numerical examples to illustrate the finite sample performance of the proposed information criteria in misspecified models.     

Parallel discretization of the Markov chain approximation for the autoregressive moving average chart

Abstract

In the Markov chain model of an autoregressive moving average chart, the post-transition states of nonzero transition probabilities are distributed along one-dimensional lines of a constant gradient over the state space. By considering this characteristic, we propose discretizing the state space parallel to the gradient of these one-dimensional lines. We demonstrate that our method substantially reduces the computational cost of the Markov chain approximation for the average run length in two- and three-dimensional state spaces. Also, we investigate the effect of these one-dimensional lines on the computational cost. Lastly, we generalize our method to state spaces larger than three dimensions.     

Asymptotic variance approximations for invariant estimators in uncertain asset-pricing models

This article derives explicit expressions for the asymptotic variances of the maximum likelihood and continuously-updated GMM estimators in models that may not satisfy the fundamental asset-pricing restrictions in population. The proposed misspecification-robust variance estimators allow the researcher to conduct valid inference on the model parameters even when the model is rejected by the data. While the results for the maximum likelihood estimator are only applicable to linear asset-pricing models, the asymptotic distribution of the continuously-updated GMM estimator is derived for general, possibly nonlinear, models. The large corrections in the asymptotic variances, that arise from explicitly incorporating model misspecification in the analysis, are illustrated using simulations and an empirical application.     

An application of information theory to the problem of statistical disclosure

By using prior knowledge it may be possible to deduce pieces of individual information from a frequency distribution of a population. If the prior information is described by a stochastic model, an information-theoretic approach can be applied in order to judge the possibilities for disclosure. By specifying the stochastic model in various ways it is shown how the decrease in entropy caused by the publication of a frequency distribution can be determined and interpreted. The stochastic models are also used to derive formulae for disclosure risks and expected numbers of disclosures.