A saddlepoint approximation to the distribution of the sum of independent non‐identically beta random variables

The exact distribution of the sum of more than two independent beta random variables has not been known. Even in terms of approximations, only the normal approximation is known for the sum. Motivated by Murakami [Statistica Neerlandica, 2014, doi:10.1111/stan.12032], we derive here a saddlepoint approximation for the distribution of sum. An extensive simulation study shows that it always performs better than the normal approximation.     

Nonlinear regression and curved exponential families. Improvement of the approximation to the asymptotic distribution

Inference in nonlinear models is usually based on the asymptotic normal distribution, based on linearizing the model. The accuracy of this approximation can in many cases be improved by a reparametrization. Systematic methods for doing this will be described. Sometimes a saddlepoint approximation can be used, and this offers several advantages compared to the asymptotic distribution and the Edgeworth expansion. The improved methods are unfortunately not commonly used. It will be discussed why this is so. The methods will be illustrated by a series of examples.     

A saddlepoint approximation to the distribution of the sum of independent non‐identically uniform random variables

Calculating the probability of the corresponding significance point is important for finite sample sizes. However, it is difficult to evaluate this probability when the sample sizes are moderate to large. Under these circumstances, consideration of a more accurate approximation for the distribution function is extremely important. Herein, we performed a saddlepoint approximation in the upper tails for the distribution of the sum of independent non‐identically uniform random variables under finite sample sizes. Saddlepoint approximation results were compared with those for a normal approximation. Additionally, the order of errors of the saddlepoint approximation was derived. © 2014 The Authors. Statistica Neerlandica © 2014 VVS.     

Saddlepoint approximations for the sum of independent non‐identically distributed binomial random variables

We discuss saddlepoint approximations to the distribution of the sum of independent non‐identically distributed binomial random variables. We examine the accuracy of the saddlepoint methods for a sum of 10 binomials with different sets of parameter values. The numerical results indicate that the saddlepoint approximations provide very accurate estimates for the probability mass function and the right‐tail probabilities for the cumulative distribution function of the sum.     

Bias in estimating multivariate and univariate diffusions

Multivariate continuous time models are now widely used in economics and finance. Empirical applications typically rely on some process of discretization so that the system may be estimated with discrete data. This paper introduces a framework for discretizing linear multivariate continuous time systems that includes the commonly used Euler and trapezoidal approximations as special cases and leads to a general class of estimators for the mean reversion matrix. Asymptotic distributions and bias formulae are obtained for estimates of the mean reversion parameter. Explicit expressions are given for the discretization bias and its relationship to estimation bias in both multivariate and in univariate settings. In the univariate context, we compare the performance of the two approximation methods relative to exact maximum likelihood (ML) in terms of bias and variance for the Vasicek process. The bias and the variance of the Euler method are found to be smaller than the trapezoidal method, which are in turn smaller than those of exact ML. Simulations suggest that when the mean reversion is slow, the approximation methods work better than ML, the bias formulae are accurate, and for scalar models the estimates obtained from the two approximate methods have smaller bias and variance than exact ML. For the square root process, the Euler method outperforms the Nowman method in terms of both bias and variance. Simulation evidence indicates that the Euler method has smaller bias and variance than exact ML, Nowman’s method and the Milstein method.     

Record values from generalized Pareto distribution and associated inference

In this paper, we derive exact explicit expressions for the single, double, triple and quadruple moments of the upper record values from a generalized Pareto distribution. We then use these expressions to compute the mean, variance, and the coefficients of skewness and kurtosis of certain linear functions of record values. Finally, we develop approximate confidence intervals for the location and scale parameters of the generalized Pareto distribution using the Edgeworth approximation and compare them with the intervals constructed through Monte Carlo simulations. Received: June 1999     

On the distribution of the sample autocorrelation coefficients

Sample autocorrelation coefficients are widely used to test the randomness of a time series. Despite its unsatisfactory performance, the asymptotic normal distribution is often used to approximate the distribution of the sample autocorrelation coefficients. This is mainly due to the lack of an efficient approach in obtaining the exact distribution of sample autocorrelation coefficients. In this paper, we provide an efficient algorithm for evaluating the exact distribution of the sample autocorrelation coefficients. Under the multivariate elliptical distribution assumption, the exact distribution as well as exact moments and joint moments of sample autocorrelation coefficients are presented. In addition, the exact mean and variance of various autocorrelation-based tests are provided. Actual size properties of the Box–Pierce and Ljung–Box tests are investigated, and they are shown to be poor when the number of lags is moderately large relative to the sample size. Using the exact mean and variance of the Box–Pierce test statistic, we propose an adjusted Box–Pierce test that has a far superior size property than the traditional Box–Pierce and Ljung–Box tests.     

A linear approximation to the power function of a test

In this paper we obtain a linear approximation to the power function of a test that is very accurate for small sample sizes. This is especially useful for robust tests where not many power functions are available. The approximation is based on the von Mises expansion of the tail probability functional and on the Tail Area Influence Function (TAIF). The goals of the paper are, first to extend the definition of the TAIF to the case of non identically distributed random variables, defining the Partial Tail Area Influence Functions and the Vectorial Tail Area Influence Function; second, to obtain exact expressions for computing these new influence functions; and, finally, to find accurate approximations to the power function, that can be used in the case of non identically distributed random variables. We include some examples of the application of this linear approximation to tests that involve the Huber statistic and also saddlepoint tests, so proving that the approximations apply not only to simple problems but also to complex ones.     

Exploring the relation between the r* approximation and the Edgeworth expansion

In this paper we study the relation between the r* saddlepoint approximation and the Edgeworth expansion when quite general assumptions for the statistic under consideration are fulfilled. We will show that the two term Edgeworth expansion approximates the r* formula up to an O(n ^?3/2) remainder; this provides a new way of looking at the order of the error of the r* approximation. This finding will be used to inspect the close connection between the r* formula and the Edgeworth B adjustment introduced in Phillips (Biometrika 65:91–98, 1978). We will show that, whenever an Edgeworth expansion exists, this adjustment approximates both the distribution function of the statistic and the r* formula to the same order degree as the Edgeworth expansion. Some numerical examples for the sample mean and U-statistics are given in order to shed light on the theoretical discussion.     

The mean and variance of the likelihood ratio criterion for the multinomial distribution

Summary The mean and variance of — 2 In A for the multinomial distribution are derived in closed form. A comparison is made betweenSchaffer's [1957] approximations to the moments and the exact moments for varying sample size.     

Exact moments of the sample autocorrelations from series generated by general arima processes of order (p, d, q), d=0 or 1

Formulae for the numerical computation of the first four exact moments of the sample autocorrelations, given a time series realisation from a general autoregressive moving average process of order (p, d, q) with d=0 or 1, are presented. The exact mean and variance of the sample autocorrelations are computed for various sample sizes and several time series models. The evaluated results are compared with those obtained from approximate formulae for the mean and variance of the sample autocorrelations. A specification of the numerical accuracy of the first two exact moments is included.     

Some robust exact results on sample autocorrelations and tests of randomness

Several exact results on the second moments of sample autocorrelations, for both Gaussian and non-Gaussian series, are presented. General formulae for the means, variances and covariances of sample autocorrelations are given for the case where the variables in a sequence are exchangeable. Bounds for the variances and covariances of sample autocorrelations from an arbitrary random sequence are derived. Exact and explicit formulae for the variances and covariances of sample autocorrelations from a Gaussian white noise are given. It is observed that the latter results hold for all spherically symmetric distributions. A simulation experiment, with Gaussian series, indicates that normalizing each sample autocorrelation with its exact mean and variance, instead of the usual approximate moments, can improve considerably the accuracy of the asymptotic N(0,1) distribution to obtain critical values for tests of randomness. The exact second moments of rank autocorrelations are also studied.     

Multivariate saddlepoint tests on the mean direction of the von Mises–Fisher distribution

This article provides P values for two new tests on the mean direction of the von Mises–Fisher distribution. The test statistics are obtained from the exponent of the saddlepoint approximation to the density of M-estimators, as suggested by Robinson et al. (Ann Stat 31:1154–1169, 2003). These test statistics are chi-square distributed with asymptotically small relative errors. Despite the high dimensionality of the problem, the proposed P values are accurate and simple to compute. The numerical precision of the P values of the new tests is illustrated by some simulation studies.     

Stable Asymptotics for M‐estimators

We review some first‐order and higher‐order asymptotic techniques for M‐estimators, and we study their stability in the presence of data contaminations. We show that the estimating function (ψ) and its derivative with respect to the parameter play a central role. We discuss in detail the first‐order Gaussian density approximation, saddlepoint density approximation, saddlepoint test, tail area approximation via the Lugannani–Rice formula and empirical saddlepoint density approximation (a technique related to the empirical likelihood method). For all these asymptotics, we show that a bounded ψ (in the Euclidean norm) and a bounded (e.g. in the Frobenius norm) yield stable inference in the presence of data contamination. We motivate and illustrate our findings by theoretical and numerical examples about the benchmark case of one‐dimensional location model.     

Normal Approximation to the Distribution of the Estimated Yield Index Spk

Process yield is the most common criterion used in the manufacturing industry for measuring process performance. A measurement index, called S_pk, has been proposed to calculate the yield for normal processes. The measurement index _pk establishes the relationship between the manufacturing specifications and the actual process performance, which provides an exact measure on process yield. Unfortunately, the sampling distribution of the estimated _pk is mathematically intractable. Therefore, process performance testing cannot be performed. In this paper; we consider a normal approximation to the distribution of the estimated _pk, and investigate its accuracy computationally. We compare the critical values calculated from the approximate distribution with those obtained using the standard simulation technique, for various commonly used quality requirements. Extensive computational results are provided and analyzed. The investigation is useful to the practitioners for making decisions in testing process performance based on the yield.     

Approximate distribution of the maximum deviation of histograms

Summary We consider a histogram, based on order statistics, and density estimators which are closely related to the histogram.When investigating the distribution of the maximum absolute deviation of density estimators it turns out that an approximation by the distribution of the largest absolute value of a normal sample is asymptotically considerably better than an approximation by the limit distribution (which is the extreme value distribution). For the one-side deviation, a corresponding approximation is less accurate. The accuracy can be improved by using an asymptotic expansion.     

Testing equality of shape parameters in several inverse Gaussian populations

Due to the strikingly resemblance to the normal theory and inference methods, the inverse Gaussian (IG) distribution is commonly applied to model positive and right-skewed data. As the shape parameter in the IG distribution is greatly related to other important quantities such as the mean, skewness, kurtosis and the coefficient of variation, it plays an important role in distribution theory. This paper focuses on testing the equality of shape parameters in several inverse Gaussian distributions. Three tests are suggested: the exact generalized inference-based test, the asymptotic test and a test that is based on parametric bootstrap approximation. Simulation studies are undertaken to examine the performances of the these methods, and three real data examples are analyzed for illustration.     

An application of Pitman-efficiency to acceptance sampling procedures*

As an exercise the concept of Pitman-efficiency has been applied to the decision problem whether to use acceptance sampling “by attributes” or “by variables”. The Pitman-efficiency has been calculated in the two cases that the variance of the underlying normal distribution is known and that it is unknown. Rather surprisingly the difference between these two cases proves to be considerable, even asymptotically. The asymptotic result is compared with the exact values of the relative efficiency in the case that s? is unknown. The asymptotic approximation appears to be rather good. The results derived also help to determine a suitable choice of the null hypothesis in order to increase the Pitman-efficiency.     

Bounds for the variance of an inverse binomial estimator

Summary  B est [1] found the variance of the minimum variance unbiased estimator of the parameter p of the negative binomial distribution. M ikulski and Sm [2] gave an upper bound to it, easier to calculate than B est's expression and a good approximation for small values of p and large values of r (the number of successes). In this paper both lower bounds and closer upper bounds are derived.     

Optimal weighted average power similar tests for the covariance structure in the linear regression model

This paper suggests a procedure for the construction of optimal weighted average power similar tests for the error covariance matrix of a Gaussian linear regression model when the alternative model belongs to the exponential family. The paper uses a saddlepoint approximation to construct simple test statistics for a large class of problems and overcomes the computational burden of evaluating the complicated integrals arising in the derivation of optimal weighted average power tests. Extensions to panel data models are considered. Applications are given to tests for error autocorrelation in the linear regression model and in a panel data framework.