Approximation of the Distribution of the Location Parameter in the Growth Curve Model

Abstract.  In this paper an Edgeworth-type approximation of order O(n ⁻² ) to the density of the estimator of the location parameter in the growth curve model has been derived. The approximation is a mixture of a normal and a Kotz-type distribution, thus being an elliptical distribution. A condition for unimodality of the mixture was found and marginal distribution of a subvector of the mixture distribution was derived. Finally, a small example was given to demonstrate an application of the approximation.     

A simple approximation relation between the symmetric generalized logistic and the students t distributions

In this paper, we obtain a new approximation of the Student's t distribution by using the symmetric generalized logistic (SGL) distribution function. The error of this approximation is shown to be 0(1/n² )where nis the degrees of freedom of thetdistribution. In comparison to similar approximations by George and Ojo and George et al. (1986), this new approximation is much simpler and more accurate. It is also shown that under some conditions, the tdistribution is a good approximation of the SGL distribution. Therefore, the complicated expressions for the cumulants and moments of the SGL can be approximated by those of the t, distribution. Finally, numerical results are given.     

负二项分布的两种近似分布及其比较

负二项分布是一个重要的离散型随机变量的分布,可以用泊松分布和正态分布作为其近似分布,通过对两种近似分布进行比较分析,结果表明：在参数q很小时,泊松近似的精度好于正态近似,而且在参数q很小时,即便r不是很大,用泊松分布也能获得负二项分布较好的近似;当参数q较大时,泊松近似效果不好,相比之下,正态近似的结果不错。     

A new approximation for the distribution of the difference of two t-variables

A mixture representation for the distribution of the difference of two independent t-varlables is provided to approximate the probabilities and percentiles The mixture of normal and standardized t is found to be quite suitable in terms of the accuracy and simplicity as it compares favorably to the best known approximation namelyt that due to Ghosh (1975). The idea of the mixture distribution is also extended to provide an approximation to the distribution of a linear combination of independent t-variables which provides an approximation to the Behrens-Fisher distribution in particular.     

Euler characteristic heuristic for approximating the distribution of the largest eigenvalue of an orthogonally invariant random matrix

The Euler characteristic heuristic has been proposed as a method for approximating the upper tail probability of the maximum of a random field with smooth sample path. When the random field is Gaussian, this method is proved to be valid in the sense that the relative approximation error is exponentially smaller. However, very little is known about the validity of the method when the random field is non-Gaussian. In this paper, as a milestone to developing the general theory about the validity of the Euler characteristic heuristic, we examine the Euler characteristic heuristic for approximating the distribution of the largest eigenvalue of an orthogonally invariant non-Gaussian random matrix. In this particular example, if the probability density function of the random matrix converges to zero sufficiently fast at the boundary of its support, the approximation error of the Euler characteristic heuristic is proved to be small and the approximation is valid. Moreover, for several standard orthogonally invariant random matrices, the approximation formula for the distribution of the largest eigenvalue and its asymptotic error are obtained explicitly. Our formulas are practical enough for the purpose of numerical calculations.     

An approximation of the inverse moments of the positive hypergeometric distribution

The negative moments of the positive hyper geometric distribution are often approximated by the inverse of the positive moments of this distribution. In this paper, a suitable approximation to the positive hypergeometric distribution is used to obtain the negative moments.     

A comment on approximating the x2 distribution in the equiprobable case

In an article appearing in this journal, Smith, et al. (1979) reported that a beta approximation of the X² distribution in the equiprobable case did not perform well. This brief note points out that they used the wrong moments and range in their approximation. It is suggested that when this problem is cor-rected, a beta approximation performs better than the chi-square or adjusted chi-square when k ≥ 3 and n is not too small.     

Estimation of the generalized exponential renewal function

When the shape parameter is a non-integer of the generalized exponential (GE) distribution, the analytical renewal function (RF) usually is not tractable. To overcome this, the approximation method has been used in this paper. In the proposed model, the n-fold convolution of the GE cumulative distribution function (CDF) is approximated by n-fold convolutions of gamma and normal CDFs. We obtain the GE RF by a series approximation model. The method is very simple in the computation. Numerical examples have shown that the approximate models are accurate and robust. When the parameters are unknown, we present the asymptotic confidence interval of the RF. The validity of the asymptotic confidence interval is checked via numerical experiments.     

An approximation to the exact distribution of the wilcoxon-mann-whitney rank sum test statistic

An approximation to the exact distribution of the Wilcoxon rank sum test (Mann-Whitney U-test) and the Siegel-Tukey test based on a linear combination of the two-sample t-test applied to ranks and the normal approximation is compared with the usual normal approximation. The normal approximation results in a conservative test in the tails while the linear combination of the test statistics provides a test that has a very high percentage of agreement with tables of the exact distribution. Sample sizes 3≤m, n≤50 were considered.     

Sample size determinations when two binomial proportions are very small

Sample size determination for testing the hypothesis of equality of proportions with a specified type I and type I1 error probabilitiesis of ten based on normal approximation to the binomial distribution. When the proportionsinvolved are very small, the exact distribution of the test statistic may not follow the assumed distribution. Consequently, the sample size determined by the test statistic may not result in the sespecifiederror probabilities. In this paper the author proposes a square root formula and compares it with several existing sample size approximation methods. It is found that with small proportion (p≦.01) the squar eroot formula provides the closest approximation to the exact sample sizes which attain a specified type I and type II error probabilities. Thes quare root formula is simple inform and has the advantage that equal differencesare equally detectable.     

Improved Measures of the Spread of Data for Some Unknown Complex Distributions Using Saddlepoint Approximations

Measures of the spread of data for random sums arise frequently in many problems and have a wide range of applications in real life, such as in the insurance field (e.g., the total claim size in a portfolio). The exact distribution of random sums is extremely difficult to determine, and normal approximation usually performs very badly for this complex distributions. A better method of approximating a random-sum distribution involves the use of saddlepoint approximations.

Saddlepoint approximations are powerful tools for providing accurate expressions for distribution functions that are not known in closed form. This method not only yields an accurate approximation near the center of the distribution but also controls the relative error in the far tail of the distribution.

In this article, we discuss approximations to the unknown complex random-sum Poisson–Erlang random variable, which has a continuous distribution, and the random-sum Poisson-negative binomial random variable, which has a discrete distribution. We show that the saddlepoint approximation method is not only quick, dependable, stable, and accurate enough for general statistical inference but is also applicable without deep knowledge of probability theory. Numerical examples of application of the saddlepoint approximation method to continuous and discrete random-sum Poisson distributions are presented.     

Asymptotic distributions of the sphericity test in a complex multivariate normal distribution

In this paper, asymptotic expansions of the null and non-null distributions of the sphericity test criterion in the case of a complex multivariate normal distribution are obtained for the first time in terms of beta distributions. In the null case, it is found that the accuracy of the approximation by taking the first term alone in the asymptotic series is sufficient for practical purposes. In fact for p - 2. the asymptotic expansion reduces to the first term which is also the exact distribution in this case. Applications of the results to the area of inferences on multivariate time series are also given.     

Saddlepoint approximations to sensitivities of tail probabilities of random sums and comparisons with Monte Carlo estimators

This article proposes computing sensitivities of upper tail probabilities of random sums by the saddlepoint approximation. The considered sensitivity is the derivative of the upper tail probability with respect to the parameter of the summation index distribution. Random sums with Poisson or Geometric distributed summation indices and Gamma or Weibull distributed summands are considered. The score method with importance sampling is considered as an alternative approximation. Numerical studies show that the saddlepoint approximation and the method of score with importance sampling are very accurate. But the saddlepoint approximation is substantially faster than the score method with importance sampling. Thus, the suggested saddlepoint approximation can be conveniently used in various scientific problems.     

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.     

Lot acceptance and compliance testing based on the sample mean and minimum/maximum

A dual acceptance criterion in terms of the sample mean and an extremum (minimum or maximum) has been used in many inspection procedures in diverse industries. An approximation is given in Vangel (Technometrics, 2002, pp. 242-248) for the joint distribution of the sample mean and an extremum when the population is normally distributed. In this paper we obtain a simple expression that depends on the distribution of the sample mean and the truncated sample mean. This expression allows us to evaluate the joint distribution exactly, in two cases, or approximately, in more general cases, making the dual acceptance criterion easier to calculate in practice. We present a saddlepoint approximation for the joint tail probability, with the application to the dual acceptance criterion under the assumption of normality.     

Approximating the Conway–Maxwell–Poisson distribution normalization constant

By adding a second parameter, Conway and Maxwell created a new distribution for situations where data deviate from the standard Poisson distribution. This new distribution contains a normalization constant expressed as an infinite sum whose summation has no known closed-form expression. Shmueli et al. produced an approximation for this sum but proved that it was valid only for integer values of the second parameter, although they conjectured it was also valid for non-integers. Here we prove their conjecture to be true and discuss for what range of parameters the approximation can be accurately applied.     

An Approximation for the Test of the Equality of the Smallest Eigenvalues of a Covariance Matrix

A limiting distribution of the likelihood ratio statistic for the test of the equality of the q smallest eigenvalues of a covariance matrix is obtained. This distribution can be used as an alternative to the chi-squared distribution which is usually used with this test. It is shown that this new method yields reasonable significance levels for those situations in which the chi-squared approximation is inadequate.     

An approximation to the distribution of the product of two dependent correlation coefficients

Many methodological studies depend on the product of two dependent correlation coefficients. However, the behavior of the distribution of the product of two dependent correlation coefficients is not well known. The distribution of sets of correlation coefficients has been well studied, but not the distribution of the product of two dependent correlation coefficients. The present study derives an approximation to the distribution of the product of two dependent correlation coefficients with a closed form, resulting in a Pearson Type I distribution. A simulation study is also conducted to assess the accuracy of the approximation.     

Approximating the difference of two t-variables for all degrees of freedom using truncated variables

A scaled t‐distribution is used to approximate the distribution of a linear combination of two independent t‐variables for any number of degrees of freedom, and in particular for low degrees of freedom where moments do not exist. The approximation is the method‐of‐moments solution to the analogous problem with truncated t‐variables. The approximation exists for all degrees of freedom, is very accurate for more than two degrees of freedom, and performs as well as other approximations of this form when they exist.     

Distributions of lrt associated with two-parameter exponential distributions

In this paper, an exact distribution of the likelihood ratio criterion for testing the equality of p two-parameter exponential distributions is obtained for unequal sample sizes in a computational form. A useful asymptotic expansion of the distribution is also obtained up to the order of n^-4 with the second term of the order of n^-3 and so can be used to obtain accurate approximations to the critical values of the test statistic even for comparatively small values of n where n is the combined sample size. In fact the first term alone which is a single beta distribution provides a powerful approximation for moderately large values of n.