DISTRIBUTION OF THE LEAST SQUARES ESTIMATOR IN A FIRST-ORDER AUTOREGRESSIVE MODEL

This paper investigates the finite sample distribution of the least squares estimator of the autoregressive parameter in a first-order autoregressive model. A uniform asymptotic expansion for the distribution applicable to both stationary and nonstationary cases is obtained. Accuracy of the approximation to the distribution by a first few terms of this expansion is then investigated. It is found that the leading term of this expansion approximates well the distribution. The approximation is, in almost all cases, accurate to the second decimal place throughout the distribution. In the literature, there exist a number of approximations to this distribution which are specifically designed to apply in some special cases of this model. The present approximation compares favorably with those approximations and in fact, its accuracy is, with almost no exception, as good as or better than these other approximations. Convenience of numerical computations seems also to favor the present approximations over the others. An application of the finding is illustrated with examples.     

Multi-Center Clinical Trials with Random Enrollment: Theoretical Approximations

ABSTRACT

We consider the problem of analyzing multi-center clinical trials when the number of patients at each center and on each treatment arm is random and follows the Poisson distribution. Theoretical approximations are made for the first two moments of the mean square errors (MSE's) for three different estimators of treatment effect difference that are commonly used in multi-center clinical trials. To construct these approximations, approximations are needed for the harmonic mean and negative moments of the Poisson distribution. This is achieved through the use of recurrence relations. The accuracy of the approximations for the moments of the MSE's were then validated through comparing the theoretical values to those obtained from a simulation study under two different enrollment environments.     

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.     

Two classes of divergence statistics for testing uniform association

The problem of testing uniform association in cross-classifications having ordered categories is considered. Two families of test statistics, both based on divergences between certain functions of the observed data, are studied and compared. Our theoretical study is based on asymptotic properties. For each family, two consistent approximations to the null distribution of the test statistic are studied: the asymptotic null distribution and a bootstrap estimator; all the tests considered are consistent against fixed alternatives; finally, we do a local power study. Surprisingly, both families detect the same local alternatives. The finite sample performance of the tests in these two classes is numerically investigated through some simulation experiments. In the light of the obtained results, some practical recommendations are given.     

A new procedure for testing equivalence in comparative bioavailability and other clinical trials

The problem of testing for equivalence in clinical trials is restated here in terms of the proper clinical hypotheses and a simple classical frequentist significance test based on the central t distribution is derived. This method is then shown to be more powerful than the methods based on usual (shortest) and symmetric confidence intervals.

We begin by considering a noncentral t statistic and then consider three approximations to it. A simulation is used to compare actual test sizes to the nominal values in crossover and completely randomized designs. A central t approximation was the best. The power calculation is then shown to be based on a central t distribution, and a method is developed for obtaining the sample size required to obtain a specified power. For the approximations, a simulation compares actual powers to those obtained for the t distribution and confirms that the theoretical results are close to the actual powers.     

BOUNDS FOR THE AVERAGE SAMPLE NUMBER FUNCTION FOR A BINOMIAL SEQUENTIAL TEST

Recently, Billard (1977) developed a partial sequential procedure for comparing a null hypothesis against a two-sided alternative hypothesis when the parameter under test is that of the binomial distribution. In that paper, approximations to the operating characteristic and average sample number function were derived. In this note, bounds to the average sample number function are derived. Using numerical results a comparison of the approximations, bounds and empirical values is made.     

PROBABILITY APPROXIMATIONS FOR DIVISIBLE DISCRETE DISTRIBUTIONS

If an integer-valued random variable can be represented as a sum of independent random variables, then powerful tools exist to derive approximations to its distribution. We apply this idea to examples in some of which it is not clear how to give a physical interpretation to the independent sum-mands. We consider bounds on the accuracy of single term approximations, Edgeworth expansions and saddlepoint approximations for both individual probabilities and cumulative probabilities.     

A proposal for plotting positions in probability plots

Probability plots allow us to determine whether a set of sample observations is distributed according to a theoretical distribution. Plotting positions are fundamental elements in statistics and, in particular, for the construction of probability plots. In this paper, a new plotting position to construct different probability plots, such as Q–Q Plot, P–P Plot and S–P Plot, is proposed. The proposed definition is based on the median of the ith order statistic of the theoretical distribution considered. The main feature of this plotting position formula is that it is independent of the theoretical distribution selected. Moreover, the procedure developed is ‘almost’ exact, reaching, without a high cost of time, an accuracy as great as we want, which avoids using approximations (proposed by other authors).     

Approximate estimation in nonlinear panel data models

This paper is concerned with the estimation of a general class of nonlinear panel data models in which the conditional distribution of the dependent variable and the distribution of the heterogeneity factors are arbitrary. In general, exact analytical results for this problem do not exist. Here, Laplace and small-sigma appriximations for the marginal likelihood are presented. The computation of the MLE from both approximations is straightforward. It is shown that the accuracy of the Laplace approximation depends on both the sample size and the variance of the individual effects, whereas the accuracy of the small-sigma approximation is 0(1) with respect to the sample size. The results are applied to count, duration and probit panel data models. The accuracy of the approximations is evaluated through a Monte Carlo simulation experiment. The approximations are also applied in an analysis of youth unemployment in Australia.     

Study of the quality of several asymptotic and near-exact approximations based on moments for the distribution of the Wilks Lambda statistic

In this paper a measure of proximity of distributions, when moments are known, is proposed. Based on cases where the exact distribution is known, evidence is given that the proposed measure is accurate to evaluate the proximity of quantiles (exact vs. approximated). The measure may be applied to compare asymptotic and near-exact approximations to distributions, in situations where although being known the exact moments, the exact distribution is not known or the expression for its probability density function is not known or too complicated to handle. In this paper the measure is applied to compare newly proposed asymptotic and near-exact approximations to the distribution of the Wilks Lambda statistic when both groups of variables have an odd number of variables. This measure is also applied to the study of several cases of telescopic near-exact approximations to the exact distribution of the Wilks Lambda statistic based on mixtures of generalized near-integer gamma distributions.     

Bootstrap estimates of the sample bivariate autocorrelation and partial autocorrelation distributions

This paper shows how the bootstrap method can be used to estimate the joint distribution of sample autocorrelations and partial autocorrelations. The exact joint distribution of sample autocorrelations is mathematically intractable and attempts at workable approximations are difficult and rely on special assumptions. The bootstrap offers an accurate solution to this problem without requiring special assumptions and in a way that avoids theoretical difficulties. The bootstrap-estimated joint distributions of the autocorrelations and partial autocorrelations of time series are shown to lead to better ARMA model identification. This is demonstrated using simulated series.     

Asymptotic approximations to the distribution of Kendall's sample τ

This paper provides a saddlepoint approximation to the distribution of the sample version of Kendall's τ, which is a measure of association between two samples. The saddlepoint approximation is compared with the Edgeworth and the normal approximations, and with the bootstrap resampling distribution. A numerical study shows that with small sample sizes the saddlepoint approximation outperforms both the normal and the Edgeworth approximations. This paper gives also an analytical comparison between approximated and exact cumulants of the sample Kendall's τ when the two samples are independent.     

Higher-order Bayesian Approximations for Pseudo-posterior Distributions

The theory of higher-order asymptotics provides accurate approximations to posterior distributions for a scalar parameter of interest, and to the corresponding tail area, for practical use in Bayesian analysis. The aim of this article is to extend these approximations to pseudo-posterior distributions, e.g., posterior distributions based on a pseudo-likelihood function and a suitable prior, which are proved to be particularly useful when the full likelihood is analytically or computationally infeasible. In particular, from a theoretical point of view, we derive the Laplace approximation for a pseudo-posterior distribution, and for the corresponding tail area, for a scalar parameter of interest, also in the presence of nuisance parameters. From a computational point of view, starting from these higher-order approximations, we discuss the higher-order tail area (HOTA) algorithm useful to approximate marginal posterior distributions, and related quantities. Compared to standard Markov chain Monte Carlo methods, the main advantage of the HOTA algorithm is that it gives independent samples at a negligible computational cost. The relevant computations are illustrated by two examples.     

A Dimensional CLT for Non-Central Wilks' Lambda in Multivariate Analysis

Abstract.  We consider the non-central distribution of the classical Wilks' lambda statistic for testing the general linear hypothesis in MANOVA. We prove that as the dimension of the observation vector goes to infinity, Wilks' lambda obeys a central limit theorem under simple growth conditions on the non-centrality matrix. In one case we also prove a stronger result: the saddlepoint cumulative distribution function (CDF) approximation for the standardized version of Wilks' lambda converges uniformly on compact sets to the standard normal CDF. These theoretical results go some way towards explaining why saddlepoint approximations to the distribution of Wilks' lambda retain excellent accuracy in high-dimensional cases.     

SOME SIMPLE APPROXIMATIONS FOR THE DOUBLY NONCENTRAL z DISTRIBUTION

We give two simple approximations for evaluating the cumulative probabilities of the doubly noncentral z distribution. These can easily be used for evaluating the cumulative probabilities of the doubly noncentral F distribution as well. We compare our results with those obtained by Tiku (1965) using series expansion. An industrial situation where a quality characteristic of interest follows the doubly noncentral z distribution is also cited. However, in this case the exact probabilities could be calculated using results on the ratio of two normal variables.     

On the biases of change point and change magnitude estimation after CUSUM test

The present paper investigates the biases of estimates of change point and change magnitude after CUSUM test. By assuming that the change point is far from the beginning and the in-control average run length of samples is large, second order approximations for the biases of both estimates are obtained by conditioning on detection, and biases of both estimates are very significant. Simulation studies show the approximations to be quite accurate in the case of detecting an increase in mean or variance when sampling from a normal distribution. The results demonstrate the fundamental differences between fixed sample size test and sequential test.     

Empirical approximations for Hoeffding’s test of bivariate independence using two Weibull extensions

The sampling distributions are generally unavailable in exact form and are approximated either in terms of the asymptotic distributions, or their correction using expansions such as Edgeworth, Laguerre or Cornish–Fisher; or by using transformations analogous to that of Wilson and Hilferty. However, when theoretical routes are intractable, in this electronic age, the sampling distributions can be reasonably approximated using empirical methods. The point is illustrated using the null distribution of Hoeffding’s test of bivariate independence which is important because of its consistency against all dependence alternatives. For constructing the approximations we employ two Weibull extensions, the generalized Weibull and the exponentiated Weibull families, which contain a rich variety of density shapes and tail lengths, and have their distribution functions and quantile functions available in closed form, making them convenient for obtaining the necessary percentiles and p-values. Both approximations are seen to be excellent in terms of accuracy, but that based on the generalized Weibull is more portable.     

Effect on probabilities and quantiles of adding a quantity with small variance

This paper gives simple approximations for the distribution function and quantiles of the sum X + Y when X is a continuous variable and Y is an independent variable with variance small compared to that of X . The approximations are based around the distribution function or quantiles of X and require only the first two or three moments of Y to be known. Example evaluations with X having a normal, Student's t or chi-squared distribution suggest that the approximations are good in unbounded tail regions when the ratio of variances is less than 0.2.     

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.     

Development and comparative study of two near-exact approximations to the distribution of the product of an odd number of independent Beta random variables

Using the concept of near-exact approximation to a distribution we developed two different near-exact approximations to the distribution of the product of an odd number of particular independent Beta random variables (r.v.'s). One of them is a particular generalized near-integer Gamma (GNIG) distribution and the other is a mixture of two GNIG distributions. These near-exact distributions are mostly adequate to be used as a basis for approximations of distributions of several statistics used in multivariate analysis. By factoring the characteristic function (c.f.) of the logarithm of the product of the Beta r.v.'s, and then replacing a suitably chosen factor of that c.f. by an adequate asymptotic result it is possible to obtain what we call a near-exact c.f., which gives rise to the near-exact approximation to the exact distribution. Depending on the asymptotic result used to replace the chosen parts of the c.f., one may obtain different near-exact approximations. Moments from the two near-exact approximations developed are compared with the exact ones. The two approximations are also compared with each other, namely in terms of moments and quantiles.