THE EFFECTS OF CORRELATION AMONG OBSERVATIONS ON THE ACCURACY OF APPROXIMATING THE DISTRIBUTION OF SAMPLE MEAN BY ITS ASYMPTOTIC DISTRIBUTION1

In this paper, we investigate the effects of correlation among observations on the accuracy of approximating the distribution of sample mean by its asymptotic distribution. The accuracy is investigated by the Berry-Esseen bound (BEB), which gives an upper bound on the error of approximation of the distribution function of the sample mean from its asymptotic distribution for independent observations. For a given sample size (n₀) the BEB is obtained when the observations are independent. Let this be BEB. We then find the sample size (n_*) required to have BEB below BEB₀, when the observations are dependent. Comparison of n_* with n₀ reveals the effects of correlation among observations on the accuracy of the asymptotic distribution as an approximation. It is shown that the effects of correlation among observations are not appreciable if the correlation is moderate to small but it can be severe for extreme correlations.     

Two Wilson–Hilferty type approximations for the null distribution of the Blum,Kiefer and Rosenblatt test of bivariate independence

The Blum et al. (Ann. Math. Statist. 32 (1961) 485) test of bivariate independence, an asymptotic equivalent of Hoeffding's (Ann. Math. Statist. 19 (1948) 546) test, is consistent against all dependence alternatives. A concise tabulation of a well-considered approximation for the asymptotic percentiles of its null distribution is given in Blum et al. and a more complete selection of Monte Carlo percentiles, for samples of size 5 and larger, appears in Mudholkar and Wilding (J. Roy. Statist. Soc. 52 (2003) 1). However, neither tabulation is adequate for estimating p-values of the test. In this note we use a moment based analogue of the classical Wilson–Hilferty transformation to obtain two transformations of type T_n=(nB_n)^h_n. The transformations T_n are then used to construct and compare a Gaussian and a scaled chi-square approximation for the null distribution of nB_n. Both approximations have excellent accuracy, but the Gaussian approximation is more convenient because of its portability.     

Distribution of smallest log-normal and gamma extremes

The asymptotic distribution of the first order statistic X₍₁₎ of log-Normal and Gamma samples is considered. The parameters of this extreme value asymptote (Weibull distribution) are approximated in terms of the initial sample size n and the parameters of the initial log-Normal and Gamma measurement models. The resulting asymptotic model of X₍₁₎ is found to be a reasonable and computationally convenient approximation to the exact model of X₍₁₎.     

The Rate of Convergence of some Asymptotic Expansions for Distribution Approximations via an Esseen Type Estimate

Some asymptotic expansions not necessarily related to the central limit theorem are studied. We first observe that the smoothing inequality of Esseen implies the proximity, in the Kolmogorov distance sense, of the distributions of the random variables of two random sequences satisfying a sort of general asymptotic relation. We then present several instances of this observation. A first example, partially motivated by the the statistical theory of high precision measurements, is given by a uniform asymptotic approximation to (g(X + μ _n )) _n∈?, where g is some smooth function, X is a random variable and (μ _n ) _n∈? is a sequence going to infinity; a multivariate version is also stated and proved. We finally present a second class of examples given by a randomization of the interesting parameter in some classical asymptotic formulas; namely, a generic Laplace's type integral, randomized by the sequence (μ _n X) _n∈?, X being a Gamma distributed random variable.     

Approximations for marginal densities of M-estimators

We develop a saddle-point approximation for the marginal density of a real-valued function p(), where is a general M-estimator of a p-dimensional parameter, that is, the solution of the system {n^-1_ljl (Y_l,) = 0}_j=1,…,p. The approximation is applied to several regression problems and yields very good accuracy for small samples. This enables us to compare different classes of estimators according to their finite-sample properties and to determine when asymptotic approximations are useful in practice.     

On the error term in bahadur's representation of an order statistic

Bahadur (1966) presented a representation of an order statistic, giving its asymptotic distribution and the rate of convergence, under weak assumptions on the density function of the parent distribution. In this paper we consider the mean(squared) deviation of the error term in Bahadur’s approximation of the q th sample quantile (q_n ). We derive a uniform bound on the mean (squared) deviation of q_n , not depending on the value of q. An application of the given result provides the corresponding result for a kernel type estimator of the q th quantile.     

An approximation for the hotelling-lawley trace in the noncentral case

A procedure for estimating power in conjunction with the Hotelling-Lawley trace is developed. By approximating a non-central Wishart distribution with a central Wishart, and using McKeon's (1974) F-type approximation, a relatively simple procedure for obtaining power estimates is obtained. The accuracy of the approximation is investigated by comparing the approximate results with those for a wide range of conditions given in Olson's (1973) extensive Monte Carlo study. Siotani's (1971) asymptotic expansion is used to provide further comparative assessments. It is demonstrated that the approximation is of sufficient accuracy to be used in practical applications.     

An empirical comparison of several methods for testing the equality of dependent proportions

Three methods for testing the equality of nonindependent proportions were compared with, the use of Monte Carlo techniques. The three methods included Cochran's test, an ANOVA F test, and Hotelling's T² test. With respect to empirical significance levels, the ANOVA F test is recommended as the preferred method of analysis.

Oftentimes an experimenter is interested in testing the equality of several proportions. When the proportions are independent Kemp and Butcher (1972) and Butcher and Kemp (1974) compared several methods for analysing large sample binomial data for the case of a 3 x 3 factorial design without replication. In addition, Levy and Narula (1977) compared many of the same methods for analyzing binomial data; however, Levy and Narula investigated the relative utility of the methods for small sample sizes.     

Further Results on the Limiting Distribution of GMM Sample Moment Conditions

In this article, we examine the limiting behavior of generalized method of moments (GMM) sample moment conditions and point out an important discontinuity that arises in their asymptotic distribution. We show that the part of the scaled sample moment conditions that gives rise to degeneracy in the asymptotic normal distribution is T-consistent and has a nonstandard limiting distribution. We derive the appropriate asymptotic (weighted chi-squared) distribution when this degeneracy occurs and show how to conduct asymptotically valid statistical inference. We also propose a new rank test that provides guidance on which (standard or nonstandard) asymptotic framework should be used for inference. The finite-sample properties of the proposed asymptotic approximation are demonstrated using simulated data from some popular asset pricing models.     

Interval estimation in discriminant analysis for large dimension

This paper is concerned with the interval estimation for the log odds of the posterior probability that the observation vector belongs to one of two homoscedastic multivariate normal distributions (Π₁ and Π₂). We give the limiting distribution of the unbiased estimator for the log odds as the sample sizes and the dimension jointly tend to infinity, and approximate the confidence interval based on the asymptotic distribution. Small-scale simulations are performed to check the precision of the approximation.     

Computation of the asymptotic null distribution of goodness-of-fit tests for multi-state models

We develop an improved approximation to the asymptotic null distribution of the goodness-of-fit tests for panel observed multi-state Markov models (Aguirre-Hernandez and Farewell, Stat Med 21:1899–1911, 2002) and hidden Markov models (Titman and Sharples, Stat Med 27:2177–2195, 2008). By considering the joint distribution of the grouped observed transition counts and the maximum likelihood estimate of the parameter vector it is shown that the distribution can be expressed as a weighted sum of independent c²₁{\chi^2_1} random variables, where the weights are dependent on the true parameters. The performance of this approximation for finite sample sizes and where the weights are calculated using the maximum likelihood estimates of the parameters is considered through simulation. In the scenarios considered, the approximation performs well and is a substantial improvement over the simple χ ² approximation.     

A family of IDMRL tests with unknown turning point

In this paper we propose a family of tests for exponentiality against the IDMRL alternative. Here we assume that the turning point or the proportion before the turning point is unknown. We derive the asymptotic null distributions of the test statistics and obtain their asymptotic critical values based on Durbin's approximation method. A simulation study is conducted to evaluate the proposed tests.     

The limiting distribution of the least-squares estimator in nearly integrated seasonal models

We consider the least-squares estimator of the autoregressive parameter in a nearly integrated seasonal model. Building on the study by Chan (1989), who obtained the limiting distribution, we derive a closed-form expression for the appropriate limiting joint moment generating function. We use this function to tabulate percentage points of the asymptotic distribution for various seasonal periods via numerical integration. The results are extended by deriving a stochastic asymptotic expansion to order O_p(T^-l), whose percentage points are also obtained by numerically integrating the appropriate limiting joint moment generating function. The adequacy of the approximation to the finite-sample distribution is discussed.     

Comparison of two measures of association in two-way contingency tables

Stuart's (1953) measure of association in contingency tables, t_C, based on Kendall's (1962) t, is compared with Goodman and Kruskal's (1954, 1959, 1963, 1972) measure G. First, it is proved that |G| ≥ |t_C|; and then it is shown that the upper bound for the asymptotic variance of G is not necessarily always smaller than the upper bound for the asymptotic variance of t_C. It is proved, however, that the upper bound for the coefficient of variation of G cannot be larger in absolute value than the upper bound for the coefficient of variation of t_C. The asymptotic variance of t_C is also derived and hence we obtain an upper bound for this asymptotic variance which is sharper than Stuart's (1953) upper bound.     

Asymptotic distribution of data-driven smoothers in density and regression estimation under dependence

We consider automatic data-driven density, regression and autoregression estimates, based on any random bandwidth selector h/_T. We show that in a first-order asymptotic approximation they behave as well as the related estimates obtained with the “optimal” bandwidth h_T as long as h_T/h_T → 1 in probability. The results are obtained for dependent observations; some of them are also new for independent observations.     

Bivariate symmetry tests for complete and competing risks data: a saddlepoint approach

A class of bivariate symmetry tests for complete data and competing risks data is considered. Saddlepoint approximation for the exact p-values of the underlying permutation distribution of these tests is derived. Several simulation studies are conducted to evaluate the performance of the saddlepoint approximation and the asymptotic approximation. The saddlepoint approximation was found to be highly accurate and superior to the asymptotic approximations in replicating the exact permutation significance.     

Projection de hájek et polynômes de bernstein

Under the hypothesis of white noise, the authors derive the explicit form of the asymptotic representation of linear rank statistics resulting from Hájek's (1968) celebrated projection lemma for linear rank statistics in the so‐called approximate score case. This representation based on Bernstein polynomials is better, in the quadratic mean sense, than the traditional one due to Hájek (1961, 1962). The polynomial representation allows for a new derivation of classical asymptotic results (asymptotic normality, Berry‐Essten bounds). Moreover, a simulation study shows that the quality of the polynomial approximation to the exact finite‐sample distributions of rank statistics is sizeably better than that resulting from the traditional approach.     

SMALL SAMPLE INFERENCE IN LOCATION SCALE MODELS WITH STUDENT(λ) ERRORS

ABSTRACT

A third order accurate approximation to the p value in testing either the location or scale parameter in a location scale model with Student(λ) errors is introduced. The third order approximation is developed via an asymptotic method, based on exponential models and the saddlepoint approximation. Techniques are presented for the numerical computation of all quantities required for the third order approximation. To compare the accuracy of various asymptotic methods a numerical example and simulation study are included. The numerical example and simulation study illustrate that the third order method presented leads to a more accurate p value approximation compared to first order methods in Student(λ) models with small samples.     

Linear least squares estimates and nonlinear means

The consistency and asymptotic normality of a linear least squares estimate of the form (X'X)^-X'Y when the mean is not Xβ is investigated in this paper. The least squares estimate is a consistent estimate of the best linear approximation of the true mean function for the design chosen. The asymptotic normality of the least squares estimate depends on the design and the asymptotic mean may not be the best linear approximation of the true mean function. Choices of designs which allow large sample inferences to be made about the best linear approximation of the true mean function are discussed.     

Asymptotic efficiency of model selection criteria: the nonzero mean gaussian ar(∞) case

Motivated by Shibata’s (1980) asymptotic efficiency results this paper dis-cusses the asymptotic efficiency of the order selected by a selection procedure for an infinite order autoregressive process with nonzero mean and unob servable errors that constitute a sequence of independent Gaussian random variables with mean zero and variance σ² The asymptotic efficiency is established for AIC–type selection criteria such as AIC’, FPE, and S_n(k). In addition, some asymptotic results about the estimators of the parameters of the process and the error–sequence are presented.