Nickos Papadatos

Heteroscedastic One-Way ANOVA and Lack-of-Fit Tests

Recent articles have considered the asymptotic behavior of the one-way analysis of variance (ANOVA) F statistic when the number of levels or groups is large. In these articles, the results were obtained under the assumption of homoscedasticity and for the case when the sample or group sizes ni remain fixed as the number of groups, a, tends to infinity. In this article, we study both weighted and unweighted test statistics in the heteroscedastic case. The unweighted statistic is new and can be used even with small group sizes. We demonstrate that an asymptotic approximation to the distribution of the weighted statistic is possible only if the group sizes tend to infinity suitably fast in relation to a. Our investigation of local alternatives reveals a similarity between lack-of-fit tests for constant regression in the present case of replicated observations and the case of no replications, which uses smoothing techniques. The asymptotic theory uses a novel application of the projection principle to obtain the asymptotic distribution of quadratic forms.

MAXIMUM VARIANCE OF ORDER STATISTICS

Yang (1982,Bull. Inst. Math. Acad. Sinica,10(2), 197–204) proved that the variance of the sample median cannot exceed the population variance. In this paper, the upper bound for the variance of order statistics is derived, and it is shown that this is attained by Bernoulli variates only. The proof is based on Hoeffdings identity for the covariance.

An extended Stein-type covariance identity for the Pearson family with applications to lower variance bounds

For an absolutely continuous (integer-valued) r.v. X of the Pearson (Ord) family, we show that, under natural moment conditions, a Stein-type covariance identity of order k holds (cf. [Goldstein and Reinert, J. Theoret. Probab. 18 (2005) 237–260]). This identity is closely related to the corresponding sequence of orthogonal polynomials, obtained by a Rodrigues-type formula, and provides convenient expressions for the Fourier coefficients of an arbitrary function. Application of the covariance identity yields some novel expressions for the corresponding lower variance bounds for a function of the r.v. X, expressions that seem to be known only in particular cases (for the Normal, see [Houdre and Kagan, J. Theoret. Probab. 8 (1995) 23–30]; see also [Houdre and Perez-Abreu, Ann. Probab. 23 (1995) 400–419] for corresponding results related to the Wiener and Poisson processes). Some applications are also given.

Bounds on expectation of order statistics from a finite population

Abstract Consider a simple random sample X1,X2,…,Xn, taken without replacement from a finite ordered population Π={x1⩽x2⩽⋯⩽xN} (n⩽N), where each element of Π has equal probability to be chosen in the sample. Let X1:n⩽X2:n⩽⋯⩽Xn:n be the ordered sample. In the present paper, the best possible bounds for the expectations of the order statistics X i : n (1⩽i⩽n) and the sample range Rn=Xn:n−X1:n are derived in terms of the population mean and variance. Some results are also given for the covariance in the simplest case where n=2. An interesting feature of the bounds derived here is that they reduce to some well-known classical results (for the i.i.d. case) as N→∞. Thus, the bounds established in this paper provide an insight into Hartley–David–Gumbel, Samuelson–Scott, Arnold–Groeneveld and some other bounds.

Exact Bounds for the Expectations of Order Statistics from Non-Negative Populations

Some new exact bounds for the expected values of order statistics, under the assumption that the parent population is non-negative, are obtained in terms of the population mean. Similar bounds for the differences of any two order statistics are also given. It is shown that the existing bounds for the general case can be improved considerably under the above assumption.

A simple method for obtaining the maximal correlation coefficient and related characterizations

We provide a method that enables the simple calculation of the maximal correlation coefficient of a bivariate distribution, under suitable conditions. In particular, the method readily applies to known results on order statistics and records. As an application we provide a new characterization of the exponential distribution: Under a splitting model on independent identically distributed observations, it is the (unique, up to a location-scale transformation) parent distribution that maximizes the correlation coefficient between the records among two different branches of the splitting sequence.

Distribution and expectation bounds on order statistics from possibly dependent variates

Let X1,X2,...,Xn be n random variables with an arbitrary n-variate distribution. We say that the Xs are maximally (resp. minimally) stable of order j (j[set membership, variant]{1,2,...,n}), if the distribution F(j) of max{Xk1,...,Xkj} (resp. G(j) of min{Xk1,...,Xkj}) is the same, for any j-subset {k1,...,kj} of {1,2,...,n}. Under the assumption of maximal (resp. minimal) stability of order j, sharp upper (resp. lower) bounds are given for the distribution Fk:n of the kth order statistic Xk:n, in terms of F(j) (resp. G(j)), and the corresponding expectation bounds are derived. Moreover, some expectation bounds in the case of j-independent-F samples (i.e., when each j-tuple Xk1,...,Xkj is independent with a common marginal distribution F) are given.

A Note on Maximum Variance of Order Statistics from Symmetric Populations

The maximum variance of order statistics from a symmetrical parent population is obtained in terms of the population variance. The proof is based on a suitable representation for the variance of order statistics in terms of the parent distribution function.

Variance inequalities for covariance kernels and applications to central limit theorems

A simple estimate for the error in the CLT, valid for a wide class of absolutely continuous r.v.s, is derived without Fourier techniques. This is achieved by using a simple convolution inequality for the variance of covariance kernels or w-functions in conjunction with bounds for the total variation distance. The results are extended to the multivariate case. Finally, a simple proof of the classical Darmois--Skitovich characterization of normality is obtained.

Intermediate order statistics with applications to nonparametric estimation

A generalization of order statistics, is presented. Using this generalization, nonparametric confidence intervals are constructed for the quantiles of an absolutely continuous distribution. Finally, an application, concerning confidence intervals for the unique median, is given.