Edgeworth expansion for signed linear rank statistics with regression constants

An Edgeworth expansion with remainder o(N^?1) is obtained for signed linear rank statistics under suitable assumptions. The theorem is proved for a wide class of score generating functions including the Chi-quantile function by adapting van Zwet's methodand Does's conditioning arguments.     

Stopping times of one-sample rank order sequential probability ratio tests

Weed, Bradley and Grovindarajulu (1974) propose one-sample probability ratio tests based on Lehmann alternatives. They also study the finite sure termination of the stopping times. Motivated by Stein's proof of (1946) of the termination of a sequential probability ratio test (SPRT) in the case of independent and identically distributed (i.i.d.) random variables and the work of Sethuraman (1970) for the two- sample rank order SPRT, we obtain a very mild condition (namely, that a certain random variable U(Z) is not identically zero) for the finite sure termination of the existence of the moment generating function (m.g.f.) for the stopping time of one- sample rank order SPRT's.     

Some new classes of multivariate survival distribution functions

The concept of the univariate mean remaining life (m.r.l.) function is generalized to the multivariate case. The multivariate mean remaining life (m.m.r.l.) function is utilized to introduce four new classes of multivariate survival distribution functions (s.d.f.'s). Each of these classes is a new generalization of the univariate decreasing mean remaining life (DMRL) class of s.d.f.'s. The duals of these classes are introduced. Some properties, physical interpretation, and relationships among these classes are investigated. Also for each case, the class of s.d.f.'s common in a class and its dual is characterized.     

An investigation of Kendall's τ modified for censored data with applications

To accommodate testing for independence in bivariate data subject to censoring, several modifications of Kendall's τ are discussed. An extensive computer simulation is done to investigate power properties of these modifications under alternatives of the bivariate normal or bivariate exponential types. The statistics are then applied to available heart pacemaker patient survival data.     

Use of the experimentwise error rate in simultaneous tests of main effects in two-way disproportionate ANOVA without interaction

It is argued that the probability of committing at least one type I error should be reported when testing the main effects simultaneously in a two-way disproportionate ANOVA without interaction. The circumstances where the two F-statistics depart appreciably from statistical independence are characterized, and it is pointed out that procedures now exist for evaluating the bivariate F-probabilities when required. The augmented analysis is illustrated with a numerical example and an extension is offered for assymmetric BIBD's with random block effects.     

Wald's approximations to the average run length in cusum procedures

Wald's approximation to the ARL(average run length in cusum) (cumulative sum) procedures are given for an exponential family of densities. From these approximations it is shown that Page's (1954) cusum procedure is (in a sense) identical with a cusum procedure defined in terms of likelihood ratios. Moreover, these approximations are improved by estimating the excess over the boundary and their closeness is examined by numerical comparisons with some exact results. Some examples are also given.     

Some complex variable transformations and exact power comparisons of two-sided tests of equality of two Hermitian covariance matrices

Power studies of tests of equality of covariance matrices of two p-variate complex normal populations σ₁ = σ₂ against two-sided alternatives have been made based on the following five criteria: (1) Roy's largest root, (2) Hotelling's trace, (4) Wilks' criterion and (5) Roy's largest and smallest roots. Some theorems on transformations and Jacobians in the two-sample complex Gaussian case have been proved in order to obtain a general theorem for establishing the local unbiasedness conditions connecting the two critical values for tests (1)–(5). Extensive unbiased power tabulations have been made for p=2, for various values of n₁, n₂, λ₁ and λ₂ where n₁ is the df of the SP matrix from the ith sample and λ₁ is the ith latent root of σ₁σ^-1₂ (i=1, 2). Equal tail areas approach has also been used further to compute powers of tests (1)–(4) for p=2 for studying the bias and facilitating comparisons with powers in the unbiased case. The inferences have been found similar to those in the real case. (Chu and Pillai, Ann. Inst. Statist. Math. 31.     

Optimal weighing designs with a string property

We consider the usual (spring balance) weighing design set-up with the design matrix having a string property meaning thereby that in every row of it, there is exactly one run of 1's (the rest of the elements being 0's). We have investigated some interesting features of such matrices and used them in deriving various optimality results.     

Characterization of certain incomplete block designs

It is shown that certain inequalities known for partially balanced incomplete block (PBIB) designs remain valid for general incomplete block designs. Some conditions for attaining their bounds are also given. Furthermore, the various types of PBIB designs are characterized by relating blocks of designs with association schemes. The approach here is based on the spectral expansion of NN' for the incidence matrix N of an incomplete block design.     

Estimation of a linear transformation and an associated distributional problem

A multivariate “errors in variables” regression model is proposed which generalizes a model previously considered by Gleser and Watson (1973). Maximum likelihood estimators [MLE's] for the parameters of this model are obtained, and the consistency properties of these estimators are investigated. Distribution of the MLE of the “error” variance is obtained in a simple case while the mean and the variance of the estimator are obtained in this case without appealing to the exact distribution.     

Orthogonal decomposition of point-symmetry for multiway tables

For multiway contingency tables, Wall and Lienert (Biom. J. 18:259–264, 1976) considered the point-symmetry model. For square contingency tables, Tomizawa (Biom. J. 27:895–905, 1985) gave a theorem that the point-symmetry model holds if and only if both the quasi point-symmetry and the marginal point-symmetry models hold. This paper proposes some quasi point-symmetry models and marginal point-symmetry models for multiway tables, and extends Tomizawa’s (Biom. J. 27:895–905, 1985) theorem into multiway tables. We also show that for multiway tables the likelihood ratio statistic for testing goodness of fit of the point-symmetry model is asymptotically equivalent to the sum of those for testing the quasi point-symmetry model with some order and the marginal point-symmetry model with the corresponding order. An example is given.     

Bounding convergence rates for Markov chains: An example of the use of computer algebra

Kolassa and Tanner (J. Am. Stat. Assoc. (1994) 89, 697–702) present the Gibbs-Skovgaard algorithm for approximate conditional inference. Kolassa (Ann Statist. (1999), 27, 129–142) gives conditions under which their Markov chain is known to converge. This paper calculates explicity bounds on convergence rates in terms calculable directly from chain transition operators. These results are useful in cases like those considered by Kolassa (1999).     

Strawderman-type estimators for a scale parameter with application to the exponential distribution

It is shown that Strawderman's [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198] technique for estimating the variance of a normal distribution can be extended to estimating a general scale parameter in the presence of a nuisance parameter. Employing standard monotone likelihood ratio-type conditions, a new class of improved estimators for this scale parameter is derived under quadratic loss. By imposing an additional condition, a broader class of improved estimators is obtained. The dominating procedures are in form analogous to those in Strawderman [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198]. Application of the general results to the exponential distribution yields new sufficient conditions, other than those of Brewster and Zidek [1974. Improving on equivariant estimators. Ann. Statist. 2, 21–38] and Kubokawa [1994. A unified approach to improving equivariant estimators. Ann. Statist. 22, 290–299], for improving the best affine equivariant estimator of the scale parameter. A class of estimators satisfying the new conditions is constructed. The results shed new light on Strawderman's [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198] technique.     

Characterization of the geometric distribution using properties of order statistics

Using certain properties of order statistics, the geometric distribution has been characterized when the components are independent and identically distributed. When the components are independent, the geometric distribution has been characterized in the class of either IFR or DFR discrete distributions. In particular, Ferguson's (1967) characterization theorem for independent components in a sample of size two has been extended in several directions.     

Goodness-of-fit tests for exponentiality based on a loss-of-memory type functional equation

Three goodness-of-fit tests for exponentiality based on the functional equation characterization 1?F(2x)=[1?F(x)]² for every x?0 are proposed and shown to compare well to several popular tests against common alternative cdf's. Small-sample critical values for α=0.10,0.05 are developed for the two superior test statistics and the asymptotic null-distributions are characterized.     

Robustness of the linear discriminant function to nonnormality: Johnson's system

The asymptotic distribution of the Errors of Misclassification in using the Linear Discriminant Function is investigated here. The purpose is to study the effects of nonnormality on these errors. The class of distributions considered is the Johnson's system. Each of the three random variables can be transformed to normality. In one particular case numerical evaluations are made, based on which it is possible to recommend whether or not it is necessary to make the transformation prior to classification. In a parallel study, we present similar results for the Edgeworth Series distribution, where the random variables cannot be transformed to normality.     

Star-shaped distributions and their generalizations

Elliptically contoured distributions can be considered to be the distributions for which the contours of the density functions are proportional ellipsoids. We generalize elliptically contoured densities to “star-shaped distributions” with concentric star-shaped contours and show that many results in the former case continue to hold in the more general case. We develop a general theory in the framework of abstract group invariance so that the results can be applied to other cases as well, especially those involving random matrices.     

Subhypotheses testing against restricted alternatives for the Cox regression model

For the Cox regression model involving both design and concomitant variates, testing of subhypotheses against restricted alternatives based on the partial likelihood scores is considered. Orthant, ordered as well as the ordered orthant alternatives are considered in this context. Roy's union-intersection principle plays a vital role in this development. Properties of the proposed tests are also studied.     

On Tiku's robust procedure — a Bayesian insight

Tiku's robust procedure for testing mean and variance from nonnormal universe is examined from the Bayesian viewpoint. The posterior distribution of the scale parameter is derived and then approximated by a Laguerre polynomial expansion while the posterior distribution of the location parameter is approximated by a linear combination of t-distributions. For the example with Darwin's data, the approximations appear to be extremely good.     

Gaussian approximation of signed rank statistics process

We consider the signed linear rank statistics of the form

$\text{S}{}_{\mathrm{\Delta N}}\text{=}\text{∑}\text{i=1}\text{N}\text{c}{}_{\mathrm{Ni}}\text{ø(}\text{R}{}^{\mathrm{\Delta }}{}_{\mathrm{Ni}}\text{(N+1)}\text{)}\text{sgn}\text{Y}{}^{\mathrm{\Delta }}{}_{\mathrm{Ni}}$

where the c_Ni's are known real numbers, Δ∈[0,1] is an unknown real parameter,R^Δ_Ni is the rank of |Y^Δ_Ni| among |Y^Δ_Nj|, 1≤j≤N, ø is a score generating function, sgn y=1 or -1 according as y≥0 or <0, and Y^Δ_Nj, 1≤j ≤N, are independent random variables with continuous cumulative distribution functions F(y?Δd_Nj), 1≤ j≤N, respectively where the d_fNi's are known real numbers. Under suitable assumptions on the c's, d's, φ and F, it is proved that the random process {S_ΔN?S_0N?ES_ΔN, 0≤Δ≤1}, properly normalized, converges weakly to a Gaussian process, and this result is also true if ES_ΔN is replaced by Δb_N, where

$\text{b}{}_{\mathrm{N}}\text{=4}\text{∑}\text{i=1}\text{N}\text{c}{}_{\mathrm{Ni}}\text{d}{}_{\mathrm{Ni}}\text{∫}\text{0}\text{∞}\text{ø′(2F(x)?1)?}{}^2\text{(x)}\text{d}\text{x}\text{and}\text{?=F′}$

. As an application, we derive the asymptotic distribution of the properly normalized length of a confidence interval for Δ.