Browsing by Subject "Nonparametric statistics -- Asymptotic theory"
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Item Aligned rank tests for repeated observation models with orthonormal design(Texas Tech University, 2004-08) Omolo, Bernard OgunaRank tests are known to be distribution-free for simple linear models, where the observations are i.i.d. For general linear models with nuisance parameters, however, the alignment principle can be applied to obtain asymptotically distribution-free rank tests. This is especially so when the centered design matrices have full rank, as is the case in Kraft and van Eeden [19], Adichie [1] and Chiang and Puri [4], among others. Motivated by the example of testing linearity in a nonparametric regression model, however, we will be dealing with models whose centered design matrices are not of full rank. More specifically, the asymptotic distribution of the aligned rank statistics will be obtained under the null hypothesis and local alternatives for testing a linear hypothesis in a repeated observation model with orthonormal design matrix. These asymptotic distributions are of the chi-square type and independent of the choice of aligner, as in the full rank case. Some simulations of the power function when the errors have a Cauchy distribution are included. The theory is presented in a self-contained manner, and based on the Chernoff- Savage rather than the Hajek approach. In principle this would allow us 1o also deal with the asymptotics under fixed alternatives, although this option is not presented to completion. This approach can be extended to scale models as well as multivariatc models. Although these topics are only briefly considered, interesting additional insight, is gained in the independence of the aligner. In the location model, this independence is obtained due to a suitable choice of the test statistic so that cancellation takes place. In the scale model, on the other hand, the aligner docs not e\en appear in the expansion of the basic components of the test statistic due to the particular form of scene functions employed for scale problems.Item Changepoint detection and estimation using nonparametric procedures(Texas Tech University, 1989-05) Balakumar, SivanandanNot availableItem The rank transform procedure in the two-way layout with interaction(Texas Tech University, 1991-12) Choi, Young HunOne particular interest of this paper is to develop the properties One particular interest of this paper is to study the asymptotic theory of the rank transformed statistic, computed on ranks or rank scores, for testing for interaction in a two-way layout. Some theorems and sufficient conditions are derived with lemmas and corollaries. Many exact, small sample results are also obtained here for the first time. These results will be useful for the other theoretical studies of the rank transform procedure in experimental designs. Let Xijn, i = 1... ,1, j = 1 . . . , J, and n = 1 . . . , N, be independent random variables such that Xijn has the continuous distribution function Fij. Further let Fi_ and Fj denote the average distribution function for block i and the treatment j , respectively. Then the results obtained in this paper can be summarized as follows. When both main effects are present {Fij ^ Fi, and Fij / F_j for at least one i and j) in a general two-way layout, without the restriction of a linear model, then under the null hypothesis of no interaction and the assumptions, which are provided as equations (2.5), (2.6) and (2.8) in this paper, the rank transformed F statistic for interaction converges in distribution to an (I-1)(J-1) degrees of freedom chi-squared random variable divided by (I-1)(J-1). For a two-by-two factorial design if only if there is one main effect ( Fij = Fi, or Fij = F,j for aU i and j ), imder the null hypothesis of no interaction the rank transformed F statistic for interaction converges in distribution to an (I-1)(J-1) degree of freedom chi-squared random variable divided by (I-1)(J-1).