Browsing by Author "Fernando, W.T.P.S."

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Convergence of multivariate isotonic regression
(University of Kelaniya, 2006) Fernando, W.T.P.S.; Kulatunga, D.D.S.
Statistical inference in the presence of order restrictions is an important area of statistical analysis. Isotonic regression theory plays a key role in this field. Let K= {1,…,k} be a finite set on which a partial order « is defined. A real vector (θ1, …,θk) is said to be isotonic if μ, υ ∈ K, μ « υ imply θμ ≤ θυ. Given real numbers x , , xk 1 K and positive numbers k w , ,w 1 K , a vector ( ) k θ θ ) K ) , , 1 is said to be the univariate isotonic regression of k x , , x 1 K with weights k w , ,w 1 K if it is isotonic and minimizes ( ) ν ν ν ν x θ w kΣ= − 1 2 under the restriction that (θ1,…,θk) is isotonic. Isotonic regression is closely related to the maximum likelihood estimate of ordered parameters of univariate normal distribution and some other univariate distributions. Various algorithms are given in the literature for computing univariate isotonic regression. Multivariate generalization of the isotonic regression and multivariate extensions of related theorems are given and proved by Sasabuchi, Inutsuka and Kulatunga (1983, 1992). A p × k real matrix ( ) k θ θ ,...,θ 1 = is said to be isotonic with respect to the partial order « , if μ, υ ∈ K, μ « υ imply μ ν θ ≤θ , where μ ν θ ≤θ means all the elements of ν μ θ −θ are nonnegative . Given p-dimensional real vectors k x , , x 1 K and p × p positive-definite matrices k Λ , ,Λ 1 K , a p × k matrix ( ) k θ θ ) K ) , , 1 is said to be the multivariate, in fact p-variate, isotonic regression of k x , , x 1 K with weights 1 1 1 , , Λ − Λ − K k if it is isotonic and satisfies min ( ) ( ) ( ) 1 ( ), 1 1 ' 1 * ' θ ν ν ν ν ν ν ν ν ν ν ν ν θ θ θ θ ) ) − Λ − = − Λ − − = − = Σ x x Σ x x k k Where min * (.) θ to denotes the minimum for all θ isotonic with respect to the partial order. An algorithm for the computation of multivariate isotonic regression is given in Sasabuchi et al. (1983, 1992). This algorithm involves iterative computation of univariate isotonic regression. The convergence of this algorithm is also studied there and it has been observed that the convergence follows only under certain conditions. (Corollary 4.1 of Sasabuchi et al (1992).) However, the simulation study conducted in Sasabuchi, Miura and Oda (2003) under special cases, has shown that the condition given in Corollary 4.1 of Sasabuchi et al. (1992) is not necessary for the convergence of this algorithm. We have written a Fortran subroutine for the computation of multivariate isotonic regression and also noted that the algorithm converges in general. This motivates us to consider a proof for the convergence of this algorithm. In this study we give a proof for the convergence of this algorithm in the bivariate case.
Tests against tree order restriction in Poisson intensities
(University of Kelaniya, 2006) Fernando, W.T.P.S.; Kulatunga, D.D.S.
We consider a situation in which one wishes to compare several Poisson intensities with a control or standard when it is believed that the intensities are higher than the control. For instance, if λ1 is the average accident rate per k.m. of a truck driver who has undergone an extensive training in driving and if λj, for j=2,3,...,k, are the average accident rates of the jth truck drivers without any prior training; and if the intensities are believed to produce at least as large an intensity as the control, then one would expect that λ1 ≤ λj for j=2,3,…,k. If xi is the number of accidents incurred by the ith of the k truck drivers and ti be the number of k.m. he drove and λi be the average accident rate per k.m., we can mathematically formulate this situation as follows: Suppose X1,…,Xk are independent Poisson variables with means μi = λiti and let Ho: λ1= λ2=…= λk and H1: λ1 ≤ λj , for j=2,3,…,k, where λ1 is the control intensity and λj, for j =2,3,...,k, are the other intensities. The ordering specified by H1 is called a tree ordering. We are interested in testing Ho versus H1-Ho . The likelihood ratio test for Ho versus H1- Ho is computed and we derive the asymptotic distribution of it. Robertson and Wegman (1978) considered order restricted tests for members of the exponential family. Their results can be applied in the testing situation considered here only if the ti are all equal. Some results are also obtained in the literature for other order restrictions (Magel & Wright (1984) and Barmi et al.(1996)). In this study we obtain explicit formulae for the null distribution of the test statistic under tree ordering.