A real data example illustrates the practical application of the methodology. The finite-sample performance of the weighted sign test is explored through a simulation study which shows that the proposed approach is very competitive. A multivariate weighted t-test is also introduced. Using Pitman asymptotic efficiency, we show that appropriate weighting can increase substantially the efficiency compared to a test that gives the same weight to each cluster. Stochastic growth processes abound in the biology of parasitism, and one mathematical tool that is particularly well suited for describing such phenomena is. Several approaches for estimating these weights are presented. These weights depend on the cluster sizes and on the intracluster correlation. We propose a new theoretical modeling framework for seismicity based on a recently introduced family of invariant Galton-Watson (IGW) stochastic branching. Optimal weights maximizing Pitman asymptotic efficiency are provided. test statistic is also given for a local alternative model under multivariate normality. Under weak assumptions, the test statistic is asymptotically distributed as a chi-squared random variable as the number of clusters goes to infinity. A family of multivariate weighted sign tests is introduced for which observations from different clusters can receive different weights. Doyle and Snell 1984 Hughes 1995 Lyons and Peres 2016 Spitzer 2001). In this paper we study simple random walks, a well-known object of the study in probability theory Spitzer ().The known results on simple random walks can be found in many sources (e.g. A natural sufficient condition is given for a Galton-Watson process in a varying environment to have a single rate of growth that obtains throughout the. Ore Oduba, classical singer Carly Paoli, Galton Blackiston and Gareth Ward. We consider the multivariate location problem with cluster-correlated data. 1.1 Formulation of the Problem and the Literature. Russell Watson, Jimmy Doherty, Lesley Waters and Lenny Carr-Roberts.
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