Asymptotic Distribution Of Likelihood Ratio Of Statistic And Wilcoxon 's Statistic

Mixing random variable sequence and association random variable sequence are two common dependent random variable sequences.Commonly used mixing sequences are:ψ-mixing,φ-mixing,β-mixing, ρ-mixing, a-mixing and sequences, where a-mixing is the weakest mixing among these mixing conditions. Rosenblatt introduced the a-mixing condition. The properties of a-mixing random variable sequence not only has been discussed, but also has been widely used in time series models.Positively associated random variables sequence and negative associated random variables are collectively called associated random variables sequences. In 1967, Esary, Proschan and Walkup introduced the concept of positively associated random vari-ables. Negatively associated random variables were introduced by Block, Savits and Sharked in 1982. The properties of associated random variables sequences were widely discussed and used.Under α-ixing samples, the asymptotic distribution of likelihood ratio statistic is established in the case of simple null hypotheses considering one dimensional parameter space. The asymptotic distribution is weighted X2-distribution. We also construct the asymptotic likelihood confidence regions for the related parameter θ and do a simulation study. The results show that the coverage probability of the asymptotic likelihood confidence regions for θ is quite good in the actual large sample testing.If the distributions of populations X and Y are unknown, the Wilcoxon two-sample statistic testing can be used to test whether the two samples were from the same popu-lation. Under the condition that sample one and sample two are independent and each sample is an associated sample, and under some mild conditions, we obtain the asymp-totic distribution of the Wilcoxon two-sample statistic. Then construct the asymptotic normally confidence regions for γ (γ= 2P(Y> X)-1) and do a simulation study.The new findings in this paper may be summarized as follows:1. This paper investigates the asymptotic distribution of likelihood ratio statistic under α-ixing samples for the first time.2. This paper improves the proof procedure of the asymptotic distribution of the Wilcoxon two-sample statistic under NA samples and investigates the asymptotic dis-tribution of the Wilcoxon two-sample statistic under PA samples. We also obtain the asymptotic distribution of the Wilcoxon two-sample statistic under mixing samples with different mixing coefficients.