Adjusted Empirical Euclidean Likelihood And Its Properties

Empirical likelihood(EL) is often viewed as one of the most popular and important methods employed in nonparametric statistics. We have demonstrated their great charm, because it has successful experience of application in different areas, such as econometrics, biomedicine, market research and other academic fields. Therefore, it has become one hot topic in both researches of statistical theory and application research.The main purpose of this paper is to investigate the adjusted empirical euclidean likeli-hood(AEEL) based on the analysis of previous studies. It has been conducted on the basis of comprehensive analysis. The concept of empirical likelihood is firstly provided by Owen(1988). Then Qin and Lawless(1994) tried to apply this method into more general semi-parametric mod-els. Empirical euclidean likelihood(EEL) use euclidean distance instead of likelihood distance in empirical likelihood, to get a kind of nonparametric methods. It has completely similar nature of empirical likelihood, therefore it can be regarded as a promotion version of empirical likelihood. We plan for one point deficiencies of empirical euclidean likelihood method's theory and applica-tion, namely, the convex hull problem, so that the new method retains all the best properties. we can overcome the shortcoming of EL, in the estimate equation Erg(y,θ)= 0, the estimation ofθexists if and only if the convex hull{g(xi,θ), i=1,…, n} contains 0 as its inner point. Then we give a simple illustration to the convex hull and our adjustment.We mainly consider the issues that we use a novel adjustment to the empirical euclidean likelihood. We choose the idea from the adjusted empirical likelihood(AEL) proposed by Chen,Variyath and Abraham(2008). In this paper, we introduce the adjusted empirical euclidean likelihood(AEEL) into semiparametric models and show that the role of it is parallel to paramet-ric likelihood. We use it to combine partial information and show the AEEL estimation for both parameter and distribution function is asymptotically efficient. In addition we discuss the adjusted empirical euclidean likelihood function's construction, distribution function and parameter esti-mation (including the point estimation and interval estimation). In theory, it gives the similar properties of the foregoing estimates and show the superiority of the proposed method. Then, in line with our simulation from the small samples, indicating that the adjusted empirical euclidean likelihood is much faster to compute. The confidence regions constucted via the adjusted empirical euclidean likelihood are found to have closer to nominal coverage probabilities. Theoretically, we discover that the adjusted empirical euclidean likelihood, comparing with the empirical likelihood or empirical euclidean likelihood, has completely similar properties. In fact, the algorithm for the AEEL is much quicker to converge, In some cases (such as two-dimensional case). Thus, the new method has the potential to improve the undercoverage problem due to the small sample size or high dimension effectively. Experience from the adjusted empirical euclidean likelihood interval estimation is possible to get high coverage rate. The one most important thing is that the adjusted empirical euclidean likelihood thoughts and calculation are quite simple to conduct. From the view point of practical angel, it has higher application value.The main significant results of this paper are illustrated as followings:1.We use a novel adjustment to the EEL so that the new method retains all the best properties. We discuss the asymptotic properties of the AEEL estimation for parameter and distribution, and also demonstrate that AEEL ratio statistics for parameters have asymptoticχ2 distributions, prove consistency and asymptotic normality of it. The result of this paper is something which have not discussed before, it is entirely new achievements.2.In fact,It is also shown to be very effective in solving some application problems associated with the use of AEEL. It is well know that, we can overcome the shortcoming of EL that the convex hull{g(xi,θ), i=1,…, n} can contain 0 as its inner point. For all cases there are solutions to reduce the difficulty of complicated computation.3.Under certain conditions, the higher coverage probabilities of the new confidence regions can be obtained in this paper compared with previous.4.This paper conclusion can be rich and improved the theory of the EEL. and provide conve-nient tool to the practical application of workers.