Research On Ties Problem About Various Likelihood

In the research about statistics fields, we should to operate one or several groups of observations (x₁,x₂,..., x_n) of sample (X₁ ,X₂,...,X_n) which come from a parametric or nonparametric model. We often suppose that the sample observations X₁,x₂,...,x_n are independent random variables with a common CDF and not equal each other, that is, X₁ ≠ X_j for i ≠ j . If x_i = X_j for i≠ j in the observations, we say that x_i and x_j are tied, in which x_j and x_j are ties in the data.It is noteworthy that, in the application and research about empirical likelihood and other likelihood (parametric likelihood, Euclidean likelihood derived from empirical likelihood, more ordinary method which include empirical likelihood and Euclidean likelihood - empirical power divergence statistics),scholars always suppose that the sample observations have no ties in the data, that is, observations are i.i.d. and not equal each other, and it has not a systematic theoretic discussion about the cases having ties. In this paper we discuss whether it has different effect on result between the cases having ties or not under various likelihood. Enlightened by the method using by Owen(1988) to treat the problem of ties when applying empirical likelihood to observations of sample, we systematically argue the ties problem about other cases of empirical likelihood and other likelihood, and draw the conclusion independently that the result has no essential difference between observations having ties or not under the method of Euclidean likelihood and empirical power divergence statistics, so we can directly applying the method of parametric likelihood, empirical likelihood, Euclidean likelihood and empirical power divergence statistics under the hypothesis that observations have no ties, namely the sample are i.i.d. and not equal each other, and it makes the theoretical system of likelihood ratios and the likelihood methods upward more integral. The principle of traditional log likelihood is to seek for the maximum about the sum of every observation's weight.In this paper,we skillfully change the sum of probability to the sum of weight.So we can draw the conclusion that the expression of likelihood ratios has no relation with the fact whether the observations of sample have ties or not, hence we can construct the profile empirical likelihood ratio function with conditional restriction and get the same confidence regions. Using this skill the thesis get the conclusion that when we applying empirical likelihood , Euclidean likelihood and empirical power divergence statistics, the result are absolutely parallel whether the observations of sample have ties or not, and proved it has no effect to result in parametric likelihood if the observations of sample have ties.