Likelihood Estimation Of Regression Model Under Transformation

In this paper, we study the modelThe motivation for the model is that after transforming the response Y via h, a linear model with homoscedastic error isassumed. Many people study this model when it is parametric forms or nonparametric forms for h. This model is popular in life testing. In this paper, we try to use a piecewise linear function to approach h when it is nonparametric form. So the problem turn nonparametric to parametric and we can get likelihood estimation. Latter we discuss quality of likelihood estimation when h is parametric. This paper has three parts.In first part, we consider h is a smooth,invertible and strictly monotonically increasing function on R and distribution of e is completely specify. We suppose(x1,y1),(x2,y2),... ,(xn,yn) is a sample from the model. Let y(1) = t0 (t1,b1),...(tm,bm and 6 is coefficient of hm. Logarithm likelihood function isIf we use hm approch to h and letthen we can obtain the following proposition.Proposition1.1 Suppose sample is fixed, if max(tj - tj-1) - 0, thenSo we can use In Ln(hm, ) approch to ln Ln(h, ) and get likelihood estimation. We need slove the problem :Because the pronlem is intractable and simulation or analytic approximation is needed. We can know merely difference between estimation with true value by simulation. It is convenient that we use appropriate likelihood function.In two part, we consider a special model for h(-) = ay + b,where a > 0. We obtain the following estimation.is invertible . In three part, we also consider such model and get conditional distribution of estimation. We can obtain the following results.Theorem2.1 Let EXXT is invertible , then a, b, are consistent estimation, that is to sayTheorem3.2 Estimation of a is a =1, ), where x2(n - p - 1, ) is noncentral x2-distribution, n - p - 1 is degrees of freedom, is noncentrality parameter. For the moreCYTheorem3.3 Estimation of is then CYn|Xn and Tn|Xn areindependent andfor the moreTheorem3. 4 Estimation of b is b = , then BYn|xn and Tn|xn areindependent andfor the more...