Statistical Inference Of Normal Linear Regression Model With Double Power Conversion

It is well known that the normal linear regression model:Y=X β+ε, ε~N n (0, σ2In)is usually used to deal with the relationship betweenrandom response variable Y and explanatory variable X, under suchcircumstances, Y should satisfy the so-called Gauss-Markovconditions:Y~N n(Xβ, σ2). It is easy to understand that not all random responsevariables can meet this requirement. When it cannot meet the requirement ofGauss-Markov, we can introduce parameter λ and let Y transfer to Y (λ), sothat Y (λ)can come from the following normal linearmodel:Y (λ)=X β+ε, ε~Nσ2n(0,). In all the possible transformations, Box-Coxtransformation has been proposed early and it is the most mature transformation. Ithas produced a large number of literature. However, after analyzing, it is easy to findthat there is a truncation problem of Box-Cox transformation. With the existence ofthis problem, on one hand we can not simulate the model, on the other hand, it makesthe opinion thatY (λ)obtained by Box-Cox transformation and came from normallinear model is not appropriate. In order to overcome the truncation problem of theBox-Cox transformation, Yang[2]introduced the dual power transformation. It isshown that this new transformation has properties similar to those of the Box-Coxtransformation. It is very useful and flexible in modeling and analysis of economicdurations and medical/engineering event-times. In this paper, the key motivation is tostudy the statistical diagnostic of parameters in linear regression model under dualpower transformation. The study includes two aspects. Firstly, we discuss theexistence and uniqueness of the maximum likelihood estimateλ MLfor the transformation parameter λ and compare the difference between the maximumlikelihood and least square estimates for parameters, then we test the transformationparameter λ and construct the confidence region of it. The results show that theonly maximum likelihood estimate for parameter λ in linear regression model underdual power transformation exist. The distribution ofλM Ldepends not only on λ alsothe regression coefficients and variance; the probability ofλ MLis strictly positivewhich indicates thatλ MLis a singular random variable without density function. Thesimulation study shows that the MLE is better than the LSE in the case that thetransformation parameter or variance is large. In addition, it is also indicating thatthere do not exists significant difference between two estimates for large sample size.The simulation study shows that the power function is closely related to the variancein linear regression model, the power function is larger and more unstable in the casethat variance is smaller.