Local Influence Analysis of Generalized Linear Model

Nearly thirty years, the diagnosis and influence analysis of linear regression model has been fully developed. So far the local influence analysis of the generalized linear model has not yet seen in the literature. In this paper, local influence is discussed. Then, concise influence matrix is obtained. At last, an example is given to illustrate our results.


Introduction
Local influence analysis is proposed from the viewpoint of differential geometry [1].Nearly thirty years, the diagnosis and influence analysis of linear regression model has been fully developed [2,3].Regarding the generalized linear model, diagnosis has some results [4].So far the local influence analysis of the generalized linear model has not yet seen in the literature, this paper attempts to study it.

Local Influence
Let  be an unknown k-dimensional parameter, whose domain is an open subset of Euclidean space .
is a object function(for example, likelihood function, punishment log-likelihood function). is a n-vector which denotes disturbed factor for example weightd or tiny shift.Let

 
M  is the disturbed model, whose where is the second derivatives of l    l  with re- spect to  , and The influence matrix is given by Formula (2.4) shows that the maximal influence curvature max , where 1  is the eigenvalue of F  whose absolute value is maximal, and max is the corresponding eigenvector which is called the direction of maximal influence curvature.Escobar and Meeker (1992) pointed out that the diagonal value of influence matrix also is the important diagnostic statistics.d

Local Influence Analysis of Model
Considering non-parametric regression model , where is the measure observations.

  
, 1 ,2 , denote that i submit to exponential distributions, the corresponding density function is where , , ,  are the first and second deriva- Supposed that the MLE of  in (3.1) is  , and  submits to , , , , then the weighted perturbation model can be shown that Substituting this result into (2.3)yields where . The second derivatives of Substituting (3.1.2) and (3.1.3)into (2.4),we obtain the corresponding influence matrix Here d  denotes the direction of maximal influence curvature.

Response Variable Perturbation Model
Suppose that Y Y     , , then the response variable perturbation model can be shown that  0 0, 0, , 0 Substituting this result into (2.3)yields The second derivatives of Here r denotes the direction of maximal influence curvature.The regression analysis of kyphosis data are as follows (Table 1).
The local influence analysis results of kyphosis data are as follows (from Figures 1-3).
Figures 1 and 2 show that the sixth, forty-third, fifty-third and the eightieth data are influential points, Figure 3 shows that the first, second, third and fourth data are influential points.Actually, the direction of maximal influence curvature also shows that the first, second,    third and fourth data are influential points.This also proves that the above method is effective.

d 4 .
An Illustrative Example(Kyphosis Data) Now we consider an example as the illustration for the above results.Considering a kyphosis data (see[6]).There are 81 patients who have been treated with chiropractic.There are four variables: kyphosis, Age, Number and Start.Wang xiaoming (2005) ultized a linear semi-parametric model to fit this test data.

Figure 1 .
Figure 1.The diagonal value of influence matrix F ω .

Figure 2 .
Figure 2. The diagonal value of influence matrix F r .

Figure 3 .
Figure 3.The direction of maximal influence curvature d ω .

Table 1 . The regression analysis of kyphosis data.
r d