Dia Gnostic Theory And Applications And In Regression Model

During past decades, the heteroscedasticity in regression is widely recognized, and residual plotting schemes are suggested for assessing heteroscedasticity and for identifying outliers. In this thesis we discuss many aspects of the regression model with heteroscedasticity .The paper is organized as follows.In Chapter l.we give introduction. In C'hapter 2.we develop diagnostic theory and method based on the quasi-residuals for a class of regression model with heteroscedasticity under fixed design. In C'hapter 3.we study the estimational and diagnostic theory and method for a class of semiparametric regression model with heteroscedasticity under random design. C'hapter 4 discusses Weighted Quasi-likelihood Estimation and Diagnosis for Heteroscedasticity in Extended Generalized linear Models.I. Diagnostic theory and method based on the quasi-residuals for a class of regression model with heteroscedasticity under fixed designConsider regression modelUsually the estimated function g(-) is bound to be drawn towards the aberrant responses that are far from the bulk of the observed responses. Therefore, it is often difficult to tell the difference between an outlier and an ordinary point due to perhaps the inherent flexibility of the regression model (O.L.I). .In this paper we suggest an estimator f/ of the residual variancecr2 in above model (0.1.1) based on a quasi-residual method. Also we suggest a diagnostic plot hoping it can identify outliers that could not be found by ordinary residual plots. Besides, we obtain asymptotic distribution properties of statisticHeuristically, if the heteroscedasticity occurs, Zn shouldn't be large, which enables .us to test the heteroscedasticity.1. Statistics of the Quasi-residual-based EstimatorIn regression model (0.1.1), we assume design forms an asymptotically regular sequence in the sense of [ Sacks and Ylvisaker (1970)] generated by a design density , which implies Definition 1. The quasi- residuals c"'s are denned as follows:Definition 2. The quasi-residuals-based estimators for the residual variance function at /,'s are defined asAssumption 1 The error variables f/'s are independent with . a constant.Assumption 2 The mean function , where 0 < n < I.Assumption 3 Residual variallce ft1l1ctioll a'' ) E Lil>scllitz,i([0. l] ).u.here;] > l.t\fe noxv consider the trade--off 1,et\\'een 1>ias al1d \,ariallce. \\'e starttxith the anal},sis of ll1ean square error NISE. u'llich is a col111,il1atio11 of thesquared bias al1d the variance of a'(ti ).Theorem 1.2. Asymptotic PropertiesIn this sectiol1 lx'e derive the as},1nptotic distri1,ution of the quasi-residuals--based statisticTheoren1 2. l-'1l(ler Assul11lJtiol1s l--f3. Defil1itio1ls 1--2 allfl tl1e col1\'(l ).3. An Illustrative Artificial ExampleIn our model (0.1 .1 ). fOllotxing the setting of Dette and Alul1k (lf)f)8),let g(t) = l + sin(t), a(t) = aexp(ct), xxl1ere a = nd al1d c = 0.5. Alsolet c .\!(0, 0.l ). t\re select ti's as equally spaced l00 poil1ts flon1 0 to l.The scatter plot of tl1e data is shotx'll ill Figure 2 (a). F1ol11 tlle data scatterplot fOr {ti. gi}. it is very hard to say xtl1ich Point is outlier tllough xx'e n1akecase 50 an outlier (the lotxer ol1e among the tlx,o in tlle top of the lniddle).This data ca'n be easily fit by ordinary least squares lnetllods. alld the relatedresidual plot fOr {ti, ci} is in Figure 2 (b), \vl1ere case 50 is the lligllest l,oillt.But there lacks a clear cut f1om the other l>oints. So it is uIlcoll\,inciIlg thatany poiht is an outlier. Also it is hard to tell \1'hetl1er lleteroscedasticit}'exists.Figure 0-1: (a) The data plot {ti. gi}: (l>) the resilot after or'liIlar3' lpasts'll.lares fit.1\b thell sin1l2l}' select a set of...