# LaTeX2HTML 96.1 (Feb 5, 1996)
# Associate images original text with physical files.
$key = q/{_inline}$gamma=fracalpha_i1-beta_i${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}Omega=left[arrayccc1;SPMamp;.2;SPMamp;0.2;SPMamp;1;SPMamp;-.10;SPMamp;-.1;SPMamp;1arrayright]{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}Sigma=left[arraycccsigma_1^2I_N;SPMamp;;SPMamp;sigma_1pI_N;SPMamp;ddots;SPMamp;sigma_p1I_N;SPMamp;;SPMamp;sigma_p^2I_Narrayright]=OmegaotimesI_N{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$(y_t,X_t)${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$hatthetatotheta^*${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}I(lambda^*)=sum_k=1^Kpartialoverpartiallambdalogp_k(lambda^*)partialoverpartiallambda'logp_k(lambda^*)p_k(lambda^*).{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$theta${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}fracpartialpartialtheta_kl_N(theta)=fracN_ktheta_k-fracleft(N-N_1-cdots-N_K-1right)left(1-theta_1-cdots-theta_K-1right)=0.{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$|H(hattheta)-H(theta^*)|to0${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$beta${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}I{tilded_i=j}=P(j|X_i,theta^*)+epsilon_i{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$mboxmed(epsilon_t|X_t)=0${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$hatlambda${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$Ktimes1${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$hattheta_N${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$tau^*_Nphantom,_LongrightarrowatopdtildeZsimN(0,sigma^2)${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}logp_k(lambda^*)=lambda^*_k-logleft[1+sum_j=1^K-1exp{lambda^*_j}right],{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$Eleft[I{x_i=k}I{x_i=l}right]=0${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}f(x_1,ldots,x_N|theta)=prod_k=1^Ktheta_k^N_k.{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}${tildeX_1,ldots,tildeX_N}${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$hatlambda_k=exp{hattheta_k}${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$(lambda_1,ldots,lambda_K-1)${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$tau${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}tau^*_N=1oversqrtNsum_i=1^Nepsilon_i=1oversqrtNsum_i=1^Nleft[I{d_i=j}-P(j|X_j,theta^*)right].{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$hattau_M${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$H_N(hattheta)toH(theta^*)${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$i=N+1,ldots,N+M${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}Eleft[widehattheta_kright]=Eleft[fracN_kNright]=frac1NEleft[N_kright]=frac1NNtheta_k=theta_k.{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}beta^*=argmin_betaE{|tildey-tildeXbeta|}.{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$Sigma=calI_N^-1(theta)${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}hattheta_k=exp{hatlambda_k}over1+sum_k'=1^K-1exp{hatlambda_k'}.{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}X=left[arraycccX_(1);SPMamp;;SPMamp;0;SPMamp;ddots;SPMamp;0;SPMamp;;SPMamp;X_(p)arrayright],qquadX_(j)=left[arraycX_1jvdotsX_Njarrayright].{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$N_k${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}fracpartial^2partialtheta_kpartialtheta_ll_N(theta)=-fracleft(N-N_1-cdots-N_K-1right)left(1-theta_1-cdots-theta_K-1right)^2{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$kneql${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}mboxmed(tildey)=argmin_muE{|tildey-mu|}.{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}sup_betainThetaleft|1overTsum_t=1^T|y_t-X_tbeta|-E{|tildey-tildeXbeta|}right|to0,{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$tilded_i${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$j=1,ldots,J${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$hattau_N=0${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$I(lambda^*)${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$i=1,ldots,N${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$SigmatimescalI_N(theta)=I${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$I{tilded_i=j}${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}Sigma=left[arraycccOmega;SPMamp;;SPMamp;0;SPMamp;ddots;SPMamp;0;SPMamp;;SPMamp;Omegaarrayright]=I_NotimesOmega{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$lambda^*_k=log(theta^*_k)${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}fracpartial^2partialtheta_k^2l_N(theta)=-fracN_ktheta_k^2-fracleft(N-N_1-cdots-N_K-1right)left(1-theta_1-cdots-theta_K-1right)^2{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$(K-1)times1${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$hattheta=p(hatlambda)${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$e_k${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$hatbeta_Ttobeta^*${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}Eleft[widehattheta_kwidehattheta_lright]=frac1N^2Eleft[N_kN_lright]=frac(N-1)theta_ktheta_lN{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$lambda_K=0${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$E{Xepsilon}=E{XE{epsilon|X}}=0${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$R^k${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}Eleft[N_kright]=sum_i=1^NEI{x_i=k}=sum_i=1^Ntheta_k=Ntheta_k{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$beta^*=(1,-.5,1)${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$sup_thetainTheta|H_N(theta)-H(theta)|to0${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$theta^*${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$X_(1)=cdots=X_(p)=Z${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$sigma^2${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$epsilon_i${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$R^2${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$beta^*=(1,-.5,1,-1,2,-.8)${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$beta^*${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}Eleft[widehattheta_k^2right]=frac1N^2Eleft[N_k^2right]=fractheta_kN+frac(N-1)theta_k^2N.{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}I(lambda^*)=mboxdiagleft[p(lambda^*)right]-p(lambda^*)p(lambda^*)'.{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$Omega${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$I{x_i=l}${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$hatlambda_N${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}widehattheta_k=fracN_kN.{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}${d_i,X_i}${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}Omega=left[arraycc1;SPMamp;.2.2;SPMamp;1arrayright]{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}N_k=sum_i=1^NI{x_i=k},qquadsum_k=1^KN_k=N{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$P(j|X_j,theta^*)${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$E{|tildey-tildeXbeta|}${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}X=left[arraycX_1vdotsX_Narrayright],qquadX_i=left[arraycccX_i1;SPMamp;;SPMamp;0;SPMamp;ddots;SPMamp;0;SPMamp;;SPMamp;X_iparrayright]{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$hattau_N${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$Theta${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}partialoverpartiallambdalogp_k(lambda^*)=e_k-p(lambda^*),{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$k^mboxth${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$mboxmed(y_t|X_t)=X_tbeta^*${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$Xbeta^*${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$hattheta_k${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}1=sum_k=1^Kexp{lambda_k}.{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}widehatbeta_GLS=(X^primeSigma^-1X)^-1X^primeSigma^-1y{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}Sigma=textcov(widehattheta)=frac1Nleft[arraycccctheta_1(1-theta_1);SPMamp;-theta_1theta_2;SPMamp;cdots;SPMamp;-theta_1theta_K-1-theta_1theta_2;SPMamp;theta_2(1-theta_2);SPMamp;cdots;SPMamp;-theta_2theta_K-1cdots;SPMamp;cdots;SPMamp;cdots;SPMamp;cdotscdots;SPMamp;cdots;SPMamp;cdots;SPMamp;cdots-theta_1theta_K-1;SPMamp;theta_2theta_K-1;SPMamp;cdots;SPMamp;theta_K-1(1-theta_K-1)arrayright].{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$k=1,ldots,K-1${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$lambda${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$mu${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}X=left[arraycccZ;SPMamp;;SPMamp;0;SPMamp;ddots;SPMamp;0;SPMamp;;SPMamp;Zarrayright]=I_potimesZ{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$chi^2${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$2times1${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$widehattheta_k${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}Xbeta^*=argmin_muE_X{|tildey-mu|},{displaymath}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$tildetheta${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$lambda=(lambda_1,ldots,lambda_K)${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$I{x_i=k}${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{_inline}$tildeX${_inline}/;
$cached_env_img{$key} = ' 
';
$key = q/{displaymath}f(x_i|theta)=prod_k=1^Ktheta_k^I{x_i=k},{displaymath}/;
$cached_env_img{$key} = ' 
';
1;