Lista de Exercícios de Inferência Estatística

Lista 2

1) $Y_i hicksim ext{Poisson}(e^{X_ieta})$.
$ ightarrow L(X_ieta) = e^{ ext{sum}(X_ieta Y_i)} e^{- ext{sum}( ext{exp}(X_ieta))} ext{prod}(Y_i!)^{-1}$
$ ightarrow ext{log}(L) = ext{Sum}(X_ieta Y_i) - ext{sum}( ext{exp}(X_ieta)) - ext{log}( ext{prod}(Y_i!))$
$ ightarrow deta = ext{sum}(X_iY_i) - ext{sum}(X_ie^{X_ieta})$
$d^2eta^2 = - ext{sum}(X_i^2 ext{exp}(X_ieta))$
$ ightarrow eta_{i+1} = eta_i - rac{deta}{d^2eta^2}$ até $|eta_{i+1} - eta_i| < ext{ extepsilon}$

2) $X hicksim ext{Gama}( ext{A}, ext{B})$
a) $ ightarrow ext{log}(L)$ = $(nA) ext{log}B - n ext{log}( ext{ extGamma}(A)) + (A-1) ext{log}( ext{Sum}X_i) - B ext{sum}(X_i)$
$dA$ = $n ext{log}B - nrac{d ext{ extGamma}(A)}{dA} + ext{log}( ext{sum}X_i)$
$d^2A^2$ = $-nrac{d^2 ext{log}( ext{ extGamma}(A))}{dA^2}$
$dB$ = $rac{nA}{B} - ext{sum}(X_i)$
$d^2B^2$ = $-rac{nA}{B^2}$
$dAdB$ = $rac{n}{B}$
b) $ ext{grad} = [dA ext{ } dB]$
$H$ = $[egin{array}{cc} d^2A^2 & dAdB \ dAdB & d^2B^2 ag{array}$]
1. Iniciar valor para $Z_0$
2. Fazer $Z_{i+1} = Z_i - ext{grad} imes H^{-1}$
3. Parar quando $|Z_{i+1} - Z_i| < ext{ extepsilon}$
c) $ ext{EP} = ext{sqrt}(I_F^{-1})$, $I_F = -H$

4) $z_{ij}$ = {1 se $x hicksim j$ e 0 c.c.}
$ ightarrow E[z_{ij}|x_i]$ = $P(z_{ij}=1|X_i) = rac{P(X_i|z_{ij}=1)P(z_{ij}=1)}{f(X_i)} = rac{W_jf_j(X_i)}{ ext{sum}_k (W_jf_j(X_i))} = p_{ij}$
$ ightarrow ext{log}(L) = ext{sum}_n ext{sum}_k z_{ij} ext{log}(W_j) + ext{sum}_n ext{sum}_k z_{ij} ext{log}(f_j(X_i| ext{ extmu}_j, ext{ extsigma}_j^2))$
$ ightarrow$ Vamos fixar $j$:
$ ext{log}(L)$ = $ ext{sum}_n p_{ij} ext{log}W_j + ext{sum}_n p_{ij} ext{log}(f_j(X_i| ext{ extmu}_j, ext{ extsigma}_j^2))$
$=$ $ ext{log}W_j ext{sum}_n p_{ij} + ext{sum}_n ig(- ext{log} ext{sqrt}(2 ext{ extpi}) - rac{1}{2} ext{log} ext{ extsigma}^2 - rac{(x_i- ext{ extmu}_j)^2}{2 ext{ extsigma}_j^2}ig)p_{ij}$
Temos $d ext{ extmu}_j$ = $rac{ ext{sum } p_{ij}}{2 ext{ extsigma}_j^2} imes 2(x_i - ext{ extmu}_j)$ e igualando a 0, $ ext{sum}_n p_{ij}x_i = ext{sum}_n ext{ extmu}_jp_{ij}$
e $d ext{ extsigma}^2$ = $ ext{sum}_n ig(-rac{1}{2 ext{ extsigma}_j^2} + rac{(x_i- ext{ extmu}_j)^2}{2 ext{ extsigma}_j^4}ig)p_{ij}$, igualando a zero, $ ext{ extsigma}^2$ = $rac{ ext{sum}_n p_{ij}(x_i- ext{ extmu}_j)^2 ext{sqrt}(2 ext{ extpi})}{ ext{sum } p_{ij}}$
$ ightarrow$ Para $W_j$, $ ext{log}L$ = $ ext{sum}_n ext{sum}_k z_{ij} ext{log}(W_j) + ext{sum}_n ext{sum}_k z_{ij} ext{log}(f_j(X_i| ext{ extmu}_j, ext{ extsigma}_j^2)) + ext{sum}_n ext{sum}_k (1-z_{ij}) ext{log}(1-W_j)$
$ ightarrow$ Fixando $j$ e $z_{ij} = p_{ij}$, $dW_j = rac{ ext{sum } p_{ij}}{W_j} + rac{ ext{sum}(1-p_{ij})}{1-W_j}$, e igualando a zero temos $W_j = rac{ ext{sum } p_{ij}}{n}$
$ ightarrow$
1. Fixar $ ext{ extmu}_j, ext{ extsigma}_j^2 ext{ e } W_j$
2. $E[z_{ij}|x_i] = p_{ij}$
3. Transmitir $Z_{ij}$ em $p_{ij}$ na $ ext{log}(v)$
4. Estimar $ ext{ extmu}_j, ext{ extsigma}_j^2 ext{ e } W_j$ com os estimadores encontrados

6) T-Student, $X_i|z_i hicksim N( ext{ extmu}, ext{ extsigma}^2/z_i)$.
a) $ ext{log}(L)$ = $-rac{ u+1}{2} ig[ ext{sum} ig(rac{1}{1 + rac{(x_i- ext{ extmu})^2}{ u ext{ extsigma}^2}}ig) imes rac{2(x_i- ext{ extmu})}{ u ext{ extsigma}^2} imes -1 ig] = ( u+1) ext{sum} rac{1}{ u ext{ extsigma}^2 + (x_i- ext{ extmu})}$
Não conseguimos isolar $ ext{ extmu}$.
b) $f_i(Z_{ij}|x_i) ext{ prop } f(X_{ij}|Z_i)f(Z_i) ext{ prop } rac{1}{ ext{sqrt}(2 ext{ extpi}) ext{ extsigma}/ ext{sqrt}(z_i)} imes e^{-rac{1}{2}(x_i- ext{ extmu})^2/( ext{ extsigma}^2/z_i)} imes rac{ u/2^{ u/2}}{ ext{ extGamma}( u/2)} imes z$
$ ext{prop } z_i^{ u/2 - 1/2} e^{-rac{z_i}{2 ext{ extsigma}^2}(x_i- ext{ extmu})^2 imes u/2 imes z_i}$
$ ext{prop } z_i^{ u/2 + 1/2 - 1/2 - 1/2} e^{-z_i [rac{1}{2}[ u + rac{(x_i- ext{ extmu})^2}{ ext{ extsigma}^2}]]}$
c)
1. Fixar $ ext{ extmu} ext{ e } ext{ extsigma}^2$
2. Calcular $E[Z_i|X_i]$
3. Log da verossimilhança e maximizar via Newton-Raphson
4. $ ext{ extmu}^* ext{ e } ext{ extsigma}^{2*}$