Não foi possível enviar o arquivo. Será algum problema com as permissões?
Diferenças
Aqui você vê as diferenças entre duas revisões dessa página.
| Ambos lados da revisão anterior Revisão anterior Próxima revisão | Revisão anterior Última revisão Ambos lados da revisão seguinte | ||
|
disciplinas:verao2007:exercicios [2007/02/17 23:42] paulojus |
disciplinas:verao2007:exercicios [2007/02/18 16:51] paulojus |
||
|---|---|---|---|
| Linha 1: | Linha 1: | ||
| ===== Exercícios ===== | ===== Exercícios ===== | ||
| - | **Atenção**: para visualizar melhor esta página voce deve se ''autenticar'' neste wiki. Para isto clique no botão ''AUTENTICAR'' no canto inferior desta página e entre com o usuário e senha ''verao2007'' | ||
| ==== Semana 1 ==== | ==== Semana 1 ==== | ||
| Linha 23: | Linha 22: | ||
| - (4) Consider the following two models for a set of responses, <m>Y_i : i=1, ... ,n</m> associated with a sequence of positions <m>x_i: i=1,...,n</m> along a one-dimensional spatial axis <m>x</m>. | - (4) Consider the following two models for a set of responses, <m>Y_i : i=1, ... ,n</m> associated with a sequence of positions <m>x_i: i=1,...,n</m> along a one-dimensional spatial axis <m>x</m>. | ||
| - <m>Y_{i} = alpha + beta x_{i} + Z_{i}</m>, where <m>alpha</m> and <m>beta</m> are parameters and the <m>Z_{i}</m> are mutually independent with mean zero and variance <m>sigma^2_{Z}</m>. | - <m>Y_{i} = alpha + beta x_{i} + Z_{i}</m>, where <m>alpha</m> and <m>beta</m> are parameters and the <m>Z_{i}</m> are mutually independent with mean zero and variance <m>sigma^2_{Z}</m>. | ||
| - | - <m>Y_i = A + B x_i + Z_i</m> where the <m>Z_i</m> are as in (a) but //A// and //B// are now random variables, independent of each other and of the <m>Z_i</m>, each with mean zero and respective variances <latex>$\sigma_A^2$</latex> and <latex>$\sigma_B^2$</latex>.\\ For each of these models, find the mean and variance of <m>Y_i</m> and the covariance between <m>Y_i</m> and <m>Y_j</m> for any <m>j != i$</m>. Given a single realisation of either model, would it be possible to distinguish between them? | + | - <m>Y_i = A + B x_i + Z_i</m> where the <m>Z_i</m> are as in (a) but //A// and //B// are now random variables, independent of each other and of the <m>Z_i</m>, each with mean zero and respective variances <latex>$\sigma_A^2$</latex> and <latex>$\sigma_B^2$</latex>.\\ For each of these models, find the mean and variance of <m>Y_i</m> and the covariance between <m>Y_i</m> and <m>Y_j</m> for any <m>j != i</m>. Given a single realisation of either model, would it be possible to distinguish between them? |
| - (5) Suppose that <latex>$Y=(Y_1,\ldots,Y_n)$</latex> follows a multivariate Gaussian distribution with <latex>${\rm E}[Y_i]=\mu$</latex> and <latex>${\rm Var}\{Y_i\}=\sigma^2$</latex> and that the covariance matrix of <m>Y</m> can be expressed as <m>V=\sigma^2 R(phi)</m>. Write down the log-likelihood function for <latex>$\theta=(\mu,\sigma^2,\phi)$</latex> based on a single realisation of <m>Y</m> and obtain explicit expressions for the maximum likelihood estimators of <m>mu</m> and <m>sigma^2</m> when <m>phi</m> is known. Discuss how you would use these expressions to find maximum likelihood estimators numerically when <m>phi</m> is unknown. | - (5) Suppose that <latex>$Y=(Y_1,\ldots,Y_n)$</latex> follows a multivariate Gaussian distribution with <latex>${\rm E}[Y_i]=\mu$</latex> and <latex>${\rm Var}\{Y_i\}=\sigma^2$</latex> and that the covariance matrix of <m>Y</m> can be expressed as <m>V=\sigma^2 R(phi)</m>. Write down the log-likelihood function for <latex>$\theta=(\mu,\sigma^2,\phi)$</latex> based on a single realisation of <m>Y</m> and obtain explicit expressions for the maximum likelihood estimators of <m>mu</m> and <m>sigma^2</m> when <m>phi</m> is known. Discuss how you would use these expressions to find maximum likelihood estimators numerically when <m>phi</m> is unknown. | ||
| - (6) Is the following a legitimate correlation function for a one-dimensional spatial process <latex>$S(x) : x \in R$</latex>? Give either a proof or a counter-example.\\ | - (6) Is the following a legitimate correlation function for a one-dimensional spatial process <latex>$S(x) : x \in R$</latex>? Give either a proof or a counter-example.\\ | ||
