Question

3. In the CRD model Xy-μ+1+ 4,i-1, independently. Then showing all steps. find ri, we assume that etj ~ Normal(0,02) ,v,, -1, (a) the distribution of Yij (b) the distribution of Y. (e) the distribution of Y (d) cov(Y.,Y,) (e) cov( (f) cov(A, A)

Answer 1

$(a)~Y_{ij}\sim N(\mu+\tau_i,\sigma^2)~since~\epsilon_{ij}\sim N(0,\sigma^2).\\ (b)~\bar{Y}_{i.}=\frac{1}{r_i}\sum_{j=1}^{r_i} Y_{ij}~is~a~linear~function~of~Y_{ij}'s~where~Y_{ij}'s~are~independent~normally~distributed\\ with~E(\bar{Y}_{i.})=\frac{1}{r_i}\sum_{j=1}^{r_i}E(Y_{ij})=\mu+\tau_i\\ Var(\bar{Y}_{i.})=\frac{1}{r_i^2}\sum_{j=1}^{r_i}Var(Y_{ij})=\frac{\sigma^2}{r_i}\\ So~\bar{Y}_{i.}\sim N\left(\mu+\tau_i,~\frac{\sigma^2}{r_i} \right )\\ (c)~\bar{Y}_{..}=\frac{1}{n}\sum_{i=1}^v\sum_{j=1}^{r_i} Y_{ij}~is~a~linear~function~of~Y_{ij}'s~where~Y_{ij}'s~are~independent~normally~distributed\\ with~E(\bar{Y}_{..})=\frac{1}{n}\sum_{i=1}^v\sum_{j=1}^{r_i}E(Y_{ij})=\mu+\frac{1}{n}\sum_{i=1}^vr_i\tau_i=\mu,\\ ~since~in~the~CRD~model,~for~estimating~\mu,~\tau_i's,~we~assume~\sum_{i=1}^vr_i\tau_i=0\\ Var(\bar{Y}_{..})=\frac{1}{n^2}\sum_{i=1}^v\sum_{j=1}^{r_i}Var(Y_{ij})=\frac{\sigma^2}{n}\\ So~\bar{Y}_{..}\sim N\left(\mu,~\frac{\sigma^2}{n} \right ),~where~n=\sum_{i=1}^vr_i\\$$(d)~Cov(\bar{Y}_{i.},\bar{Y}_{j.})=\frac{\sigma^2}{r_i}~when~i=j\\ =0~if~i\neq j~since~Y_{ij}'s~are~independent.\\\\ (e)~Cov(\bar{Y}_{i.},\bar{Y}_{..})=Cov\left(\bar{Y}_{i.},\frac{1}{n}\sum_{i=1}^vr_i\bar{Y}_{i.}\right)=\frac{1}{n}\times r_iVar(\bar{Y}_{i.})=\frac{\sigma^2}{n}\\ (f)~For~CRD~model,~the~least~square~estimate~of~\mu,~\tau_i's~are\\ \hat{\mu}=\bar{Y}_{..},~\hat{\tau}=\bar{Y}_{i.}-\bar{Y}_{..}\\ Cov(\hat{\mu},\hat{\tau}_i)=Cov(\bar{Y}_{..}.\bar{Y}_{i.}-\bar{Y}_{..})=Cov(\bar{Y}_{..},\bar{Y}_{i.})-Var(\bar{Y}_{..})=\frac{\sigma^2}{n}-\frac{\sigma^2}{n}=0$