Question:
For $\lambda \subset \mathbb{R}$, define the function $$ g_\lambda(y)=e^{-(y-\lambda)} _{(y\geq\lambda)} $$ Let $Y=(Y_1, Y_2, \ldots, Y_n)$ with $Y_1, Y_2, \ldots, Y_n$ iid random variables. Assume that each $Y_i$ has a continuous distribution pdf $g_\lambda$ where the parameter $\lambda$ is unknown.
(a) Find the maximum likelihood estimator $T(Y)$ for $\lambda$.
I have found out that the $\hat{\lambda_{MLE}}= max(Y_1,...,Y_n)$.
(b) Compute $P_\lambda(T(Y)>t)$ for $t\in\mathbb{R}$.
My attempt:
$P_\lambda(T(Y)>t)$
$=P_\lambda(max(Y_1,...,Y_n>t))$
$=1-P_\lambda(max(Y_1,...,Y_n)\leq t)$
$=1-P_\lambda(Y_1\leq t, ...,Y_n\leq t)$
since $Y_1,...,Y_n$ are iid,
$=1-[P_\lambda(Y_1\leq t)...P_\lambda(Y_n\leq t)]$
$=1-[(1-e^{-(t-\lambda)})...(1-e^{-(t-\lambda)})]$
$=1-(1-e^{-(t-\lambda)})^{n}$
(c) Based on the expression found in (b), compute the cdf $F_\lambda$ of $T(Y)$.
My attempt:
$F_\lambda(t)$
$=P_\lambda(T(Y)\leq t)$
$=1-P_\lambda(T(Y)>t)$
$=1-[1-(1-e^{-(t-\lambda)})^{n}]$
$=(1-e^{-(t-\lambda)})^{n}$
(d) Based on $F_\lambda$, compute the pdf of $T(Y)$.
$f_\lambda(t)$
$=\frac{\partial}{\partial t}F_\lambda(t)$
$=n(1-e^{-(t-\lambda)})^{n-1}e^{-(t-\lambda)}$
(e) Compute the bias of the estimator $T(Y)$. You may use known expressions for the mean and variance of an exponential distribution without proof.
My attempt:
bias($T(Y)$)
$=\mathbb{E}[T(Y)]-T(Y)$
$=\mathbb{E}[max(Y_1,...,Y_n)]-max(Y_1,...,Y_n)$
I'm having difficulty determining the distribution of max(Y_1, ..., Y_n) by examining the cumulative distribution function constructed in (c). I'm wondering if I made any mistakes in the previous parts of this question. Any help would be greatly appreciated!"