Question

5 let i.. , Xn lean i i d random sample from a population with unknown parameter ? If-1, then the population pdf is f(r) 1 for 0 < r < 1 and zero otherwise. If ? 0, then the population pdf is for 0 <?< 1 and zero otherwise. a. (1 point) Find the likelihood function l( ri "). Note that this defined for t-0 and ?- 1, and is zero for any other possible value of ?. b. (1 point) what is the maximim likelihood estimator of ?? Tp: the estimate trill be either 0 or 1 Finding the MLE requires determining uhether L(l) or L(0) has the larger alue for your observed sample.

Answer 1

(a)

$L(\theta\ |\ X_1, X_2, ..., X_n)= \begin{cases} \prod_{i=1}^nf(X_i\ |\ \theta=1)&if\ \theta=1\\ \prod_{i=1}^nf(X_i\ |\ \theta=0)&if\ \theta=0\\ 0&otherwise \end{cases}\ \ \ = \begin{cases} 1&if\ \theta=1\ \&\ 0<X_1,X_2,...,X_n<1\\ \prod_{i=1}^n\frac{1}{2\sqrt{X_i}}&if\ \theta=0\ \&\ 0<X_1,X_2,...,X_n<1\\ 0&otherwise \end{cases}$

(b)

For 0 < X₁, X₂, ..., X_n < 1,

$\\ L(1)>L(0)\ \ \Leftrightarrow\ \ 1>\prod_{i=1}^n\frac{1}{2\sqrt{X_i}} \ \ \Leftrightarrow \ \ 2^n>\prod_{i=1}^n\frac{1}{\sqrt{X_i}}\\ \Leftrightarrow \ \ 2^{2n}>\prod_{i=1}^n\frac{1}{X_i}\ \ \Leftrightarrow \ \ \prod_{i=1}^nX_i>2^{-2n}$

$\\ MLE,\ \hat{\theta}=I\left(\prod_{i=1}^nX_i>2^{-2n}\right)\\ \\ (where\ I(a>b)=1\ if\ a>b\ and\ 0\ otherwise)$