Question

Let X be a random variable with the following probability distribution.
f(x)= (θ+1)x^θ for 0≤x≤1
f(x)= 0 for x<0 , x>1


Find the maximum likelihood estimator of θ based on a random sample size n.

Answer 1

I will try this. Give me a moment

Answer 2

\[L(\theta)=\prod_{i=1}^{n}f(x_i)=\prod_{i=1}^n(\theta+1)x_i^{\theta}=(\theta+1)^n\left(\prod_{i=1}^{n}x_i\right)^{\theta}\]\[l(\theta)=\log L(\theta)=n\log(\theta+1)+\theta\sum_{i=1}^n\log x_i\]\[S(\theta)=\frac{\partial}{\partial \theta}l(\theta)=\frac{n}{\theta+1}+\sum_{i=1}^n\log x_i\]
Set \(S(\theta)=0\)
\[\frac{n}{\hat{\theta}+1}+\sum_{i=1}^n\log x_i=0\]
\[\hat{\theta}=\frac{-n}{\sum_{i=1}^n\log x_i}-1\]

Verify that this is a maximum by checking \[I(\hat{\theta)}=-\left.\frac{\partial^2}{\partial \theta^2}l(\theta) \right|_{\theta=\hat{\theta}}>0 \]