Question

ID No Page 8 oi 8 this oints) Get the maximum likelihood estimator of θ and compute the estimate f data.

Answer 1

Here we have

$f(x_{1})=\theta (x_{1})^{\theta-1}$

$f(x_{2})=\theta (x_{2})^{\theta-1}$

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$f(x_{n})=\theta (x_{n})^{\theta-1}$

So the likelihood function is

$L(\theta)=f(x_{1})f(x_{2})f(x_{3})...f(x_{n})=\theta^{n} \left ( \prod_{i=1}^{n}x_{i} \right )^{\theta-1}$

Taking log of both sides gives

$ln\left [L(\theta) \right ]=nln\left [\theta\right ]+(\theta-1) \sum_{i=}^{n}ln(x_{i})$

Differentiating both sides gives

$\frac{d}{d\theta}ln\left [L(\theta) \right ]=\frac{n}{\theta}+\sum_{i=1}^{n}ln(x_{i})$

Equating it equal to zero gives

$\frac{n}{\theta}+\sum_{i=1}^{n}ln(x_{i})=0$

$\theta=-\frac{n}{\sum_{i=1}^{n}ln(x_{i})}$

Hence, required estimate is

$\tilde{\theta}=-\frac{n}{\sum_{i=1}^{n}ln(x_{i})}$