Question

4. Let X be continuous random variable whose PDF is given by for r>1 otherwise, where 0 is an unknown scalar parmeter. (a) Find the maximum likelihood estimator of . (b) Find a method-of-moments estimator of θ for the case when θ > 1. (c) Why can we not find a method-of-moments estimator when θ < 1? 151 151 151

Answer 1

(a)

The likelihood function for independent samples X1, X2, ..., Xn is,

$L(\theta) = f(x_1;\theta).f(x_2;\theta).....f(x_n;\theta) = \theta^n \prod_{i=1}^{n}x_i^{-(\theta&plus;1)}$ for xi > 1

The log-likelihood function is,

$lnL(\theta) = n ~ln(\theta) -(\theta&plus;1) \sum_{i=1}^{n}~ln(x_i)$

For maximum likelihood estimate,

$\frac{\partial }{\partial \theta}lnL(\theta) = n/\theta -\sum_{i=1}^{n}~ln(x_i)= 0$

$\Rightarrow \theta = n/ \sum_{i=1}^{n}~ln(x_i)$

$\frac{\partial^2 }{\partial \theta^2}lnL(\theta) = -n/\theta^2 < 0$

Thus, the maximum likelihood  estimate of $\theta$ is,

$\theta_{MLE} = n/ \sum_{i=1}^{n}~ln(x_i)$

(b)

First moment (mean) of f(x) is,

$E(x) = \int_{1}^{\infty }xf(x)~dx = \int_{1}^{\infty }\theta x^{-\theta}~dx = \theta \left [x^{-\theta &plus; 1}/(-\theta &plus; 1) \right ]^{\infty }_1$ ----(1)

$\theta$ > 1 =>  $-\theta$ + 1 < 0 => $x^{-\theta &plus; 1} \rightarrow 0$ as $x \rightarrow \infty$ .

$E(x) = \theta \left [x^{-\theta &plus; 1}/(-\theta &plus; 1) \right ]^{\infty }_1 = \theta / (\theta -1)$

Thus, $\sum_{i=1}^{n} X_i /n= \bar{X} = E(x) = \theta / (\theta -1)$

$\Rightarrow \bar{X} (\theta -1) = \theta$

$\Rightarrow \bar{X} \theta -\bar{X} = \theta$

$\Rightarrow \bar{X} \theta -\theta = \bar{X}$

$\Rightarrow \theta(\bar{X} -1) = \bar{X}$

$\Rightarrow \theta = \bar{X} /(\bar{X} -1)$

So, the method of moments estimator is,

$\theta_{MOM} = \bar{X} /(\bar{X} -1)$ where $\bar{X} = \sum_{i=1}^{n} X_i /n$

(c)

For $\theta = 1$, the first moment (mean) of f(x,) $E(x) = \theta / (\theta -1)$ is undefined.

For $\theta < 1$,

$-\theta$ + 1 > 0 => $x^{-\theta &plus; 1} \rightarrow \infty$ as $x \rightarrow \infty$

So, by equation (1), the first moment (mean) of f(x) is undefined for $\theta < 1$. Hence we cannot find a method of moments estimator for $\theta \le 1$.