Method of moment estimator based on the given density

by RAHUl JHa   Last Updated January 14, 2018 10:19 AM

I have the following density function for which I want to obtain the method of moment estimator:

$f(x,\theta)=\frac {1}{3}\left [ \frac {1}{\theta}exp\left ( -\frac{x}{\theta} \right)+\frac {1}{\theta^2}exp\left ( -\frac{x}{\theta^2} \right)+exp\left ( x \right ) \right]$, $x>0$, $\theta>0$

The Sample Values has been provided in the problem as $4,5,1,3$.

I obtained the mean of the above density as follows:

$E[X]=\frac{\theta^2+\theta+1}{3}$

Then according to moment method, the above population mean should be equal to Sample mean. Hence, I got the following equation:

$E[X]=\frac{\theta^2+\theta+1}{3}=\frac{13}{4}$

After solving the above equation for $\theta$, I got the following answer:

$\theta=\frac{-4\pm24}{8}$

Theta can take only positive Values. Hence, the required answer is $\theta=2.5$

Have I done it correctly?. Actually, my answer is not matching but I think I have done it correctly. Any help would be great. Thanks in advance.



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