Question

Exam.(Jan 15) Question 7. Suppose thatX,x,, Xo is a simple random samp distribution with the following p.d.f Circle out yoar Class: Mee& Wed or Mon.Evening le from the f(x0) otherwise where 0>0, a random sample of size 10 yields data 092 0.79 0.9 0.65 0.86 0.47 0.73 0.97 0.94 0.77 1) (6 points) Get the moment estimator of θ, and compute the estimate for this data;

Answer 1

(1)

Let us first find the mean of X. So

$\mu=E(X)=\int_{0}^{1}xf(x)dx=\int_{0}^{1}\theta x^{\theta }dx=\left [ \frac{\theta x^{\theta +1}}{\theta +1} \right ]=\frac{\theta }{\theta +1}$

Now the sample mean is

$\bar{x}=\frac{x_{1}+x_{2}+...+x_{n}}{n}=\frac{1}{n}\sum_{i=1}^{n}x_{i}$

Since according to method of moments estimators, first theortical moment is equal to first sample moment so we have

$\mu=\bar{x}$

$\frac{\theta }{\theta +1}=\bar{x}$

$\theta =\bar{x}\theta +\bar{x}$

$\theta =\frac{\bar{x}}{1-\bar{x}}$

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For the given data, the sample mean is

$-=-=0.80$

So the required MOM is