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I have been attempting this question for a while and whenever I get to the standard error question of part (a) I keep getting 0.

Ok so ai) $$\ E(X)= 3-2\theta $$ $$\ Var(X) = 2\theta -2\theta^2 $$

for aii)

I found my MME by equating first theoretical moment (mu) to the 1st sample moment.

First moment is $$ E(X) = 3-2\theta$$

and sample moment is

$ 1/n\sum_{i=1}^n X_i =M_k $ ,where n=20 . This can be referred to as X.bar

by equating these two I get an estimator of Theta, Theta.hat $$ Theta.hat = (3-2X.bar)/2$$

To get an estimate I plug in Xbar of 1.75 into here and get estimate of $$ \theta=0.625$$

Now to find Standard error I first need to find variance

$$Var(theta.hat)= E(theta.hat^2)- E(theta.hat)^2$$

However now as I substitute in my estimator into the equation I find they just cancel, giving me a variance of 0

Mathman
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  • Edit in your calculation of the standard error. – J.G. Aug 28 '19 at 08:05
  • Yes, please show your results of the first two parts of (a), your definition of standard error, and what happens when you try to compute it. // Also (b) seems unrelated to your question. – BruceET Aug 28 '19 at 08:15

1 Answers1

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Equating sample mean with population mean you get

$$\overline X=3-2\theta$$

Method of moments estimator of $\theta$ is simply $$\hat\theta(\mathbf X)=\frac{3-\overline X}{2}$$

Based on the given sample, your estimated value of $\theta$ is some $\hat\theta$.

Now $$\operatorname{Var}(\hat\theta(\mathbf X))=\frac{\operatorname{Var}(\overline X)}{4}=\frac{\theta(1-\theta)}{2n}$$

So the estimated standard error of $\hat\theta(\mathbf X)$ is $$\widehat{\text{S.E}(\hat\theta(\mathbf X))}=\sqrt{\frac{\hat\theta(1-\hat\theta)}{2n}}$$

StubbornAtom
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