Did I calculate this method of moments estimator correctly?

Question

The question

My Solution: $$\bar{x}=\int_0^1(\theta+1)x^{\theta+1}dx$$ $$=\frac{\theta+1}{\theta+2}$$ Therefore $\theta=\frac{2\bar{x}-1}{1-\bar{x}}$

Is this the correct way to do this kind of question? Thank you

Answer 1

Yeah, since you have one paramater, the only moment you want is the mean, and you have correctly calculated the mean in terms of the paramater and inverted that function to solve for the parameter in terms of the mean.

Answer 2

Yes. The Answer is correct and maybe you want some explanation?

Assume $X_i$ is i.i.d. to $X$,

$$EX = \int x dF(x) = \int_0^1 x f(x; \theta) dx = \frac{\theta+1}{\theta+2}$$

We use the mean $\overline{x}$ as a statistical estimate for $EX$,

$$E\overline{x} = \frac{\sum EX_i}{n} = EX$$

so we solve

$$\overline{x} = \frac{\theta+1}{\theta+2}$$

and get the estimate for $\theta$.