Let $X_1, ..., X_n$ be independent random variables on some probability space such that for each $i = 1, ..., n$ we have that $X_i \sim N(a_i\theta, 1)$, where $a_1,...,a_n$ are given constants .Find the Method of moments estimator $\hat{\theta}$ of $\theta$
I got confused because of the $a_i$. \begin{equation} \begin{split} E_{\theta}(X_i) &= \int_{-\infty}^{\infty}\frac{x}{\sqrt{2\pi}}e^{-\frac{(x-a_i\theta)^2}{2}} dx\\ &=\frac{1}{\sqrt{2\pi}}\sqrt{2\pi} a_i \theta = a_i\theta \end{split} \end{equation} From which we deduce that $\hat{\theta} = \frac{\overline{X_n}}{a_i}$
But I don't think it's right that I have $a_i$ in that formula.
Any hint is much appreciated.
You are correct that $\mathbf E_\theta X_i = a_i\theta$. Thus $\mathbf E_\theta \bar X_n = \bar a_n \theta $ where $\bar a_n = \sum_{i = 1}^n a_i$.
The method of moments now says to determine $\hat \theta $ such that $\bar X_n = \mathbf E_{\hat\theta} \bar X_n = \bar a_n \hat\theta$, thus $\hat\theta = \frac{\bar X_n}{\bar a_n}$.