A consistent estimator for theta is?

Question

Let $X_1,X_2,.....X_n$ be a random sample from a $N(2\theta,\theta^2)$ population,$\theta>0$. A consistent estimator for $\theta $ is ?

(A) $\dfrac{1}{n}\sum_{i=1}^{n}X_i$

(B) $\bigg(\dfrac{5}{n}\sum_{i=1}^{n}X_i^{2}\bigg)^{\frac{1}{2}}$

(C) $\dfrac{1}{5n}\sum_{i=1}^{n}X_i^{2}$

(D)$\bigg(\dfrac{1}{5n}\sum_{i=1}^{n}X_i^{2}\bigg)^{\frac{1}{2}}$

My input : First idea came in my mind during my test is that Mle's are consistent and i instantaneously marked (A).Now i realized that Mle for $\theta$ is not $ \bar x$.we have variance unknown as well.I tried to figure out other options i tried to calculate if their variance is zero as n is $\infty$ but lengthy calculations telling me i am doing something wrong this question came in 2 marks.Any ideas someone?

Answer 1

The answer is (D). Because $E(X_1^2) = (2θ)^2 + θ^2 = 5θ^2$, by the law of large numbers, there is$$ \frac{1}{n} \sum_{k = 1}^n X_k^2 \xrightarrow{P_θ} 5θ^2. $$ By the continuous mapping theorem,$$ \left( \frac{1}{5n} \sum_{k = 1}^n X_k^2 \right)^{\frac{1}{2}} \xrightarrow{P_θ} θ. $$