What kind of distribution is $f_X(x) = 2 \lambda \pi x e^{-\lambda \pi x ^2}$?

What kind of function is:

$f_X(x) = 2 \lambda \pi x e^{-\lambda \pi x ^2}$

Is this a common distribution? I am trying to find a confidence interval of $\lambda$ using the estimator $\hat{\lambda}=\frac{n}{\pi \sum^n_{i=1} X^2_i}$ and I am struggling to prove if this estimator has Asymptotic Normality.

Thanks

Answer

It is a square root of exponential distribution with rate $\pi\lambda$. This means that if $Y\sim\exp(\pi\lambda)$, then $\sqrt{Y}\sim f_X$.

Since your estimate is maximum likelihood estimate it should be asymptotically normal. This follows immediately from the properties of maximum likelihood estimates. In this particular case:

$$\sqrt{n}(\hat\lambda-\lambda)\to N(0,\lambda^2)$$

since

$$E\frac{\partial^2}{\partial \lambda^2}\log f_X(X)=-\frac{1}{\lambda^2}.$$

Attribution
Source : Link , Question Author : Mitch , Answer Author : mpiktas

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