# Maximum likelihood estimator of $P(X < y)$ for fixed $y$

I'm having a problem understanding the following question.

Given the following density function $f_X(x; \theta) = (\theta +1)x^\theta$ on $0<x<1$, find the maximum likelihood estimator for $P(X<y)$ for $0<y<1$ fixed. I always thought that the point of MLE was to estimate parameters, appearently I was wrong. The most compelling solution to this question (in my mind), is finding the MLE for the parameter $\theta$ and plugging it into an expression for $P(X<y)$.

You can easily check that the MLE for $\theta$ is $\frac{-n}{\sum_{i=1}^n \ln(X_i)}-1$. The expression for $P(X<y)= y^{\theta+1}$. Swapping $\theta$ with its MLE gives: $y^{\frac{-n}{\sum_{i=1}^n \ln(X_i)}}$.

As for calculating the assymptotic variance of this estimator , since we actually just transformed the MLE for $\theta$, I suggest using the delta method. By calculating the fisher information number we of course know the assymptotic variance of the MLE of $\theta$.

What do you think of this answer? Am I interpreting this question correctly?

Thanks!

What you wrote is intuitively correct. That is to find the maximum likelihood estimator for the parameter $\theta$ and replace it where it is asked. However the maximum likelihood estimator estimates the parameter $\theta$ such that the likelihood function is maximized (with respect to $\theta$). Let $Z=P(X<y)=y^{\theta+1}$. Then, what is your likelihood function in terms of $Z$? and what is the estimator $\hat{Z}(\bf x)$ that maximizes the likelihood function based on $f_X(x;Z)$?

• Mmh... I don't quite understand. Are you trying to show that my intuition coincides with the correct answer? This is what I think you mean that i should maximize $L(x; Z)= (y^\theta +1)^n \prod x_i^{y^\theta}$. But that does not seem to make much sense...
– Leo
Commented Jan 8, 2014 at 13:54
• That makes sense. Your formulation of $L(x;Z)$ is incorrect. Check it once again. Commented Jan 8, 2014 at 14:04
• I really can't figure out what is wrong with it.$L(x; Z)= \prod f(x_i; Z) = (Z +1)^n \prod x_i^Z$. Where do I go wrong?
– Leo
Commented Jan 8, 2014 at 14:30
• $Z=y^{\theta +1}$ if you solve it for $\theta$ you find $\theta=f(Z)$ insert this $f(Z)$ into your likelihood function. Commented Jan 8, 2014 at 15:09
• Ok I think I get it now.
– Leo
Commented Jan 8, 2014 at 15:13