# Thread: statistical test problem

1. ## statistical test problem

Hey, I have an interesting problem to solve. Any ideas how to estimate the parameter?

Thank you for your help!

2. ## Re: statistical test problem

we have to assume the samples are independently taken.

let $p_\theta(j)$ be the distribution shown in the problem

Doing that we have the a posteriori density as

$p_{ap}(\theta)=p_\theta(0)^2 p_\theta(1)^3 p_\theta(2) = \Large -\frac{\theta ^5}{4}-\frac{\theta ^4}{16}+\frac{5 \theta ^3}{64}+\frac{\theta ^2}{256}-\frac{\theta }{128}+\frac{1}{1024}$

we can apply the usual process to find extrema, differentiate with respect to $\theta$ and solve for that being $0$

$\dfrac{dp_{ap}}{d\theta} = \Large -\frac{5 \theta ^4}{4}-\frac{\theta ^3}{4}+\frac{15 \theta ^2}{64}+\frac{\theta }{128}-\frac{1}{128}$

with roots of $\theta = \{-\dfrac 1 2, -\dfrac 1 5, \dfrac 1 4\}$

The second derivative test must be applied to see which, if any, are maxima.

I leave it to you to show that $\theta = -\dfrac 1 5$ is the MAP estimate of $\theta$

3. ## Re: statistical test problem

Originally Posted by romsek
we have to assume the samples are independently taken.

let $p_\theta(j)$ be the distribution shown in the problem

Doing that we have the a posteriori density as

$p_{ap}(\theta)=p_\theta(0)^2 p_\theta(1)^3 p_\theta(2) = \Large -\frac{\theta ^5}{4}-\frac{\theta ^4}{16}+\frac{5 \theta ^3}{64}+\frac{\theta ^2}{256}-\frac{\theta }{128}+\frac{1}{1024}$

we can apply the usual process to find extrema, differentiate with respect to $\theta$ and solve for that being $0$

$\dfrac{dp_{ap}}{d\theta} = \Large -\frac{5 \theta ^4}{4}-\frac{\theta ^3}{4}+\frac{15 \theta ^2}{64}+\frac{\theta }{128}-\frac{1}{128}$

with roots of $\theta = \{-\dfrac 1 2, -\dfrac 1 5, \dfrac 1 4\}$

The second derivative test must be applied to see which, if any, are maxima.

I leave it to you to show that $\theta = -\dfrac 1 5$ is the MAP estimate of $\theta$
Hello,
What's the meaning of MAP estimate?

4. ## Re: statistical test problem

Originally Posted by Vinod
Hello,
What's the meaning of MAP estimate?
https://en.wikipedia.org/wiki/Maximu...ori_estimation

5. ## Re: statistical test problem

Originally Posted by Vinod
Hello,
What's the meaning of MAP estimate?
Maximum A Posteori

The best estimate you can make given the A Posteori distribution