Thread: Developing a theory for estimation and hypothesis testing

Hi all, this is my first post.

Here's the challenge:

f(x; θ) =1/θ x^((1/θ) −1)


for x ∈ [0, 1]



Is the function we're dealing with, so depending on theta, we can have either uniformly distributed (namely if theta = 1) or skewness to the right (theta < 1) or skewness to the left (theta > 1)

I've been trying to develop a statistic for theta, the most obvious one that comes to mind is the mean. From there I attempted to find a sufficient estimator of theta (or perhaps I should be trying to use the maximum likelihood estimator?)

I hit a snag in my algebra, here's what I got after simplifying their products:

((1/θ)^n) x^(∑1/θ - n)

I'm not sure whether x should be raised to the Nth power or what i'm doin...

Normally I'd break this down into ((1/θ)^n) x^(∑1/θ - n) equalling g(Θ,θ) and h(x.....xn) = 1

help out if ya'll can!

Attached is my project if you'd like more information. I'm not asking for a way out, I'm asking for advice. I've hit a snag and I'd like to do the project myself.

Okay, I've made some progress;

f(x1|θ) x f(x2|θ) x ..... (fxn|θ) = 1/θ Prod<xsubi> ^ ((n/θ)-1)

sorry for the crappy notation, if anyone knows an easier way to type, lemme kno


Edit:


(1/θ)^n ∏x ^ ((n/θ) - n)


I'm checking to see if theta is a sufficient estimator, but can't deal with the product of x's unless they're factored out to h(x1,x2....xn)


Should I replace the values of x with my statistic estimate of ?

n ln(theta) + ((1/theta) - n) ( ln(xsub1) + ln (xsub2) + . . . ln(xsubn))

is my newest breakthrough, since I decided to go with the MLE of the product of the moment generating functions