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Math Help - Developing a theory for estimation and hypothesis testing

  1. #1
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    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!
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  2. #2
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    Re: Developing a theory for estimation and hypothesis testing

    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.
    Attached Files Attached Files
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  3. #3
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    Re: Developing a theory for estimation and hypothesis testing

    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 ?
    Last edited by VanBuren; May 7th 2012 at 09:52 AM.
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  4. #4
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    Re: Developing a theory for estimation and hypothesis testing

    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
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