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Parametric hypothesis, uniform distribution

  1. Dec 15, 2013 #1
    1. The problem statement, all variables and given/known data

    We are given a sample of size 100. After some tests (histogram, Kolmogorov) we deduce the sample X is distributed uniformly. The next task is to presume the parameters are equal to values of your choice, and test if such hypothesis is true.

    2. Relevant equations



    3. The attempt at a solution

    Uniform distribution has two parameters, a and b. My estimated parameters are a=1.01 (minimum value in the sample) and b=3 (max value in the sample).

    I'm testing null hypothesis: b=3. The value of parameter a is known (1.01).

    Mn= ((Xmax-b)*100)/(b-a) = 0

    The percentiles are calculated using this formula:

    hp=ln(p). So
    h0.025=ln(0.025)=-3.6888794541139363
    h0.975=ln(0.975)=-0.0253178079842899

    The value of Mn should fall in the interval between h0.975 and h0.025 for the hypothesis to be accepted as correct.

    This must be wrong, because I chose b value that is equal to the max value of the sample X, which should be a good estimate, and so the hypothesis should be accepted. What am I missing?
     
  2. jcsd
  3. Dec 15, 2013 #2
    I think I might have got it. The reason values like 2.99 or 2.999 won't work is because b value is the maximum, and we already have a maximum of 3 in our sample of 100 elements. So the real value of parameter b can't be smaller than the one we already have in our sample, only equal or larger. I calculated, using the confidence interval formulas, that the real value of b should be in the interval of 3.001 and 3.076. It's because there might be larger elements in the general sample. Kinda makes sense, though it's a pity my estimated paramater b value of 3 doesn't fall in the interval and has to be rejected..
     
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