Parametric hypothesis, uniform distribution

[Deimantas]

Homework Statement

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.

Homework Equations

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).

M_n= ((X_max-b)*100)/(b-a) = 0

The percentiles are calculated using this formula:

h_p=ln(p). So
h_0.025=ln(0.025)=-3.6888794541139363
h_0.975=ln(0.975)=-0.0253178079842899

The value of M_n should fall in the interval between h_0.975 and h_0.025 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?

 

[Deimantas]

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 parameter b value of 3 doesn't fall in the interval and has to be rejected..