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School of Mathematics and Statistics

Computer Exercise Week 8

STAT3023: Statistical Inference

Semester 2, 2021

Two-sided tests for a normal variance

Suppose are iid In the week 7 Tutorial it was noted that the statistic  (where  and  are the sample mean and variance) has a distribution (note we are not multiplying by  as we did in the week 7 Tutorial!). Consider testing

1. One possible level-α test is the “equal-tailed” test based on Y , where we reject for Y < a or Y > b where

(a) Taking α = 0.04 and n = 5, find appropriate values a and b.

(b) Defining sig.sq=(50:150)/100 plot the power of the test against sig.sq. Add a horizontal dotted line to indicate the level.

2. In Tutorial week 7 we also saw that the UMPU test rejects for large values of

which is equivalent to rejecting for small values of the statistic

to see this, write log() = log Y − log(n − 1), multiply through by n − 1 and ignore the (n − 1) log(n − 1) term.

If the test is to have level α, we reject for Y ≤ c or Y ≥ d where

and

(a) Write a function of the form

which

● computes the appropriate d so that c and d satisfy (2);

● then computes and outputs the difference between the left-hand side and right-hand side in (3).

(b) Use the R function uniroot() to find the root (in c) of the equation fn(c,0.04,5)=0. In your code you will need a command along the lines of

Consult the week 7 exercise for some hints as to how to choose the upper=.... When you have worked out the right commands, wrap it all in a function of the form

which returns a list containing elements $c and $d.

(c) Recreate your plot from part (b) of the previous question and add to it the power function of the UMPU test.

3. The GLRT test of (1) above uses the statistic

which is an increasing function of Y − n log Y (as opposed to the UMPU which rejects for large Y − (n − 1) log Y ). Adapt your code for the previous question to compute the power of the exact GLRT, recreate your earlier plot and add a power curve to it so it shows all 3 power curves on the 1 graph. Add an informative heading, legend, etc.. Comment on the main differences between the 3 tests.

4. As a final step, recreate your last plot but use an extended range for the parameter: sig.sq=(1:400)/100.