Feel free to use any software you are comfortable with, both for the computations and to type up your answers.
statistics
Description
Practicalities. Feel free to use any software you are comfortable with, both for the computations and to type up your answers. Partly handwritten answers are fine too, as long as they are reasonably legible; just scan or take a picture of those parts. Your submission needs to contain both your computer code and your answers to all parts of all three questions; there are fifteen such parts, worth one mark each. Please submit your solutions using the submission tool on Canvas. In order to comply with our anonymous marking policy, make sure that your submission contains your SID but not your name. The deadline is final and no extensions can be allowed, as I will be publishing sample solutions shortly after this deadline.
Question 1. Assume that we have a random sample X1, X2, . . . , Xn. We do not wish to assume anything
about the distribution from which the sample was generated, other than that it satisfies E[X] ≥ 0 and
E
X2
≤ 1. For the sake of brevity, denote θ1 = E[X] and θ2 = E
X2
. Sensible estimators are We will be interested in testing the joint null hypothesis (θ1, θ2) = (θ1,0, θ2,0), using three different tests,
which will be outlined in the remainder of this question. All of these tests use the test statistic
and we will obviously reject H0 if this statistic is large. All tests should have a nominal level of 5%,
should use the boundary values θ1,0 = 0 and θ2,0 = 1 in the null hypothesis, and should be based on
samples of size n = 25. In each of part (a), (b), and (f), perform a simulation study based on 10 000
generated data sets to assess the size of each test, as well as the power against each of the following three
alternatives: (i) θ1 = 0 and θ2 = 0.8; (ii) θ1 = 0.2 and θ2 = 1; (iii) θ1 = 0.2 and θ2 = 0.8. For the sake
of convenience, use a normal distribution when generating the data sets for your simulation studies.
