LOYOLA COLLEGE (AUTONOMOUS), CHENNAI-600 034.
M.Sc. DEGREE EXAMINATION – STATISTICS
second SEMESTER – APRIL 2003
ST 2802/ S 817 testing of hypothesis
24.04.2003
1.00 – 4.00 Max: 100 Marks
SECTION – A (10 ´ 2 = 20 Marks)
Answer ALL questions. Each carries TWO marks.
- Let X be an observation from
Find the power of the test
for testing H: q =1 against K: q = 2
- Give the test function
for testing H : q £ vs K: q > based on an observation drawn from B (3,q),
find the probability of rejecting H when q =.
- When do you say that a family of density functions has MLR property?
- What is a similar test?
- Define: Confidence set.
- Examine the validity of the statement “A MPT is always unbiased”.
- Give an example of a family having MLR property but not a member of one
parameter exponential family.
- Define UMPUT.
- Suppose a test function is of the form
for a family having MLR property in T(x). Can such a test function satisfy
the condition bf (q1) = bf (q2) (q1¹q2) ?
- Define: Maximal Invariance
Section – B (5 ´ 8 = 40 Marks)
Answer any FIVE. Each carries EIGHT marks
- Let = (X1 , X2,….., xn) where Xi’s are i i d with pdf pq(x) = e-(x–q),x >q, q >0 .
Show that the family of densities pq() has MLR property and hence derive
the UMPT of level a for testing H : q £ q0 Vs K: q >q0 .
- For each q0 ÎW, let A (q0) be the acceptance region of a level – a test for testing.
H(q0): q = q0 and for each sample point x let S(x) denote the set of parameter
values S(x) = {q|x ÎA(q), qÎW}
- Show that s(x) is a family of confidence sets for q at confidence level 1-a.
- If A(q0) is UMP for testing H (q0) at level a against K(q0), then for each q0ÎW,
Show that S(x) minimizes pq(q0ÎS(x)) ” q ÎK(q0) among all level (1-a) families of confidence sets for q .
- Solve the problem of minimizing ò f fm+1dm subject to ò f fi d m = ci , i = 1,2,….,m ,
where f1,f2, …..,fm , fm+1 are (m+1), m integrable functions.
- Let the distribution of X be given by
| X |
0 1 2 3 |
| Pq (X = x) |
q 2q 0.9.-2q 0.1-q |
Where 0 < q < 0.1. For testing H: q =0.05 against K: q > 0.05 at
level a =0.05, determine which of the following tests (if any) is UMP.
(i) f (0) = 1, f (1) = f(2) = f(3) = 0
- f (1) = 5, f(0) = f(2) = f(3) = 0
- f (3) = 1, f(0) = f(1) = f(2) = 0
- Let X be an observation drawn from a population with pdf
pq (x) = q e–qx, x >0, q >0
Derive the UMPT of level a = 0.05 for testing H : q £ 1, q ³ 2 Vs K : 1< q < 2.
- State and prove a necessary condition for all similar tests to have Neyman structure.
- Let X and Y be independent Poisson varictes with means l and m
H : l £ m Vs K: l >m.
- Write a descriptive note on invariant tests.
SECTION – C (2 ´ 20 = 20 Marks)
Answer any TWO. Each carries twenty marks.
- State and prove Neyman – Pearson lemma.
- Derive the UMPUT for testing H: q =q0 Vs K: q ≠ q0 in
pq() = c(q) eqT(x) h(x)
- Let (X1,X2, ….,Xm) and (Y1,Y2,…,Yn) be samples of sizes m and n respectively
from N(x, s2) and N (ך,s). Derive the UMPUT
(unconditional) for testing (i) H :ך £ x Vs K: ך > x and
(ii) H: ך = x Vs K: ך ¹ x
- Illustrate, with an example, the steps involved in developing unconditional
UMPUT’s for one-sided testing problems in the multi-parameter
exponential setup
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