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.

 

1. Let X be an observation from

Find the power of the test

 

 

for testing H: q =1 against K: q = 2

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 =.

3. When do you say that a family of density functions has MLR property?
4. What is a similar test?
5. Define: Confidence set.
6. Examine the validity of the statement “A MPT is always unbiased”.
7. Give an example of a family having MLR property but not a member of one

parameter exponential family.

8. Define UMPUT.
9. 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    b_f (q₁) = b_f (q₂) (q₁¹q₂) ?

10. Define: Maximal Invariance

 

 

 

 

 

 

 

Section – B                       (5 ´ 8 = 40 Marks)

Answer any FIVE.  Each carries EIGHT marks

11. Let = (X₁ , X_2,….., x_n) where X_i’s  are  i i d  with pdf  p_q(x) = e^-(x–q),x >q, q >0 .

Show that the family of densities p_q() has MLR property and hence derive

the UMPT of level a for testing H :  q £ q₀ Vs K: q >q₀ .

12. 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 p_q(q₀ÎS(x)) ” q ÎK(q₀) among all level (1-a) families of confidence sets for q .

13. Solve the problem of minimizing ò f f_m+1dm subject to ò f f_i d m = c_i , i  = 1,2,….,m ,

where f₁,f_2, …..,f_m , f_m+1     are (m+1), m integrable functions.

14. 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

15. Let X be an observation drawn from a population with pdf

p_q (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.

16. State and prove a necessary condition for all similar tests to have Neyman structure.
17. Let X and Y be independent Poisson varictes with means l and m

H : l £ m Vs K:  l >m.

18. Write a descriptive note on invariant tests.

SECTION – C                      (2 ´ 20 = 20 Marks)

Answer any TWO.  Each carries twenty marks.

19. State and prove Neyman – Pearson lemma.
20. Derive the UMPUT for testing H: q =q₀ Vs K: q ≠ q₀ in

p_q()  = c(q) e^qT(x)      h(x)

21. Let (X₁,X_2, ….,X_m) and (Y_1,Y_2,…,Y_n) 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

22. 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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