Loyola College M.Sc. Statistics April 2003 Testing Of Hypothesis Question Paper PDF Download

 

 

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

  1. 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 =.

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

parameter exponential family.

  1. Define UMPUT.
  2. 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) ?

  1. Define: Maximal Invariance

 

 

 

 

 

 

 

Section B                       (5 ´ 8 = 40 Marks)

Answer any FIVE.  Each carries EIGHT marks

  1. Let = (X1 , X2,….., xn) where Xi’s  are  i i d  with pdf  pq(x) = e-(xq),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 .

  1. 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 .

  1. 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.

  1. 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
  1. Let X be an observation drawn from a population with pdf

pq (x) = q eqx, 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.

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

H : l £ m Vs K:  l >m.

  1. Write a descriptive note on invariant tests.

SECTION C                      (2 ´ 20 = 20 Marks)

Answer any TWO.  Each carries twenty marks.

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

pq()  = c(q) eqT(x)      h(x)

  1. 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

  1. 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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