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 b_{f} (q_{1}) = b_{f} (q_{2}) (q_{1}¹q_{2}) ?

- Define: Maximal Invariance

**Section ****–**** B** (5 ´ 8 = 40 Marks)

# Answer any FIVE. Each carries EIGHT marks

- Let = (X
_{1 }, 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 £ q_{0} Vs K: q >q_{0} .

- For each q
_{0} ÎW, let A (q_{0}) be the acceptance region of a level – a test for testing.

H(q_{0}): q = q_{0} 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(q
_{0}) is UMP for testing H (q_{0}) at level a against K(q_{0}), then for each q_{0}ÎW,

Show that S(x) minimizes p_{q}(q_{0}ÎS(x)) ” q ÎK(q_{0}) among all level (1-a) families of confidence sets for q .

- Solve the problem of minimizing ò f f
_{m+1}dm subject to ò f f_{i }d m = c_{i} , i = 1,2,….,m ,

where f_{1},f_{2,} …..,f_{m }, f_{m+1 } are (m+1), m integrable functions.

- Let the distribution of X be given by

X |
0 1 2 3 |

P_{q} (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

p_{q} (x) = q e^{–}^{q}^{x}, 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 =q
_{0 }Vs K: q ≠ q_{0} in

p_{q}() = c(q) e^{q}^{T(x) } h(x)

- Let (X
_{1},X_{2,} ….,X_{m}) and (Y_{1,}Y_{2,…,}Y_{n}) be samples of sizes m and n respectively

from N(x, s^{2}) 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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