LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

NO 39

SECOND SEMESTER – APRIL 2008

ST 2809 – TESTING STATISTICAL HYPOTHESIS

 

 

 

Date : 22/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

SECTION A

Answer all questions.                                                                                      (10 x 2 = 20)

 

1. Define Test function.
2. Let X be a random variable with pdf .

Consider the problem of testing H₀: θ = 1 Vs H₁: θ = 2.

Define. Find the level and power of the above test.

1. Show that one parameter exponential family has MLR property.
2. Define Uniformly Most Powerful Test.
3. Show that a test function having Neyman Structure is similar.
4. Comment on the following: “UMP always exist”.
5. What is meant by shortest length confidence interval?
6. What is the connection between confidence interval and hypotheses test?
7. Define Maximal Invariant function.
8. What is meant by nuisance parameter? Give an example.

 

SECTION B

Answer any FIVE questions.                                                                    (5 x 8 = 40)

 

1. Let .Derive a UMP test of size 0.05 for testing Vs
2. Let X₁,X₂,…,X_n be a random sample of size n from U(0, θ). Show that the above family of distributions has MLR in X_(n).
3. Let X₁,X₂,…,X_n be a random sample of size n from N(0,σ²). Derive UMPUT of size α for testing H₀:σ² = σ₀²  Vs  H₁: σ² ≠ σ₀².
4. Consider a (k+1) parameter exponential family with density . Define V = h(U,T) such that V is independent of T for θ = θ₀ and V is increasing in U for a fixed T= t. Derive an unconditional UMPUT of size a for testing  Vs .
5. Write short notes on a.) Locally most powerful test   b.) Similar test.
6. Let X₁,X₂,…,X_n be a random sample of size n from N(μ,σ²). Construct a (1- a) level confidence bound for m when i.) s is known and ii.) s is unknown.
7. Define Likelihood Ratio Test and show that the critical region provided by the Neyman – Pearson test and Likelihood Ratio Test are same when H₀ and H₁ are simple.
8. Find maximal invariant function under the group of i.) location transformations and ii.) scale transformations.

 

SECTION C

Answer any TWO questions.                                                                         (2 x 20 = 40)

 

1. State and prove the necessary and sufficient conditions of Neyman – Pearson lemma.
2. Consider a one parameter exponential family with density f(x) = c(θ)e^Q(θ)T(x)h(x). Assume Q(θ) is strictly increasing in θ. Show that for testing  Vs , there always exist UMP test of level α and is of the form

where c_i and g_I (i=1,2) are selected so that .

1. Let X and Y be independent Poisson variables with means l and m respectively. Derive UMPUT of size a for testing H₀: l ≤ am  Vs  H₁: l > am  where a > 0.
2. a.) Derive Likelihood ratio test for testing H₀: m = m₀ Vs H₁: m ≠ m₀ when a random sample of size n is drawn from N(m,s²) where s² is unknown.

b.) Derive LMP test for testing H₀: θ ≤ 0 Vs H₁: θ > 0 based on a random sample of size n from C(θ,1).                                                                                       (14+6)

 

 

 

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