LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 33

SECOND SEMESTER – APRIL 2007

ST 2809/ST 2807/ST 2802 – TESTING STATISTICAL HYPOTHESIS

 

 

 

Date & Time: 19/04/2007 / 1:00 – 4:00          Dept. No.                                                             Max. : 100 Marks

 

 

 

SECTION-A (10 x 2 = 20)

Answer ALL the questions.   Each carries 2 marks.

 

1. Distinguish between randomized and non-randomized tests.
2. What are the two types of errors in testing of hypothesis?
3.  Give an example of a family of distributions, which has MLR property.
4. State the necessary condition of Neyman Pearson Fundamental Lemma.
5. Use Graphical illustration to differentiate between MPT and UMPT.
6. Define the (k+1) parameter exponential family and give an example.
7. What do you mean by Unbiasedness?
8. When do you say that a test function is similar?
9. When do you say that a function is maximal invariant?
10. Explain briefly the principles of LRT.

 

SECTION-B  (5 x 8 = 40)

Answer any FIVE questions.  Each carries 8 marks.

 

11. If X ≥ 1 is the critical region for testing H₀: θ = 1 against H₁: θ = 2 on the basis of a single observation

from the population with pdf

f(x ,θ) =  θ exp{ – θ x },  0 < x <∞;  0 otherwise.

Obtain the size and power of the test.

 

12. State and prove MLR theorem of Karlin-Rubin.

 

13. Suppose there exists UMPT of size a for testing a composite H₀ against composite H₁ then show that it is

unbiased.

 

14. Let X_1, X₂, …, X_n be i.i.d random variables each with density

 

f(x, θ)  =   exp    { – (x_i-θ)}, θ< x < ∞,  -∞<θ<∞

0,  elsewhere.

 

Find the UMPT of size α for testing H₀: θ≤ θ₀ against H₁: θ > θ₀.

Also, obtain the cut-off point when α = 0.05, n=15 and θ₀ = 5.

 

 

 

 

 

 

15. Let X₁,X₂,…,X_n be iid C(θ, 1). Derive LMPT of size a for testing H₀:θ ≤ 0  against  H₁: θ > 0 and show

that it is biased.

 

16. Show that a function T is invariant under G if and only if T is a function of the maximal invariant.

 

 

17. Let the p.d.f. of X be f(x) =        (2/θ²)   (θ-x) ;  0< x < θ ,

 

0, otherwise

Construct 100(1-α)% confidence interval for θ.

 

18. Let X be a binomial variate with parameters n and p. Derive the likelihood ratio test of level α for testing

H₀: p ≤ p₀ against H₁: p > p₀.

 

 

SECTION-C (2 x 20 =40)

Answer any TWO questions.   Each carries 20 marks.

 

 

19. a) State and prove the sufficiency part of Neyman- Pearson Generalized theorem.            (12)

 

1. b) Show that UMPT of size α does not exist for testing H₀: μ= µ₀ against H₁: μ ≠ µ₀

when the sample of size ‘n’ is drawn from N(μ, 1).                                                     (8)

 

20. Let X and Y be independent Poisson variates with parameters λ and μ respectively. Derive the

unconditional UMPUT of size a for testing H₀: λ ≤ aμ against H₁: λ> aμ, where a > 0.             (20)

 

21. a) Consider the ( k+1) parameter exponential family. Suppose there exists a function

V =h(u,t) such that V is independent of T when q = q₀  and V is increasing in U for every fixed T then

derive the UMPT of size a for testing H₀ : q £ q₀  against H₁ : q > q_0.                                   (10)

 

1. b) Why do we require Locally optimal tests? How do you derive it using the

Generalized Neyman-Pearson theorem?                          (10)

 

22 a)   Let X_1, X_2,…..X_n be iid N(m,s²). Consider the problem of testing H₀: s £ s₀  against H₁: s > s_0.

 Derive  UMPIT for the above testing problem under the appropriate group of transformations.     (12)

 

1. b) Let X₁, X₂, …, X_n be iid U(0, θ) random variables.  Construct (1-α) – level UMA

confidence  interval for θ.                                                     (8)

 

 

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