LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 34

SECOND SEMESTER – APRIL 2006

                                         ST 2809 – TESTING STATISTICAL HYPOTHESIS

(Also equivalent to ST 2807/2802)

 

 

Date & Time : 21-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

SECTION  A                           Answer all the questions                            10 x 2 = 20

1. Define test function and randomized test function.
2. Let X be B(1, q), q = 0.2,0.4,0.5. For testing H: q = 0.2,0.4 Vs K: q = 0.5, a test is given by

f(x)   =  0.3,    x = 0

=  0.6,    x =  1.

Find the size of the test.

3. Show that a UMP level a test is unbiased.
4. Define MLR property and give an example.
5. Show that a test with Neyman structure is similar.
6. Describe Type I and Type II right censoring.
7. Give two examples for multiparameter exponential family.
8. Define location family and give an example.
9. Describe likelihood ratio test.
10. Explain UMA and UMAU confidence sets.

 

SECTION B                            Answer any five questions                           5 x 8 = 40

11. Let X be DU{1,2,…, q }, q = 1,2. For testing H: q = 1 Vs K: q = 2,  find MP level a test using LP technique.
12. Give an example of a testing problem for which UMP test does not exist.
13. Given a random sample of size n from E(0, q ), q > 0, derive UMP level a test for testing H: q £ q ₀ Vs K: q > q ₀.Examine whether the test is consistent.
14. If the power function of an unbiased test is continuous, show that the test is similar.

15.Given a random sample of size n from P( q ), q > 0, derive UMPU level a test for testing H: q = q ₀ Vs K: q ¹ q _0.

16.Show that a statistic is invariant if and only if it is a function of a maximal invariant statistic.

17.Derive likelihood ratio test for testing H: q = q ₀ Vs K: q > q ₀ based on a random sample from E(0,q), q >0.

18.Explain shortest length confidence interval and illustrate with an example.

 

 

 

SECTION C                           Answer any two questions                         2 x 20 = 40

19 a).   State and establish the sufficient part of Neyman-Pearson lemma.

1. b) Let X₁,X₂,…X_n denote a random sample of size n from E(q ,1), q e Examine if there exists UMP level a test for testing H: q = q ₀ Vs K: q ¹ q ₀.

20 a)  In the case of one-parameter  exponential family show that there exists UMP level a  test for testing one-sided hypothesis against one-sided alternative. State your assumptions.

1. b) Derive UMPU level a test for testing H:  q₁ £ q £ q₂ Vs K: q < q₁  or q > q                               based on a random sample from N(q , 1), q e R. Explain the determination of the constants.Is the test unique?

21 a)   Discuss the relation between similar tests and tests with Neyman structure.

3. b) Let X₁,X₂,…X_n  be a random sample from P( ) and Y₁,Y₂,…Y_m be a random sample from an independent  Poisson population P( ).Derive UMPU level a test for testing H:l £ m  Vs K:l > m. Determine the constants when  n = 2 and m = 1, X₁ = 1, X₂ = 2 and Y₁ = 3.

22 a)   State and establish the asymptotic null distribution of the likelihood ratio statistic.

1. b) For testing H:(X₁ , X₂ ) is BVN(q, q ,1,1, 0.5) Vs K: (X₁ , X₂ ) is BVN(q, q,1, 4, 0.5), derive UMPI level a test with respect to location transformations.

 

 

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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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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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 37

SECOND SEMESTER – April 2009

ST 2812 / 2809 – TESTING STATISTIACAL HYPOTHESIS

 

 

 

Date & Time: 22/04/2009 / 1:00 – 4:00       Dept. No.                                                       Max. : 100 Marks

 

 

 

SECTION A

Answer all questions.                                                                                    (10 x 2 = 20)

 

1. Define level and power of a test.
2. Let X be a random variable with pdf .

Obtain the Most Powerful Test of size for testing H₀: θ = 1 Vs H₁: θ = 2.

1. Give the general form of (k+1) parameter exponential family of distributions.
2. Define Uniformly Most Powerful Test.
3. Let. Consider the test function

for testing H₀: θ = 0.2 Vs H₁: θ > 0.2.Obtain the value of power function at

θ = 0.4.

1. What are the circumstances under which Locally Most Powerful test is used?
2. What is meant by shortest length confidence interval?
3. Define maximal invariant function.
4. What is meant by nuisance parameter? Give an example.
5. Define Likelihood Ratio Test.

 

 

SECTION B

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

 

1. Let  denote a random sample fromDerive a Most Powerful test of  level 0.05 for testing Vs. Also obtain the       cut-off point.
2. Show that the family of densities possesses MLR property.
3. Let denote a random sample of size n from. Consider the problem of testing Vs. Show that UMP test of  does not exist.
4. For (k+1) parameter exponential family of densities, derive an unconditional UMPUT of level for testing  Vs  clearly stating the assumptions.
5. State and prove the sufficient part of Generalized Neyman-Pearson lemma.
6. Show that any test having Neyman structure is similar. Also show that the converse is true under certain assumptions (to be stated).

 

 

1. Derive the Locally Most Powerful test for testing Vs based on a random sample of size n from, where  and  are known pdf’s.
2. 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. a.) Derive a UMP test of level  for testing  Vs  for the family of densities that possess MLR in T(x). Show that the power function of the above testing problem increases in

b.) Show that any UMP test is always UMPUT.                                          (16+4)

1. Consider a one parameter exponential family with density. Assume  is strictly increasing in. Derive a UMP test of level  for testing  Vs.
2. Let X and Y be independent Binomial variables with parameters and  respectively, where m and n are assumed to be known. Derive a conditional UMPUT of size  for testing  Vs.
3. Let anddenote independent random samples from  and respectively. Derive the Likelihood Ratio Test for testing Vs.

 

 

 

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