LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AC 56

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – APRIL 2007

ST 4900 – MATHEMATICAL STATISTICS – II

Date & Time: 25/04/2007 / 9:00 – 12:00 Dept. No. Max. : 100 Marks

SECTION-A (10 x 2 = 20)

Answer ALL the questions. Each carries 2 marks.

Distinguish between point and interval estimation.
Examine whether the sample variance is a biased estimator of s² when a random sample of size ‘n’ is drawn from N(m, s² ).
When do you say that a statistic is a consistent estimator of a parameter?
Distinguish between power function and power of the test.
What is degree of freedom?
Illustrate graphically the meaning of UMPT.
What do you understand by likelihood ratio tests?
Define F statistic.

Define: Markov Chain.

Classify the stochastic processes with respect to time and state space.

SECTION-B (5 x 8 = 40)

Answer any FIVE questions. Each carries 8 marks.

Let Y₁ and Y₂ be two independent unbiased estimators of θ. Let the variance of Y₁ be twice the variance of Y₂. Find the constants k₁ and k₂ so that k₁ Y₁ + k₂ Y₂ is an unbiased estimator with smallest possible variance for such a linear combination.
State and prove Rao-Cramer Inequality.

Let X₁, …, X_n be i.i.d., each with distribution having p.d.f.

f(x; θ₁, θ₂ ) = (1 / θ₂) exp{ – ( x- θ₁ ) /θ₂}, -θ₁ ≤ x< ∞, -∞ < θ₁< ∞, 0 <θ₂< ∞,

0; elsewhere.

Find the maximum likelihood estimators of θ₁ and θ₂.

Let X ≥ 1 be 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.

Let X₁, X₂…X_n be iid U(0,q), q>0. Show that the family of distributions has MLR in X_(n).

A random sample of size 14 drawn from a normal population provides a sample mean of 3.22mm with an unbiased standard deviation of 0.34mm. Can you conclude at 5% level of significance that it meets the company’s specification of 2.7mm against more than 2.7mm? Construct 95% confidence limits for the population mean.
Let X_1,X_2,…,X_nbe iid N(m,s²). Derive LRT for testing H₀: m = m ₀against

H₁: m ¹ m_0,s² is unknown.

Describe Poisson process.

SECTION-C (2 x 20 =40)

Answer any TWO questions. Each carries 20 marks.

a) State and prove factorization criterion for determining sufficient statistics. (12)
b) Show that the first order statistic Y₁ of a random sample of size n from the

distribution having p.d.f.

f(x: θ) = e^{–(x – θ)}, -¥ < x < ¥, – ¥ < θ < ¥, zero elsewhere,

is a complete sufficient statistic for θ.

Find the unique function of this statistic, which is the unbiased minimum variance

estimator of θ. (8)

State and prove the necessary and sufficient conditions of Neyman-Pearson

Fundamental Lemma. (10+10)

a) Let X_1,X₂and X₃ be a sample of size 3 from Poisson distribution with mean θ.

Consider the problem of testing H₀: θ = 2 against H₁: θ = 3. Find the randomized

MPT of level α =0.05. (12)

b) Prove or disprove:

“UMPT of level α always exists for all types of testing problems”.

Justify your answer. (8)

a) A number is to be selected from the interval (x : 0 < x < 2) by a random process.

Let A_i = {(x : (i-1)/2 < x < i/2}, i = 1,2,3, and let A₄ = {x : 3/2 < x < 2}. A certain

hypothesis assigns probabilities p_io to these sets in accordance with

p_io = ò_Ai (1/2) (2-x) dx, i = 1,2,3,4. If the observed frequencies of the

sets A_i, i = 1,2,3,4, are respectively, 30, 30, 10, 10, would H_o be accepted at the

(approximate) 5 percent level of significance ? (10)

b) Consider a Markov chain having state space S = {0,1,2} and transition probability

matrix

1/3 1/3 1/3

¼ ½ ¼

1/6 1/3 ½

Show that this chain has a unique stationary distribution p and find p. (10)

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