AHK/KW/19/1390
Master of Science (M.Sc.) Semester—I (CBCS) (Statistics) Examination
ESTIMATION THEORY
Paper—3
Paper—III
Time : Three Hours] [Maximum Marks : 80
N.B. :— All questions are compulsory and carry equal marks.
EITHER
1. (a) Define unbiased estimator of a parameter. If X1, X2 is a random sample from Bernoulli distribution
with parameter θ, obtain unbiased estimator of ψ(θ) = θ(1 – θ).
(b) Describe method of moments with suitable example. 8+8
OR
(c) Define the following terms :
(i) Likelihood function
(ii) Maximum likelihood estimator.
Show that MLE need not always be unbiased by giving suitable example.
(d) If T1 is MVUE of θ with variance σ 2 and T2 is another unbiased estimator of θ with variance
σ 2/E, where E is efficiency of the estimator T2, then prove that correlation coefficient between
T1 and T2 is ρ = E . 6+10
EITHER
2. (a) State Cramer-Rao inequality. Let X ~ N(µ, 1). Obtain CRLB for variance of UE of ψ(µ) = µ2.
(b) Define sufficient statistic and minimal sufficient statistics. Obtain sufficient statistic for a parameter
of Bernoulli distribution. 8+8
OR
(c) Define Fisher’s information. Show that generally there is a loss of information when data is
reduced by defining a statistic. Give the condition on statistic such that there is no loss of
information.
(d) State and prove Factorization theorem. 8+8
EITHER
3. (a) State Lehmann-Schffe theorem. Obtain MVUE of λ on the basis of a random sample of size
n from exponential distribution with mean λ using above theorem.
(b) Describe the meaning of completeness and show that the family of Poisson distribution is
complete. 8+8
OR
(c) Let X be a r.s. with pdf
p θ ( x ) = θ2 · (1 − θ) x ; x = 0, 1, 2, ....
=1− θ ; if x = − 1, 0 < θ < 1
Show that this family of distribution is not complete.
(d) Define Pitman family. Let {f(x, θ) ; θ ∈ Ω} belong to a Pitman family; then show that for
a(θ) = a = constant, X(n) is minimal sufficient statistic; symbols carry usual meanings. 8+8
CC—9941 1 (Contd.)
EITHER
4. (a) Discuss the problem of interval estimation. Let X1, X2 be a r.s. from exponential distribution with
mean θ. Obtain level of confidence interval (X1, X1 + X2).
(b) Construct a(1 – α) 100% C.I. for the parameter p of Binomial distribution when sample size is
large. 8+8
OR
(c) Define :
(i) Pivot
(ii) Confidence interval
(iii) Level of confidence.
(d) Explain shortest length C.I. Obtain it for σ 2 on the basis of r.s. taken from N(µ, σ 2) where
(i) µ is known (ii) µ is unknown. 6+10
5. (a) Show that unbiased estimator need not always exist.
(b) Describe Blackwellization.
(c) If the distribution belongs to one-parameter exponential family, show that M = Σk(Xi) is always
minimal sufficient statistic.
(d) If X1, ..... Xn is a random sample from uniform distribution over (0, θ), obtain pivot.
4+4+4+4
CC—9941 2 AHK/KW/19/1390
