LINKÖPINGS UNIVERSITET
Institutionen för datavetenskap
Statistik, ANd
732A23 THEORY OF STATISTICS, 6 CDTS
Master’s program in Statistics and Data Mining
Spring semester 2011
Written exam
Written exam August 11, 2011
Rules:
Hours: From 09.00 until 13.00
Means allowed: The textbook “Garthwaite PH, Joliffe IT, Jones B. (2002). Statistical Inference. 2nd ed.
Oxford University Press.”, Calculator.
In your solutions, provide details of your derivations. Do not only provide the answer.
Grades: The grading on this home exam uses the whole ECTS-scale, i.e. the grades may range from F to
A. Grades are given on basis of the total impression from the solution of tasks, no specific points or intervals
of points are used.
Good luck!
Task 1
Assume a sample of observations x=(x1 , . . . , xn ) is from the Gamma distribution with p.d.f.
3 x
x
f (x; λ) =
e− λ , x > 0
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λ · 3!
(a) Find a minimal sufficient statistic for λ.
(b) Find the maximum likelihood estimator of λ.
Task 2
Assume the bivariate distribution for the random variable (X, Y ) has p.d.f.
f (x, y; θ) = θ−2 e−y/θ , 0 < x < y < ∞ , θ > 0
(a) Investigate if the marginal distribution of the random variable X belongs to the oneparameter exponential family of distributions
Assume a sample of n pairs of values (xi , yi ) from the bivariate distribution.
(b) Find an unbiased point estimator of θ based on the y-values of the sample only and
determine a lower bound for the variance of that point estimator.
(c) Find an approximate 95% confidence interval for θ when ȳ = 3 and n = 100
Task 3
Let x=(x1 , . . . , xn ) be a random sample from the Geometric distribution with p.m.f.
f (x; π) = (1 − π)x−1 π , x = 1, 2, . . . ; 0 < π < 1
(a) Find the form of the best test for H0 : π = π0 vs. H1 : π = π1 , where π1 > π0 .
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(b) The sum of n independendent geometrically distributed random variables all with
parameter π follows a negative binomial distribution with p.m.f
x+n−1
f (x; π) =
· (1 − π)n π x
x
If the size of the test is supposed to be 5%, give the equation for finding the critical
region of the best test when π0 = 0.2 and π1 = 0.1.
(c) Find the MLRT for H0 : π = 0.2 vs. H1 : π 6= 0.2.
Task 4
Let x=(2, 1, 9) be a random sample from an exponential distribution with p.d.f. f (x; θ) =
θe−θx , x > 0; θ > 0.
(a) Show that g(θ) = (1/4)e−θ/4 is a conjugate prior for θ.
(b) Find using the above prior the Bayes estimator of θ under quadratic error loss. Calβ α xα−1 e−βx
culate its value. [Hint: The gamma distribution with p.d.f. f (x; α, β) =
Γ(α)
has mean α/β and variance α/β 2 ]
Task 5
In a chemical analysis the measured amount of a certain substance is assumed to be
N (µ, σ = 1)-distributed. The kinds of materials measured are such that µ can be assumed
to possess a N (2, σ = 1) prior distribution. Assume a sample of 10 measurements has the
sample mean 2.7.
(a) Find a 90% equal-tailed credible interval for µ.
(b) If it is assumed that µ is different from 2 the standard deviation of the prior distribution
is rather assumed to be 1.5 than 1. Use this information to determine the posterior
odds of H0 : µ = 2 vs. H1 : µ 6= 2 when the prior odds is equal to one.
Task 6
Assume a sample x=(12, 10, 11, 15, 22, 13, 16) and a sample y=(16, 28, 25, 32, 18, 29, 30)
independent of x, both from continuous populations. Use the Mann-Whitney test to decide
upon whether the two samples come from populations with different locations. Assume the
samples are large enough for applying appropriate approximations, although this might not
be true.
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