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This is the Past Exam of Likelihood Inference which includes Quantiles, Table, Independent, Identically Distributed, Exponential Random Variables, Expectation, Probability Density Function, Distribution Function, Maximum Likelihood etc. Key important points are: Quantiles, Table, Independent, Identically Distributed, Exponential Random Variables, Expectation, Probability Density Function, Distribution Function, Maximum Likelihood, Expected Information
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PART II (Third or Fourth Year)
MATHEMATICS & STATISTICS
Math 350 Likelihood Inference 2 hours
You should answer all Section A questions and TWO Section B questions.
In Section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is capped at 40.
The following table of quantiles of a χ^2 d may be required.
d 95% quantile 97.5% quantile 1 3.8 5. 2 6.0 7. 3 7.8 9.
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A1. Let X 1 ,... , Xn be independent and identically distributed Exponential random variables, with expectation θ > 0, with probability density function
f (x; θ) =
θ exp(−x/θ)^ for^ x >^0 0 otherwise, and distribution function
F (x; θ) =
1 − exp(−x/θ) for x > 0 0 otherwise. . (a) Find the maximum likelihood estimator of θ. [5] (b) Show that the expected information on θ is nθ−^2. [5] (c) If the width of the 95% confidence interval for θ is 0.24 from 100 observations, approxi- mately what sample size is required for this width to be 0.12? [5] Let φp be the pth^ quantile of the Exponential distribution, i.e. P (X < φp) = p. (d) Show that φˆp = −x¯ log(1 − p). [5] (e) Show that as n → ∞ φ^ ˆp ∼ N
φp, φ^2 p n
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SECTION A continued
A4. The left panel of the figure below shows 25 observations drawn independently from a Normal(μ=5,σ^2 =9) distribution and a histogram of the data. It also shows 4 density func- tions corresponding to different Normal distributions. The right panel gives the contours of the log-likelihood function (μ, σ) assuming a Normal(μ,σ^2 ) model for the data. Each of the 4 points (labelled “0” to “3”) corresponds to one of the densities (labelled “a” to “d”) in the left panel.
− 2 0 2 4 6 8 10 12
b
d c
a
mu
sigma
2 4 6 8
2
3
4
5
2
01 3
Match the points in the right panel with the densities in the left panel (you do not need to explain your reasoning). [8]
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B1. Let X 1 ,... , Xn, Y 1 ,... , Yn be independent random variables with
Xi ∼ N (μ, σ^2 ) and Yi ∼ N (0, σ^2 ) for i = 1,... , n. (a) Show that up to an additive constant the log-likelihood of (μ, σ) is
(μ, σ) = − 2 n log(σ) − (^21) σ 2
∑^ n i=
(xi − μ)^2 − (^2) σ^12
∑^ n i=
y^2 i. [4]
(b) Show that (ˆμ, σˆ) =
⎝x,¯
2 n
∑^ n i=
(xi − x¯)^2 + (^21) n
∑^ n i=
y^2 i
(c) Find the expected information matrix of (μ, σ), and explain why this shows that these parameters are orthogonal. [6] (d) Briefly list the general benefits for inference of having orthogonal parameters over non- orthogonal parameters. [3] (e) Derive the asymptotic distribution (using the expected information) of (ˆμ, σˆ). [3] (f) Show that, up to an additional constant, the profile log–likelihood for μ is
P (μ) = −n log
σˆ μ^2
where σˆ μ^2 = (^21) n
( (^) ∑n
i=
(xi − μ)^2 +
∑^ n i=
y^2 i
(g) A 95% confidence interval for σ obtained using just the data from the X variables is (0. 9 , 1 .2). Would you expect this interval to widen, narrow or remain unchanged by using all the data on X and Y variables? Justify your answer. [3]
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SECTION B continued
Question B2 continued
B2. (b) (iii) We wish to test whether the probability of a win is identical to the probability of a loss. Show that this corresponds to
θ 1 =^12 (1 − θ 2 ).
[5] (iv) Show that the likelihood ratio test of statistic W of the test
H 0 : θ 1 =^12 (1 − θ 2 ) v H 1 : θ 1 , θ 2 unconstrained
is
W = 2
x 1
log
( (^) x 1 n
− log
( (^) x 1 + x 3 2 n
log
( (^) x 3 n
− log
( (^) x 1 + x 3 2 n
(v) If W = 4.2, explain what decision you would make concerning the validity of H 0. [3]
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SECTION B continued
B3. (a) Let X 1 ,... , Xn be independent and identically distributed Poisson (μ) random variables with probability mass function
P(Xi = x; μ) = μ
x (^) exp(−μ) x! for μ > 0. (i) Show that ˆμ = ¯x. [4] (ii) For data on the number of accidents at a factory each day ˆμ = 1.5 and 95% con- fidence interval for μ, found using the deviance method, is (1. 2 , 1 .8). Estimate the probability of no accidents tomorrow and give a 95% confidence interval for the estimate. [6] (b) The number of accidents Y 1 ,... , Yn in n different parts of the factory are thought to be independent of each other following Poisson distributions with different mean values. Specifically in part i of the factory the mean number of accidents is taken to be
μi = exp(α + βzi)
where zi is a known covariate. (i) Show that (ˆα, βˆ) satisfy αˆ = log
( (^) ∑n ∑ i=1^ yi ni=1 exp( βzˆ i)
and (^) ( (^) n ∑ i=
yizi
)( (^) n ∑ i=
exp( βzˆi)
( (^) n ∑ i=
yi
)( (^) n ∑ i=
zi exp( βzˆ i)
(ii) If IE
α,ˆ βˆ
and (α,ˆ βˆ)^ = (1, 1) estimate the expected number of accidents in a part of the factory with covariate z = 0.5 and find the standard error of this estimate. [7]
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