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This is the Past Exam of Statistical Inference which includes Identically Distributed, Sample, Likelihood, Estimating, Asymptotic Distribution, Deviance Function, Statistical Model, Regularity Conditions, Values, Linear Regression Model etc. Key important points are: Asymptotic Distribution, Maximum Likelihood, Single Parameter, Mentioning, Likelihood, Same Parameter, Distribution, Null Hypothesis, Deviance, Function
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PART II (Third year)
MATHEMATICS & STATISTICS 2 hours
MATH331: Statistical Inference
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. Note there is a formula sheet at the back of this exam. Throughout the paper “MLE” will stand for “maximum likelihood estimator”.
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.
SECTION A
A1. (a) State the asymptotic distribution of the maximum likelihood estimator for a model with a single parameter θ. [2] (b) Indicate how the result in Part (a) can be used to find an approximate 95% con- fidence interval (CI), mentioning a drawback of this method when the likelihood function is asymmetric. [3] (c) Define the deviance for the same parameter and state the distribution of its deviance under the null hypothesis as n → ∞. [3] (d) How would a deviance based 95% CI would be calculated? (Plot the deviance function together with the asocciated C.I.) [5]
please turn over
SECTION A continued
A2. Let X 1 ,... , Xn be independent with Xi ∼ N (α + βzi, 1) for known constants {zi}. (a) Calculate the log-likelihood (α, β). [3] (b) Show that the MLEs, (ˆα, βˆ), satisfy the equation ¯x = ˆα + βˆ z,¯ where ¯x and ¯z and the sample means of x and z respectively. [3] (c) Show that Fisher’s information matrix IE (α, β) can be expressed as
IE (α, β) =
n ∑ni=1 zi ∑n i=1 zi
∑n i=1 z^2 i
(d) Give a condition which leads to the orthogonality of α and β. (^) [3]
A3. An observation x is taken from a binomial distribution X ∼ Binomial (n, π). The probability of success, π, is allocated a beta prior π ∼ Beta (α, β). (a) Find and identify the posterior distribution of π. [3] (b) Define the marginal likelihood m(x) and show that for the likelihood and prior in (a) that m(x) =
n x
B(α + x, β + n − x) B(α, β) , where B(α, β) = Γ( Γ(αα)Γ(+ββ)). [5] (c) Why is the marginal likelihood a useful quantity for Bayesian statisticians? [2]
A4. (^) (a) Define Jeffreys’ prior and explain briefly why it is often preferred over other methods for specifying prior ignorance. [4] (b) Calculate the Jeffrey’s prior in the case where X 1 ,... , Xn are independently distributed with a Geometric (θ) distribution, and determine the corresponding posterior distribution for θ. Note that the probability mass function for the geometric distribution is given by f (x | θ) = (1 − θ)x−^1 θ x = 1, 2 ,... and E(X) = (^1) θ. [6] (c) This is an improper distribution. Explain what this means and why this is the case. [4]
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SECTION B continued
B2. (a) Give one advantage and one disadvantage of using a conjugate family of prior distributions in a Bayesian analysis. [4] (b) The variables Y 1 ,... , Yn are independently distributed as Yi ∼ Poisson (xiθ), where x 1 ,... , xn are known constants. The prior distribution for θ is chosen to be Gamma(p, q). (i) Show that the posterior distribution of θ|y, x is Gamma(P, Q) where P = p +
∑^ n i=
yi, Q = q +
∑^ n i=
xi,
and y = y 1 ,... , yn, x = x 1 ,... , xn.
(ii) Show that the posterior mean, E[θ|x, y], satisfies
E[θ|x, y] = γμ 0 + (1 − γ)
∑ (^) y ∑ i xi^ , where μ 0 is the prior mean and γ is a constant to be found. [4] (iii) Give an interpretation of the result in (ii) and find under what conditions the posterior mean will be greater than the prior mean. [5]
(c) We assume that the data, X 1 ,... , Xn are independent Normal variables such that Xi|θ ∼ Normal
θ, σ^2
, i = 1,... , n, (i) Assume θ is assumed known and the unknown precision, τ = (^) σ^12 , has the prior distribution τ ∼ Gamma (α, β). Show that the posterior distribution of τ can be written as τ |x 1 ,... , xn ∼ Gamma (A, D) and find the quantities A and D. [5] (ii) Use the same likelihood but this time assume σ^2 known and θ unknown with prior θ ∼ Normal
μ, σ^2
. The posterior distribution of θ can be shown to be θ | x 1 ,... , xn ∼ Normal
μp, ψ^2
where ψ^2 is the posterior variance and μp is the posterior mean. Let Y be a future observation from the same likelihood. Show that the predictive. distribution of Y can be written as Y |x 1 ,... , xn ∼ Normal
μp, σ^2 + ψ^2
State all necessary assumptions. [6] please turn over
SECTION B continued
B3. (a) (From Antleman, 1997). Suppose that a trucking company owns a large fleet of well-maintained trucks and assume that breakdowns appear to occur at random times. The president of the company is interested in learning about the daily rate R at which breakdowns occur. For a given value of the rate parameter R, it is known that the number of breakdowns y on a particular day has a Poisson distribution with mean R. (i) Suppose that one observes the number of truck breakdowns for n consec- utive days and denote them by y 1 ,... , yn. If one assumes that these are independent measurements, derive an expression for the likelihood of these observations y 1 ,... , yn. [2] (ii) Find the MLE of R and its associated asymptotic standard error. [5] (b) Suppose that an employee of the trucking firm wishes to learn about the break- down rate. She is uncertain about the value of R and so assigns the non infor- mative prior density: p(R) ∝ (^) R^1 , R > 0 (i) Show that the posterior distribution of R is from the gamma family and show that the posterior mean of R is the same as the MLE. [3] (ii) Explain in what way the interpretation of a classical confidence interval is different to the interpretation of a Bayesian credible interval. [4] (c) The president has some knowledge about the location of the Poisson rate pa- rameter R based on the observed number of breakdowns from previous years. His prior beliefs about R are represented by means of the gamma density
g(R) ∝ R^4 −^1 exp(− 2 R) R > 0.
(i) Find the mean, variance and mode of the president’s prior knowledge about the rate parameter, R. Draw a rough sketch of these beliefs. [4] (ii) Using this prior, find the posterior density of R. [2] (iii) The sample mean over n = 12 observations is ¯y = 2. Explain a difference and a similarity of this posterior distribution from the one derived in part (b) under a non-informative prior. Interpret this result. [5]
Question B3 is continued over the page
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Formula Sheet
You may use the following:
∗ A univariate Normal with mean μ ∈ IR and variance σ^2 , is denoted by Normal
μ, σ^2
and the corresponding density function is:
p (y | μ, σ) = √^1 2 πσ^2 exp
− (^21) σ 2 (y − μ)^2
for y ∈ R.
In terms of the precision τ = (^) σ^12 the univariate normal can be expressed as
p (y | μ, τ ) = τ^
(^12) √ 2 π exp
− τ 2 (y − μ)^2
for y ∈ R.
∗ A Gamma distribution with shape parameter α > 0 and rate parameter β > 0 is denoted by Gamma (α, β), and the corresponding density is:
p(y | α, β) = β
α Γ(α) y
α− (^1) exp(−βy) for y > 0.
The Gamma distribution, Gamma (α, β), has mean α β and variance (^) βα 2.
∗ An Exponential distribution with rate parameter λ > 0 is denoted by Exp (λ) and the corresponding density is:
p(y|λ) = λ exp(−λy) for y > 0.
∗ A Beta distribution with parameters α > 0 and β > 0 is denoted by Beta (α, β), and the corresponding density is:
p(y | α, β) = (^) B(α, β^1 ) yα−^1 (1 − y)β−^1 for 0 < y < 1.
where B(α, β) = Γ( Γ(αα)Γ(+ββ)) where Γ(n + 1) = n! for integer n. The Beta distribution, Beta (α, β), has mean (^) αα+β.
∗ A Binomial distribution with parameter 0 ≤ p ≤ 1 is denoted by Binomial (n, p), and the corresponding probability mass function is:
p(y | n, p) =
n y
py(1 − p)n−y^ for y = 0, 1 ,... , n.
∗ A Poisson distribution with parameter λ > 0 is denoted by Poisson (λ), and the corresponding probability mass function is:
p(y | λ) = e
−λ y! λ
y (^) for y = 0, 1 ,....
The mean and variance of a Poisson distribution is λ.
∗ A Negative binomial distribution with parameters 0 ≤ θ ≤ 1 and k ∈ { 1 ,.. .} is denoted by Negative-Binomial (θ, k), and the corresponding probability mass function is:
p(y|θ, k) =
y + k − 1 k − 1
θk(1 − θ)y, y = 0, 1 , 2 ,...