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This is the Exam of Statistical Science which includes Continuous Local Martingale, Filtered Probability, Space Satisfying, Usual Conditions, Qubit, Pure States, Bloch Vectors, Two States, Determine etc. Key important points are: Autocorrelation Function, Time Series, Weakly Stationary Proces, Weakly Stationary, Expression, Obtained, Density Function, Spectral Density Function, Gain, Taking
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Wednesday 4 June 2008 9.00 to 12.
Attempt no more than FOUR questions. There are SIX questions in total. The questions carry equal weight.
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1 Time Series
Explain what is meant by a weakly stationary process {Xt}. Define the autocovariance function and the autocorrelation function of {Xt}.
Let Xt = α (Xt− 1 − Xt− 2 ) + t , (1)
where α is a real constant and {t} is a white noise process with mean zero and variance σ^2. Determine the range of possible values of α for which (1) has a unique weakly stationary solution.
For α = − 1 /12 , find the Wold representation of {Xt} and determine the autocovariance function of {Xt}.
[Results from lectures may be quoted and used without proof.]
2 Time Series
Let {Xt}t∈Z be a weakly stationary process with autocovariance function γk and spectral density function fX (λ). Write down an expression for γk in terms of fX (λ).
The process {Yt} is obtained from {Xt} by applying the filter {ar}r∈Z , with ar ∈ R for all r ∈ Z and
r∈Z |ar|^ <^ ∞^ , so that^ Yt^ =^
r∈Z ar^ Xt−r^. Show that^ {Yt}^ is weakly stationary and find its spectral density function fY (λ) in terms of fX (λ) and a(λ) =
r∈Z ar^ e irλ (^).
∑ Let^ {Zt}^ be obtained from^ {Yt}^ by applying the filter^ {br}, with^ br^ ∈^ R^ for all^ r^ ∈^ Z^ and r∈Z |br|^ <^ ∞^. Write down the spectral density function^ fZ^ (λ) of^ {Zt}. Show that^ {Zt}^ can be obtained from {Xt} by applying a linear filter {cr}, and find cr in terms of the ak’s and the bk’s.
Let the gain of a filter {ar} be Ga(λ) = | a(λ)| , λ ∈ [0, π].
(a) Suppose that Yt = Xt − Xt− 1. Find fY (λ). Sketch the gain of the filter taking {Xt} to {Yt} and comment.
(b) Suppose that Zt = Yt − Yt− 12. Find fZ (λ). Sketch the gain of the filter taking {Yt} to {Zt} and comment.
(c) Find the filter that takes {Xt} onto {Zt} and find its gain.
Paper 47
4 Monte Carlo Inference
(a) Describe the jackknife and nonparametric bootstrap methods for estimating the variance of an estimator θˆ of some parameter θ(F ), on the basis of a random sample x 1 ,... , xn of distinct observations from F. Your description of the nonparametric bootstrap should include the form of the empirical distribution function Fˆn used in the algorithm.
(b) Find the probability that a bootstrap sample contains at least one repeated value.
(c) Consider the following R code where x is a vector of length n containing the random sample x 1 ,... , xn , where x and n have been set earlier in the code.
R1a> mat <- matrix(NA, nrow=n, ncol=n-1) R2a> for(i in 1:n) mat[i,] <- x[-i] R3a> vect <- apply(mat, 1, mean) R4a> (n-1)mean((vect - mean(vect))^2) R5a> (n-1)(mean(vect) - mean(x))
Explain what is being calculated in lines R4a and R5a. Give the numerical value of the expression in line R5a, and justify your answer. Now consider another piece of R code below (with the same x as above).
R1b> alpha <- 0. R2b> B <- 199 R3b> mat <- matrix(NA, nrow=B, ncol=n) R4b> for(b in 1:B) mat <- sample(x, n, replace=TRUE) R5b> vect <- apply(mat, 1, mean) R6b> s <- sort(vect) R7b> c(s[(B+1)alpha/2], s[(B+1)(1-alpha/2)])
Explain what is being calculated in the code, with particular attention paid to the value of the expression in line R7b.
(d) Suppose that we had another random sample y 1 ,... , ym from a distribution G 6 = F , where θ = EF {X} = EG{Y } and Cov(X, Y ) < 0. Give an algorithm for constructing an efficient, unbiased estimator θ˜ of θ that uses the combined sample x 1 ,... , xn, y 1 ,... , ym. How could you estimate Var(θ˜)?
Paper 47
5 Monte Carlo Inference
(a) (i) Describe the Gibbs Sampler for obtaining a dependent sample from some distribution π(θ), θ ∈ Rp. (ii) Suppose that we observe data y = (y 1 ,... , yn)>, with corresponding known (scalar) covariates x = (x 1 ,... , xn)>^ and that we want to fit a polynomial regression model of order k to the data. Then we can express the model in the form
y = Xkβk + ε
for design matrix
Xk =
1 x 1 · · · xk 1 .. .
1 xn · · · xkn
where βk = (β 0 , β 1 ,... , βk)>^ and ε = (ε 1 ,... , εn)>, with ε ∼ Nn( 0 , σ^2 I), where I is the n × n identity matrix. For independent priors σ^2 ∼ Γ−^1 (a, b) and βk ∼ Nk+1(μk, Σk) the posterior distribution is given by
π(βk, σ^2 |x, y) ∝ (σ^2 )−n/^2 exp
2 σ^2
(y − Xkβk)>(y − Xkβk)
× (σ^2 )−(a+1)^ exp
b σ^2
× exp
(βk − μk)>Σ− k 1 (βk − μk)
Show that the conditional distributions π(βk|σ^2 , x, y) and π(σ^2 |βk, x, y) are mul- tivariate normal and inverse gamma, respectively, and calculate the parameters of each distribution. (iii) Hence describe how we can use the Gibbs Sampler to obtain a dependent sample from the joint posterior distribution of π(β, σ^2 |x, y).
(b) Now suppose that the order of the polynomial is unknown, and that we wish to use a reversible jump procedure to update the order of the polynomial model. We propose to move from the model of order k, with parameters βk, to the model of order k + 1 with parameters β′ k+1 (keeping σ^2 fixed) using the following procedure,
β i′ = βi for i = 1,... , k β′ k+1 = z for z ∼ N (0, σ^2 β ) and σ^2 β known
β 0 ′ = β 0 − z n
∑^ n
i=
xk i +1.
(i) Calculate an explicit expression for the corresponding acceptance probability for this move. (ii) Define the reverse move, for moving from the model of order k + 1 to the model of order k. (iii) What is the corresponding acceptance probability for this reverse move, from the model of order k + 1, to the model of order k?
Paper 47 [TURN OVER
