Semester 2, 2021
Seat Number: …………………….. Last Name: ……………………….. Other Names: …………………….. SID: ……………………………….
The University of of Science
STAT3023, STAT3923 and STAT4023: Statistical Inference, Statistical Inference (Advanced) and Theory and Methods of Statistical Inference
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Lecturers: and Time allowed: Two hours
This booklet contains 7 pages.
CONFIDENTIAL
There are 6 questions of equal value over pages 2, 3 and 4. Attempt all questions. Show all working. Pages 5, 6 and 7 have a list of useful formulae.
Semester 2, 2021 page 2 of 7 1. Let X be a random variable with probability mass function
P(X=j)=θ(1−θ)j, j=0,1,..,
(a) Show that X is a member of the one parameter exponential family.
where 0 < θ < 1.
(b) Given n independent observations from this population, X1, X2, .., Xn, show
f(x,y)=e−(x+y), R = X and
x≥0, y≥0. T = X + Y.
that T = n Xis sufficient for θ.
(c) Show that X ̄ = ni=1 Xi/n is not an unbiased estimator for θ.
2. X and Y are two independent exponential random variables. Their joint pdf is
(a) State the distribution of T .
(b) Derive the joint pdf of (R, T ).
(c) ShowthatR∼U(0,1).
(d) Are R and T independent?
3. Suppose X1, . . . , Xn are independent normal random variables with mean zero and variance θ, and thus common density
fθ(x)=√2πθe 2θ , for−∞
[,1] [,2] [,3]
[1,] 1 0.02380952 0.02380952
[2,] 2 0.23809524 0.26190476
[3,] 3 0.47619048 0.73809524
[4,] 4 0.23809524 0.97619048
[5,] 5 0.02380952 1.00000000
5. Suppose X1, . . . , Xn are modelled as independent and identically distributed ran- dom variables from a Pareto distribution with unit scale and unknown shape pa- rameter θ, with density given by
fθ(x) = θx−θ−1 for x > 1 0 for x ≤ 1.
Consider Bayes procedures based on the “flat” prior (or weight function) given by w(θ)=1 forθ>0,
(a) Derive the corresponding posterior distribution and hence explain why the
Bayes estimator using squared-error loss is given by n+1 .
ni=1 log Xi
(b) Noting that log X1 has an exponential distribution with rate θ, identify the distribution of the estimator in the previous part and hence show that its mean-squared error is given by
θ2(n + 7) . (n−1)(n−2)
P M ( X = x ) = Nn , forallintegersxsuchthat0≤x≤M and0≤n−x≤N−M.
MN−M x n−x
turn to page 4
Semester 2, 2021 page 4 of 7
6. Suppose X1, . . . , Xn are independent U(0, θ) random variables for some unknown θ > 0 and it is desired to provide an interval estimate of θ of width n2 . The sample maximum X(n) = maxi=1,…,n Xi has the CDF
0 for x ≤ 0, PθX(n) ≤x= xθn for0
E(X) = (a + b)/2, V ar(X) = (b − a)2/12.
• Normal X ∼ N(0,1), then X has density fX(x) = (2π)−1/2e−x2/2. E(X) = 0, Var(X) = 1.
Y ∼N(μ,σ2),then(Y −μ)/σ∼N(0,1).
• Gamma X ∼ Gamma(α, β), then X has density
fX(x) = 1 xα−1e−x/β for x > 0, βαΓ(α)
Γ(·) is the Gamma function, Γ(α + 1) = αΓ(α); for integer α ≥ 1, Γ(α) = (α − 1)!. E(X) = αβ, V ar(X) = αβ2. Here β is a scale parameter; 1/β is also called the rate parameter.
• Exponential X ∼ Exponential(β) is the same as X ∼ Gamma(1, β). Here the scale parameter β is also the mean.
• Inverse Gamma X has an Inverse Gamma(α, λ) distribution, then X has density λα e−λ/x
fX(x) = xα+1Γ(α) for x > 0.
Note then that Y = X−1 has an ordinary gamma distribution with shape α and rate λ; E(X) =
λ/(α − 1), V ar(X) = λ2/ (α − 1)2(α − 2). • Beta X∼Beta(α, β), X has density
xα−1(1 − x)β−1
fX(x) = B(α,β) for 0 < x < 1,
B(α,β)= Γ(α)Γ(β) isthebetafunction;E(X)=α/(α+β),Var(X)=αβ(α+β)2(α+β+1). Γ(α+β)
• Pareto X has a Pareto(α, m) distribution, then X has density αmα
fX(x) = xα+1 for x ≥ m, E(X)=αm/(α−1)forα>1(+∞otherwise),Var(X)=αm2(α−1)2(α−2) forα>2(+∞
for 1 < α ≤ 2, undefined otherwise).
Convergence
• Convergence in distribution: A sequence of random variables X1 , X2 , . . . is said to converge in distribution to the continuous CDF F if for any sequence xn → x and real x as n → ∞,
P(Xn ≤xn)→F(x).
If this holds then it also holds with ≤ replaced by <. If F(·) is the N(0,σ2) CDF we also write
Xn →d N(0,σ2).
• Central limit theorem: If X1,...,Xn are iid random variables with mean μ and variance σ2, thenasn→∞, ni=1Xi−nμ d
√nσ2 → N(0,1).
• AsymptoticallyNormal: If√n(Xn−μ)→d N(0,σ2)thenwewriteXn ∼ANμ,σ2andsay
the sequence {Xn} is asymptotically normal with asymptotic mean μ and asymptotic variance σ2 .
• One variable: Suppose X has density f(x), consider y = u(x) where u(·) is a differentiable and either strictly increasing or strictly decreasing function for all values within the range of X for which f(x) ̸= 0. Then we can find x = w(y), and the density of Y = u(X) is given by
g(y) = f(w(y)) · |w′(y)|
for all y with corresponding x such that f(x) ̸= 0, and 0 otherwise.
• Extension of one variable: Suppose (X1, X2) has joint density f(x1, x2), consider Y = u(X1, X2). If fixing x2, u(·, x2) satisfies the conditions in the one-variable case, then the joint density of (Y, X2)
• Delta Method If Xn ∼ AN μ, σ2 and the function g(·) has derivative g′(μ) at μ then n
g (Xn) ∼ AN g(μ), g′(μ)2σ2 . n
Transformation of random variables
is given by
where x1 needs to be expressed in terms of y and x2. Fixing x1 is similar.
g(y,x )=f(x ,x )·∂x1, 2 12∂y
Exponential family
• A one-parameter exponential family is a set of probability distributions whose density function or probability mass function can be written in the form
f(x;θ) = eη(θ)T(x)−ψ(θ)h(x)IA(x),
IA is an indicator for the support of the distribution, and A does not depend on θ.
Sufficient statistic
• Factorisation theorem: For random variables X = (X1 , . . . , Xn ), if their joint density function
can be written as
f (x; θ) = g(T (x); θ)h(x), where x = (x1, . . . , xn), then T (X) is a sufficient statistic.
Cram ́er- Bound
• If l(θ;X) is a log-likelihood depending on a parameter θ, then under regularity conditions the
variance of any unbiased estimator of θ is bounded below by
1 . Varθ ∂l(θ;X)
• Suppose that for a sequence {Ln(·|θ)} of loss functions and for any a < b, the corresponding sequence of Bayes procedures {dn(·)} based on the U(a,b) prior is such that for each a < θ < b,
lim E L d(X)θ=S(θ) n→∞ θ n n
for some continuous function S(·). Then for any other sequence of procedures {dn(·)}, and any c < d,
lim max Eθ [Ln (dn(X)|θ)] ≥ max S(θ) .
Asymptotic Minimax Lower Bound
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