- Information
- AI Chat
This is a Premium Document. Some documents on Studocu are Premium. Upgrade to Premium to unlock it.
Was this document helpful?
This is a Premium Document. Some documents on Studocu are Premium. Upgrade to Premium to unlock it.
Principles of Statistics 2016-2017 Example Sheet 1
Module: Principles of Statistics
5 Documents
Students shared 5 documents in this course
University: University of Cambridge
Was this document helpful?
This is a preview
Do you want full access? Go Premium and unlock all 2 pages
Access to all documents
Get Unlimited Downloads
Improve your grades
Already Premium?
PRINCIPLES OF STATISTICS – EXAMPLES 1/4
Part II, Michaelmas 2016, Quentin Berthet (email: q.berthet@statslab.cam.ac.uk)
Throughout, the abbreviations ‘i.i.d.’, ‘pdf/pmf’ and ‘MLE’ stand for ‘independent and iden-
tically distributed’, ‘probability density/mass function’ and ‘maximum likelihood estimator’, re-
spectively. A normal distribution in Rdwith mean vector µand covariance matrix Σ is denoted
by Nd(µ, Σ), and N(µ, σ2) corresponds to the univariate case d= 1.
1. Consider an i.i.d. sample X1, . . . , Xnof random variables. For each of the following
parametric models of pmf/pdf’s, find the MLE of the unknown parameter, the score equation
and the Fisher information.
a) Xi∼i.i.d. Bernoulli(θ), θ ∈[0,1],
b) Xi∼i.i.d. N(θ, 1), θ ∈R,
c) Xi∼i.i.d. N(0, θ), θ ∈(0,∞),
d) Xi∼i.i.d. N(µ, σ2), θ = (µ, σ2)T∈R×(0,∞),
e) Xi∼i.i.d. P oisson(θ), θ∈(0,∞),
f) Xi∼i.i.d. from model {f(·, θ) : θ∈(0,∞)}with pdf f(x, θ) = (1/θ)e−x/θ , x ≥0.
g) Xi∼i.i.d. from model {f(·, θ) : θ∈(0,∞)}with pdf f(x, θ) = θe−θx, x ≥0.
2. In which of the examples of the previous exercise is the MLE unbiased (i.e., does one have
Eθˆ
θ=θfor all θ∈Θ)? When unbiased, deduce whether the variance of the MLE attains the
Cram`er-Rao lower bound or not.
3. Let X1, . . . , Xnbe i.i.d. Poisson random variables with parameter θ > 0, and let ¯
Xn=
(1/n)Pn
i=1 Xi, S2
n= (n−1)−1Pn
i=1(Xi−¯
Xn)2. Show that V ar(¯
Xn)≤V ar(S2
n).
4. Find the MLE for an i.i.d. sample X1, . . . , Xnarising from the models a) N(θ, 1) where
θ∈Θ = [0,∞) and b) N(θ, θ) where θ∈Θ = (0,∞).
5. Consider an i.i.d. sample X1, . . . , Xnarising from the model
{f(·, θ) : θ∈R}, f(x, θ) = 1
2e−|x−θ|, x ∈R,
of Laplace distributions. Assuming nto be odd for simplicity, show that the MLE is equal to the
sample median. Discuss what happens when nis even. Can you calculate the Fisher information?
6. Consider observing an n×1 random vector Y∼N(Xθ, I) where Xis a non-stochastic
n×pmatrix of full column rank, where θ∈Θ = Rpfor p≤n, and where Iis the n×nidentity
matrix. Compute the MLE and find its distribution. Calculate the Fisher information for this
model and compare it to the variance of the MLE. Deduce, as a special case, the form of the
MLE and Fisher information in the case when p=nand X=I.
7. Let (X, Xn:n∈N) be random vectors in Rk.
a) Prove that Xn→PXas n→ ∞ if and only if each vector component Xn,j , j = 1, . . . , k,
of Xnconverges in probability to the corresponding vector component Xjof Xas n→ ∞.
Formulate and prove an analogous result for random symmetric k×k-matrices.
b) Suppose EkXn−Xk → 0 as n→ ∞ where k · k is the Euclidean norm on Rk. Deduce
that Xn→PXas n→ ∞.
c) Show that the converse in b) is false, that is, give an example of real random variables
Xn→PXas n→ ∞ but E|Xn−X| 6→ 0.
1
Why is this page out of focus?
This is a Premium document. Become Premium to read the whole document.