A complete summary of the weekly course material for statistics for econometrics students. The course is taught in the first year and this summary explains thoroughly what needs to be understood each week, including examples.
Law of Large Numbers (LLN)
suppose {Xn }∞
n=1 a sequence iid random variables, then there will be almost sure
convergence to X̃ iff
P lim |Xn − X̃| < ϵ = 1, ∀ϵ > 0
n→∞
n
1X
lim Xi = lim X̄n → E[X1 ] almost surely
n→∞ n n→∞
i=1
lim P |Xn − X̃| < ϵ = 1, ∀ϵ > 0 Convergence in probability
n→∞
Central limit theorem (CLT) √
n(X̄−µ)
For finite expectation and variance we have. σ
→ N (0, 1)
1
,Week 2: Statistical models
fX (x1 , · · · , xn ) = nk=1 fXk (xk )
Q
A histogram gives the first insights on whether we have possibly chosen the
correct probability distribution for our dataset.
Let {aj }m j=1 be a partition over range xi . It holds that aj − aj−1 = c
Choose y ∈ (aj−1 , aj ]. then
hn (y) = #{1 ≤ i ≤ n|aj−1 < xi ≤ aj } = ni=1 1{xi ∈(aj−1 ,aj ]}
P
A scaled histogram is then:
hn˜(y) =
#{1≤i≤n|aj−1 <xi ≤aj }
cn
Transformations
How do we get the distribution of Y = h(X) from X?
FY (y) = P(Y ≤ y) = P(h(X) ≤ y) = P(X ≤ h−1 (y)) = FX (h−1 (y)
∂
fY (y) = ∂y FY (y)
fY (y) = fX (h−1 (y)) · ∂ −1
∂y
h (y)
Location-scale family Let µ ∈ R, σ > 0,
x−µ
Hµ,σ (x) = H
σ
Y , a random variable with cdf H, define Zµ,σ = µ + σY , Then Zµ,σ has cdf Hµ,σ
P(Zµ,σ ≤ y) = P(µ + σY ≤ y)
y−µ y−µ
=P Y ≤ =H
σ σ
Week 3: Maximum Likelihood
Definition An estimate for θ0 is any function of the data W (⃗x). The corresponding
estimator is a stochastic variable obtained by filling in the random vector.
Method of moments
n
1X
lim Xi = E(X1 ) → X̄ ≈ E(X1 )
n→∞ n
i=1
n
1X 2
lim Xi = E(X12 ) → X̄ 2 ≈ E(X12 )
n→∞ n
i=1
..
.
n
1X k
lim Xi = E(X1k ) → X¯k ≈ E(X1k )
n→∞ n
i=1
Sample mean: X̄ = n1 ni=1 P
P
Xi
Sample variance: S = n−1 ni=1 (Xi − X̄)2
2 1
2
, Definitions on maximum likelihood
Likelihood function: θ → L(θ|⃗x) = fθ (⃗x)
Maximum likelihood estimate: W (⃗x) = argmaxθ∈Θ L(θ|⃗x), The parameter value in
the parameter space at which the likelihood functino attains its maximum.
L(θ|⃗x) = fθ (⃗x) = Πni=1 gθ (xi )
Log likelihood: θ → log(L(θ|⃗x))
Suppose that the log likelihood is differentiable on Θ ⊆ Rk . Then the maximum
can be attained at two different kinds of points:
i) boundary points
ii) stationary points: is a point θ̃ that satisfies ∂θ∂ j log(L(θ|⃗x))|θ=θ̃ = 0, ∀j ∈ {1, · · · , k}
Week 4: Evaluating estimators
Definition Biasθ (W ) = Eθ W (X) ⃗ − τ (θ) . We say that an estimator is unbiased
⃗ = τ (θ).
if Eθ (W (X))
⃗ − τ (θ)||
M AE(θ, W ) = Eθ ||W (X)
⃗ − τ (θ)||2 = Varθ (W (X))
M SE(θ, W ) = Eθ ||W (X) ⃗ + Bias2 (W )
θ
Where the variance is called the precision and the bias squared is called the accuracy.
Definition An estimator W ∗ is a UMVU estimator if it is unbiased and, for any
other estimator W that is unbiased, we have Varθ (W ∗ ) ≤ Varθ (W ).
Cauchy-Schwarz Lemma E(Y Z)2 ≤ E(Y 2 )E(Z 2 )
2
∂
Iθ = Eθ log(fθ (X)) Fisher information
∂θ
2
∂ ∂
iθ = Eθ log(gθ (X1 )) = Var log(fθ (X)) for an individual observation
∂θ ∂θ
Cramer-Rao:
′ 2
⃗ ≥ τ (θ)
Varθ (W (X))
Iθ
′ 2
⃗ ≥ τ (θ)
Varθ (W (X))
niθ
Week 5: Exponential families
Definition A set of univariate distributions {gθ |θ ∈ Θ} is called an exponential
family if we can rewrite it as:
Pm
wj (θ)tj (x)
gθ (x) = h(x)c(θ)e j=1
3
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