Topic 4 Fitting probability Distributions

Question 1

What is statistical inference?

Answer 1

Inferring the nature of a population distribution based on a sample.

Question 2

Fitting a probability model to data is known as _______.

Answer 2

statistical inference

Question 3

Match summary stats ↔ ______
Maximise likelihood ↔ ______

Answer 3

Method of Moments (MoM)
Maximum Likelihood Method (MLM)

Question 4

What is the main principle of the Method of Moments?

Answer 4

Match population moments (mean, variance, etc.) to sample moments.

Question 5

METHOD OF MOMENTS (MoM)
E(X) = __
Var(X) = __

Answer 5

x̄
s²ₓ

Question 6

Minimise weighted difference:
Min_θ ∑ₖ wₖ (Mk − mk)²
Where:

Mk = __
mk = __

Answer 6

• Population moment
• Sample moment

Question 7

Estimated parameters are denoted with a ___.

Answer 7

hat (e.g. ω̂, ε̂)

Question 8

If x̄ and s²ₓ are sample mean and variance:
p̂ = ____
n̂ = ____

Answer 8

• 1 − s²ₓ / x̄
• x̄² / (x̄ − s²ₓ)

Question 9

Geometric MoM Estimator
p̂ = ____

Answer 9

x̄ / s²ₓ

Question 10

Gamma MoM Estimators
ε̂ = _____
ω̂ = _____

Answer 10

• s²ₓ / x̄
• x̄² / s²ₓ

Question 11

What does MLM aim to do?

Answer 11

Find the parameter values that maximise the probability of observing the sample

Question 12

Formula
Likelihood function:

Answer 12

L(θ) = ∏ fₓ(xᵢ | θ)

Question 13

Condition for Maximum

First derivative =
Second derivative <

Answer 13

0

Question 14

Why use log-likelihood?

Answer 14

• Easier to differentiate
• Converts product to sum

Question 15

Log-Likelihood
Formula

Answer 15

ln(L) = ∑ ln(fₓ(xᵢ | θ))

Question 16

List 3 graphical methods to assess fit:

Answer 16

• Empirical histogram vs PMF/PDF
• Empirical vs theoretical CDF
• QQ plot

Question 17

In QQ plots, the theoretical quantile is plotted against the ______ quantile.

Answer 17

Empirical

Question 18

QQ Plot Formula
p = (j − 0.5)/n → used to find ______ quantiles

Answer 18

theoretical

Question 19

x̄ → E(X) as n → ∞ is known as the ______

Answer 19

Law of Large Numbers

Question 20

What makes an estimator efficient?

Answer 20

It has the lowest possible variance (Cramér-Rao lower bound)

Question 21

Use s²ₓ instead of σ²ₓ and ĝ₁(x) instead of g₁(x) to ensure _______ sample moments.

Answer 21

Unbiased

Question 22

What are the parameter estimators obtained by MoM and MLM?

Answer 22

• Consistent
• Biased
• Inefficient

Question 23

When is a an estimator consistent?

Answer 23

An estimator is consistent if the sample moments
converge to the population moments as the sample size
tends to infinity e.g., σ²ₓ → Var(X)

Question 24

What is a biased estimator?

Answer 24

A biased estimator has a mean sample moment that is
not equal to the population moment

Question 25

Do MoM and MLM produce efficient or inefficient estimators?

Answer 25

Inefficient

Question 26

What does it mean by the MLM approach being asymptotically normal?

Answer 26

For large sample sizes the estimators are normally-distributed random variables