Week 2

Question 1

Sufficiency principle

Answer 1

Question 2

If T is a sufficient statistic then

Answer 2

Question 3

Factorisation theorem

Answer 3

Question 4

Proving sufficiency

Answer 4

Use factorisation theorem

Question 5

YOU ARE GIVEN PDFS ETC DURING EXAM

Answer 5

Yay

Question 6

Minimal sufficient statistic

Answer 6

Question 7

How to find minimal sufficient statistic

Answer 7

When asked to prove that a statistic is minimal sufficient you must prove the condition in red

Question 8

Proof of method to find minimal sufficient statistic

Answer 8

You do not need to prove this theorem for this module

Question 9

Conventional estimators for variance and difference

Answer 9

This is a biased estimator, for unbiased we must have n-1

Question 10

Bias of an estimator

Answer 10

Question 11

Def of pos def matrix

Answer 11

Question 12

Quickest way to compute FIM

Answer 12

Question 13

Def parametric stat model

Answer 13

Question 14

Derive χ² dist

Answer 14

Arises from a sum of k squares of standard independent gaussians

K is deg of freedom

Question 15

When to use t dist?

Answer 15

To estimate population parameters from sample size

Use when pop dist is assumed to be normal

Question 16

Relate t dist to χ dist

Answer 16

S² = (1/(n-1))*sum((x_i)- x^-)²)
RV of T dist is (below) for which S²(n-1) follows a χ² dist with n-1 deg of freedom

Question 17

Assumptions of t dist

Answer 17

Data are sampled from population that follows normal dist

Observations are independent

Pop var is unknown and estimated from data

Question 18

Relate F dist to χ

Answer 18

F is ratio of 2 χ² distributions (each divided by their respective DoF)

DoF of F are k1, k2

Question 19

Relate χ² to Γ

Answer 19

χ² is a special case of Γ where the shape parameter is an integer (equal to the DoF)

Question 20

The gamma dist is

Answer 20

Sum of exponential RVs

Question 21

Alternative way to show sufficiency

Answer 21

In this example T(X) is sum(xi) expo RVs, therefore takes gamma dist

Question 22

What can you say about the MSE of an unbiased estimator

Answer 22

This is the variance of the estimator