Week 2 Flashcards
Sufficiency principle
If T is a sufficient statistic then
Factorisation theorem
Proving sufficiency
Use factorisation theorem
YOU ARE GIVEN PDFS ETC DURING EXAM
Yay
Minimal sufficient statistic
How to find minimal sufficient statistic
When asked to prove that a statistic is minimal sufficient you must prove the condition in red
Proof of method to find minimal sufficient statistic
You do not need to prove this theorem for this module
Conventional estimators for variance and difference
This is a biased estimator, for unbiased we must have n-1
Bias of an estimator
Def of pos def matrix
Quickest way to compute FIM
Def parametric stat model
Derive χ2 dist
Arises from a sum of k squares of standard independent gaussians
K is deg of freedom
When to use t dist?
To estimate population parameters from sample size
Use when pop dist is assumed to be normal
Relate t dist to χ dist
S2 = (1/(n-1))*sum((xi)- x-)2)
RV of T dist is (below) for which S2(n-1) follows a χ2 dist with n-1 deg of freedom
Assumptions of t dist
Data are sampled from population that follows normal dist
Observations are independent
Pop var is unknown and estimated from data
Relate F dist to χ
F is ratio of 2 χ2 distributions (each divided by their respective DoF)
DoF of F are k1, k2
Relate χ2 to Γ
χ2 is a special case of Γ where the shape parameter is an integer (equal to the DoF)
The gamma dist is
Sum of exponential RVs
Alternative way to show sufficiency
In this example T(X) is sum(xi) expo RVs, therefore takes gamma dist
What can you say about the MSE of an unbiased estimator
This is the variance of the estimator
