Principles of data reduction

Question 1

Define a sufficient statistic

Answer 1

A sufficient statistic is a summary of the observed data containing all the
information needed to infer the parameter of interest. - contains sufficient information to calculate the likelihood function.

Question 2

What is the fisher neyman factorisation for sufficient statistics

Answer 2

A statistic T (X) is sufficient for θ if and only if there exist two functions g and
h such that:
Lx (θ) = h(x) · g (T(x), θ)
Note x and theta can be vectors

Question 3

How to find a sufficient statistic

Answer 3

1. write down liklihood
2. examine how it depends on the data
3. determine form this what the sufficient statistic could be
   Try to rewrite liklihood at fisher neyman factorisation with this sufficient statistic

Question 4

Define a minimal sufficient statistic

Answer 4

A sufficient statistic is called minimal sufficient statistic if it can be written as
a function of any other sufficient statistic I can create. This information the statistic contains is enough to calculate the likelihood but is as summarised as possible, least info needed

Question 5

Are minimal sufficient or sufficient statistics unique

Answer 5

No in fact

they maintain their property after linear transformations

Question 6

What two statements have to be equivalent for T to be a minimal statistic for theta

Answer 6

P (X = x|θ) /P (X = y|θ) does not depend on θ;

T (x) = T (y)

Question 7

How will you know in a question if the likelihood ratio does not depend on theta is the same as Tx=Ty

Answer 7

The dependency on the data and/or parameters disappears if and only if tx=ty

Question 8

What theorem gives a strong rationale for basing estimators on sufficient
statistics, if they exist

Answer 8

Rao blackwell theorem

Question 9

How does rao blackwell theorem suggest that sufficient statistics should be the basis for estimators

Answer 9

If an estimator is not constructed using a sufficient statistic, then
it can be theoretically improved by considering its expectation conditionally on
the sufficient statistic and could have a smaller MSE in theory

Question 10

What estimator should we use to incorporate the sufficient statistic

Answer 10

Using the maximum likelihood estimator, this is naturally constructed using a
sufficient statistic.