An introduction to statistical decision theory Flashcards
In statistical decision theory who are the two players in a game
In statistical decision theory, we have one player playing against nature
(“nature” is a player that makes a random decision) so the player makes a decision in a condition of uncertainty
How can we evaluate what the optimal decision is?
Make a decision based on minimising the expected loss for each option available to us.
Step 1 : define the loss for every scenario
Step 2 : average out the losses with respect to the randomness of the sample.
Step 3 : Find out the admissable and optimal estimators
4. one value summaries
What is a statistical decision
A decision is a rule that determines the value of the parameter of interest
for any outcome of the experiment.
A decision is an estimator
Define the loss function
Given a statistical decision problem, a loss function is a function
L : Θ × Θ → R, where L(d(x), θ) describes the loss occurred when the decision
d(x) is taken and the true value of the parameter is θ
Risks are functions of the unknown parameter θ. How do we compare the two functions ex: d1 and d2?
We say that d1 dominates d2 if, for all θ ∈ Θ: Rd1 (θ) ≤ Rd2 (θ). We will want to select the estimator that has the lower risk
What can risk also be known as
Expected loss
Define an admissable decision
There is no other decision that dominates it - happy to choose either this one or other one
Within a set of unbiased estimators what result can help depict the optimal estimator
If an estimator obtains rao lower bound then it is optimal in that set.
How can we further analyse any admissable decisions
Using one number summaries such as examining min max decision or the bayes decision
Under quadratic loss what is the expected loss or risk of the estimator the same as
The MSE
