Contents
When is a statistic is a sufficient statistic?
The statistic is sufficient if someone observing the distribution of T ∘ X learns no more about θ when observing the distribution of X directly. So the distribution of X conditional T, which you can calculate for each θ separately, doesn’t show you anything about θ, it is independent of θ.
When is a statistic sufficient for a conditional distribution?
A statistic is sufficient for θ if the conditional distribution of X given T does not depend on θ . You’re right in that θ is not a random variable. It’s not the probability of θ being in any particular set; it’s a probability of some event given θ.
Do you know the underlying probabilities of Statistics?
In statistics, you don’t know the underlying probabilities- you usually want to learn them. Now the statistician has no direct information about θ, but observes the distribution μ θ X − 1 of a random variable X that provides the data. Suppose now, that you have a statistic T, a measurable function of X.
What do you need to know about statisitcal decision theory?
In statisitcal decision theory, you don’t work with a single probability space. You have a measurable sample space ( Ω, Σ) and a whole family of probability measures, ( μ θ), indexed by θ on ( Ω, Σ). In statistics, you don’t know the underlying probabilities- you usually want to learn them.
A statistic is sufficient if you can write the following joint pdf for functions g (t|θ) and h ( x ): f ( x |θ) = g (T ( X )|θ)h ( x) Where: θ is the unknown parameter belonging to the parameter space Q, and the pdf exists for all values of x, and θ ∈ Q.
When is y a sufficient statistic for P?
The definition of sufficiency tells us that if the conditional distribution of X 1, X 2, …, X n, given the statistic Y, does not depend on p, then Y is a sufficient statistic for p. The conditional distribution of X 1, X 2, …, X n, given Y, is by definition:
What do you call statistics that summarize all information?
In this lesson, we’ll learn how to find statistics that summarize all of the information in a sample about the desired parameter. Such statistics are called sufficient statistics, and hence the name of this lesson. To learn a formal definition of sufficiency. To learn how to apply the Factorization Theorem to identify a sufficient statistic.
How to figure out the statistic F ( x1 x2 xn )?
A statistic T = r(X1,X2,···,Xn) is sufficient if and only if the joint density can be factored as follows: f(x1,x2,···,xn|θ) = u(x1,x2,···,xn)v(r(x1,x2,···,xn),θ) (2) where u and v are non-negative functions.