How do you show minimal sufficient?
To show minimal sufficiency, suppose X’ and Y’ were another sample from the same distribution. If the ratio of pdfs, L(θ;x,y)L(θ;x’,y’) is independent of θ iff T1(X,Y)=T1(X’,Y’) and T2(X,Y)=T2(X’,Y’), then (T1,T2) is minimal sufficient.
Is there always sufficient statistic?
Hence, a sufficient statistic always exists. We can compute the density of the sufficient statistics. Many statistical problems can be phrased in the language of decision theory. Suppose as usual that we have data X whose distribution depend on a parameter Θ.
Which is an example of a sufficiently sufficient statistic?
For example, for a Gaussian distribution with unknown mean and variance, the jointly sufficient statistic, from which maximum likelihood estimates of both parameters can be estimated, consists of two functions, the sum of all data points and the sum of all squared data points (or equivalently, the sample mean and sample variance ).
Which is a sufficient statistic for the parameter θ?
A statistic t = T ( X) is sufficient for underlying parameter θ precisely if the conditional probability distribution of the data X, given the statistic t = T ( X ), does not depend on the parameter θ. As an example, the sample mean is sufficient for the mean ( μ) of a normal distribution with known variance.
Is the probability density function t sufficient for θ?
Fisher’s factorization theorem or factorization criterion provides a convenient characterization of a sufficient statistic. If the probability density function is ƒ θ ( x ), then T is sufficient for θ if and only if nonnegative functions g and h can be found such that
Which is an implication of the sufficient statistic theorem?
An implication of the theorem is that when using likelihood-based inference, two sets of data yielding the same value for the sufficient statistic T ( X) will always yield the same inferences about θ. By the factorization criterion, the likelihood’s dependence on θ is only in conjunction with T ( X ).