Lecture 23 : Sufficiency - I

[@cheolhunkim1927]

[Lecture 23 Summary] [+] Concept of Sufficiency (or sufficient statistic) >> A new concept that is called “sufficiency” in the context of a statistical inference >> Useful to retain the necessary data without losing any information >> In practical problems, it is difficult to retain data >>>> due to storage space (or etc.) >> Minimum amount of data which is enough(sufficient) >>>> for whatever information or whatever useful inferences >>>> not losing an important part [+] Definition of Sufficiency is given. >> usually, mention the word “sufficient” for “the family of probability distributions”. >>>> loosely, “sufficient” for the “parameter theta” [+] Physical interpretation of the definition is given. >> Data reduction >>>> No original observation => generate sample [+] Application >> if there is any inference made in terms of estimation, testing of hypothesis, etc. >>>> inference based on the “sufficient statistics”, we are better off. [+] Examples >> ex) binomial(p) dist.: independent of p (intuition: dart game) >> ex) Poisson(lambda) dist.: independent of lambda [+] Remarks >> 1. Let T sufficient, T be a function of U, then U is also sufficient. >> 2. “Trivial sufficient statistics”: Full sample is always sufficient [+] Theorems >> Thm. With a given sufficient statistics, we can always generate the original sample. (formal proof is shown) >> Thm. Rao-Blackwell theorem: CR Rao(1945, Indian statistician), David Blackwell(1947) (with proof) >>>> Very significant thm. because the thm. implies >>>>>> Based on the sufficient statistics is always better(or at least equal) than do not base it. >>>> applications of this result will be given in the next lecture.