3. Warning: This question is HARDER so expect to spend a little longer on thisone.Let X = (X₁,..., Xn) be an iid sample with Xi~ foo. We have seen in lecturesthat if we have a statistic T(X) of the data instead of the full data thenIT(X) (00) ≤ Ix (00)A statistic is a summary of the data and so intuitively you can lose informationabout the unknown model parameters if you get given the statistic instead ofthe full data. However, this is not always the case.Definition 0.1 (Sufficient Statistic). A statistic T is said to be sufficient for astatistical model {fe : 0 } of X if the conditional distribution of X givenT = t is independent of 0 for all t.What this means is that once you know the value of the statistic the distributionof X has no aspect that any longer depends on 0.What this really means is revealed in the following Theorem:Theorem 0.1 (Factorization Criterion). Let P = {fe : 0 € O} be a statisticalmodel. A statistic T is sufficient for P if and only if there exists non-negativefunctions ge() and h() such thatfo(x) = go(T(x))h(x)(1)Thus this is saying that the bit of the distribution that involves is interlockedwith the sufficient statistic T and ONLY T (which could in many exampled bemulti dimensional!).(a) Suppose that fe is a probability mass function i.e. the data is discrete.Prove in this special case that if equation (1) holds then the distributionof X doesn't depend on once you condition on T(X) == t.(b) Prove that if T is a sufficient statistic for the model then the score functiononly involves the data through the sufficient statistic.(c) Use the answer in the previous part to explain why if (R1)-(R4) hold and Tis a sufficient statistic then the MLE only depends on the sufficient statisticand no other function of the data.(d) Suppose that T(X) is a sufficient statistic for some statistical model withX foo for some unknown o. Assume further that X is a discrete RV.Prove thatIT(x) (00) = Ix (00)Note: This is significant because it shows that we lose no informationabout o by only being told the sufficient statistic.(e) Show that 1X₂ is a sufficient statistic for the setting in Question 1.(f) Find a Sufficient Statistic in the setting of Question 2. Hint: This will be2 dimensional.

A statistic T is considered sufficient for a model with parameter of X if the conditional…

Answered: (a) Suppose that fe is a probability…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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