PART 4 - MINIMAL SUFFICIENT STATISTICS Intuitively, a minimal sufficient statistic most efficiently captures all possible information about the parameter θ.
Definition : A set of jointly sufficient statistics is defined to be minimal sufficient iff it is a function of every other set of sufficient statistics.
🏴 Theorem 6.1.4 Casella & Berger. Let f (x |θ) be the pmf/pdf of a sample X.Suppose that there exists a function T(x) such that for sample points x & y, the ratio 
iff T(x)=T(y). Then T(x) is a minimal sufficient statistics for θ.
🖊 Worked Example [Normal minimal sufficient statistic with 2 parameters]
🖊 Worked Example [Uniform minimal sufficient statistic]
❤.. EXPONENTIAL FAMILY ...
........ONE - PARAMETER....... Definition 1: A one parameter family of densities that can be expressed as ⭐ f (x;θ) = a(θ) b(x) exp [c(θ)d(x)] is defined to be belong to the exponential family.
🚩 Under random sampling, if a density belongs to the one-parameter exponential family, then there is a ✅sufficient statistic. In fact, it can be shown that it is also ✅minimal sufficient statistic. ❤ ❤ ❤ (this what makes Exp Method the best!)
🖊Worked Example [ Check for Exponential Family]
........K - PARAMETER....... Definition 2: A family of densities . f( .; θ1,..., θk) that can be expressed as ⭐ f (x;θ1,..., θk) = a(θ1,..., θk) b(x) exp [ summation c(θ1,..., θk)d(x)] is defined to be belong to the exponential family.
🖊Worked Example [ 2-parameter Exponential Family]
ANCILLARY STATISTICS A complementary purpose to sufficient statistics 
A statistics S(x) whose distribution does not depend on the parameter θ is called an ancillary statistics.
⚠ Note: 🚩 Ancillary statistics contains no information about θ. 🚩 It is an observation on a random variable whose distribution is fixed and known, unrelated to θ. 🚩 when used in conjunction with other statistics, sometimes does contain valuable information for inferences about θ.
🖊 Worked Example [Ancillary statistic Using Tranformation]
🖊 Worked Example [Ancillary statistic For Location Parameter Family]
🖊 Worked Example [Ancillary statistic For Scale Parameter Family]
COMPLETE STATISTICS 
🏴 Definition 1: The family of densities of T defined to be complete iff Eθ[z(T)]=0 for all θ, implies that Pθ[z(T)=0]=1. Also, statistic T is said to be complete iff family densities is complete.
TOOL FOR ARGUING COMPLETENESS
🖊 Worked Example [Complete Bernoulli Density]
🖊 Worked Example [Complete Uniform Density]
✅ Theorem 1 : If f (x;θ) = a(θ) b(x) exp [c(θ)d(x)] is a member of one parameter exponential family, then summation d(Xi) is a complete minimal sufficient statistics
✅ Theorem 2: (BASU's Theorem) If T(X) is a complete & minimal sufficient statistics, then T(X) is independent of every ancillary statistics.
🖊 Worked Example [Complete Statistics]
❗ Basu's Theorem Useful - allow us to deduce the independence of 2 statistics WITHOUT ever finding the joint distribution. ⭐
NOTE: Please Click "Worked Example" to view