ESTIMATION THEORY

Basically i ll be talking about characteristics of an estimator

UNBIASEDNESS

•Lets assume T_n is an estimator of some parameter θ from a population having density function

D(θ) .

If E(T_n )= θ for all θ∈θ(parameter space for θ), then T_n will be called an Unbiased estimator.

§ What if an estimator is not unbiased(i.e biased) ,we can calculate the amount of bias it has:

b(θ) = E(T_n )- θ,

this biasedness can be negative ,positive or zero.

CONSISTENCY

An estimator T_n is said to be consistent estimator of parameter θ if

〖 T〗_n converges to θ in probability

What do we mean by Convergency with probability

• Can any Estimator be biased and consistent at the same time ?( THINK )
• well i would say ..NO!
• EFFICIENCY
• •Let T_1 and T_2 are two estimators of some parameter θ and if
• V(T_1) < V(T_2) for all n
• then we will say T_1 is an efficient estimator for the given unknown parameter.
• •Efficiency….(it is a relative measure )
• E = v(T_1)/v(T_2)
• E >1 when T_2 is more efficient than T_1.
• E< 1 when T_1 is more efficient than T_2.
• SUFFICIENCY
• In lame words ,we can say an estimator is sufficient if it contains all the information for the parameter
• FORMAL DEFINATION
• Let T =t(x_1, x_2, x_3,…., x_n) is an estimator of parameter θ based on the sample
• x_1, x_2, x_3,…., x_n and if the conditional distribution of 〖(x〗_1, x_2, x_3,…., x_n) given
• T is Independent of parameter θ ,then
• T will be called as a sufficient estimator for parameter θ
• FACTORIZATION THEOREM
• T = t(x) is sufficient estimator for θ ó if the joint density function( likelihood function) L ,of the sample values can be expressed as
• L = G_θ[t(x)].h(x)
• Where G_θ[t(x)] depends on x and parameter θ
• h(x) depends only on x
• NOTE: whole sample is always a sufficient estimator for its parameter .

MVUE(minimum variance unbiased estimator)

§If in the class of all unbiased estimator for a parameter ,the one having minimum variance is called MVUE.

§RAO –BLACKWELL THEOREM

Basically it tells us how to obtain MVUE from any unbiased estimator with the help of sufficient estimator.

Let U = u(x_1,x_2, x_3……, x_n) be any unbiased estimator of θ and let V = v(x_1,x_2, x_3……, x_n) be any sufficient est. for θ, then if

ϕ(t)=E(U/V = v)

Where ϕ(t) is independent from θ

and E[ϕ(t)] = θ and var(ϕ(t)) ≤ var(U)

THEN ,we can say ϕ(t) is a MVUE for θ