Statistical Distributions Let X1X2 Xn be a random sample fro

Statistical Distributions

Let X1,X2, ...,Xn be a random sample from a distribution with PDF fXi(x) of Xi given by: where we assume that lambda > 1 is known but beta > 0 is not. Hint: Notice the range of the Xi\'s (a) Find the CDF FX(1) (xi) of the random variables Xi and also give the support of this CDF. (b) Find the CDF FX(1) (x) and PDF fX(1) (x) of the smallest observation (c) Find the likelihood function L(beta|x1, ...,xn, lambda) and a sufficient statistic for estimation of beta.

Solution

(a)

The cdf is given by

F_X(x) = P[Xx]

= Integration (from=,to=x, t^-(+1) dt)

          = *int( ,x, t^-(+1) dt)

          = [ t^-/(-) ]^x

          = (t^- – t^-x )

Hence

F_X(x) = 0              if x<

= (t^- – t^-x )     if x>

The support of the cdf is given by S= { x : x> }

(b)

F_X1(x) = P[X₍₁₎x]

          = 1 – P[X₍₁₎ > x]

          =1- P[X₁ > x, X₂ >x, …… , X_n > x]   (Since smallest observation >x => all

                                                                                    observations are > x)

            = 1- Product(i=1, to =n, P[X_i >x])

          = 1- { P[X₁ > x] }ⁿ   (Since the random variables are i.i.d)

          = 1 – { 1 – P[X₁ x] }ⁿ

          = 1- {1 - F_X(x) }ⁿ

          = 1 – {1-0}          if x<

                       1- { 1- (t^- – t^-x )} ⁿ    if x>

Therefore the cdf of X₍₁₎ is

F_X1(x) = 0 if x<

          = 1- { 1- (t^- – t^-x )} ⁿ    if x>

(c)

L(| x₁ ,x₂ ,……,x_n , ) = Product(i=1,n, f(x_i))

          = prod(i=1,n, x_i^-(+1) )

          = ⁿⁿ prod(i=1,n, x_i^-(+1) )        if x_i >

To find sufficient statistics we use Neyman Fisher factorisation theorem.

Now,

L(| x₁ ,x₂ ,……,x_n , ) = ⁿⁿ prod(i=1,n, x_i^-(+1) ) if x_i >

          = ⁿⁿ prod(i=1,n, x_i^-(+1) ) if x₍₁₎ >

          = ⁿⁿ prod(i=1,n, x_i^-(+1) ) I(,x₍₁₎)    where I(a,b) = 1 if a<b

                                                                                                0 if a>b

          = g(t,)*h(x₁ ,x₂ ,……,x_n)

Where

t= x₍₁₎  

g(t,)= ⁿⁿ I(,t)

h(x₁ ,x₂ ,……,x_n) = prod(i=1,n, x_i^-(+1) )

Then using the theorem we find that T= X₍₁₎ is sufficient for .