Question

Let N be a random variable with PMF P(N Also XINnBinomial (n, 0) i) Pi, i-1, 2, il pi -1,0 Pi 1. I. Show that (X,N) is a minimal sufficient statistic for θ. 2. Show that is unbiased for θ.

Answer 1

1. A statistic is said to be minimal sufficient statistic for the parameter $\theta$ if it satisfies the following conditions:

a) if it is a sufficient for the parameter $\theta$

b) if it is a function of every other sufficient statistic

The pmf of X|N=n is given by :

P[X|N=n] = (ⁿC_x )*$\theta$^x(1-$\theta$)^n-x , x=0(1)n

= 0 , otherwise

Since we have a single random variable as our sample we can consider it to be our statistic

Using Neyman Fisher Factorisation Theorem, we can say that since we can write:

f(x,n|$\theta$) = (ⁿC_x )*$\theta$^x(1-$\theta$)^n-x

  = h(x).g(x,n ; $\theta$)

we can say that (X,N) are our sufficient statistic for $\theta$.

Then we know that any further statistic of $\theta$ will always be a function of X, we can say that X is the minimal sufficient statistic for $\theta$.

2. Since we know that X|N=n ~ Bin(n,$\theta$), we know that

$\fn_cm \small E[\frac{X}{N}]=\sum_{i=1}^{\infty }\sum_{x=0}^{n}\frac{x}{n}.f(x,n)$

$\fn_cm \small E[\frac{X}{N}]=\sum_{i=1}^{\infty }\sum_{x=0}^{n}\frac{x}{n}.f(X=x|N=n).f(N=i)$

$\fn_cm \small E[\frac{X}{N}]=\sum_{i=1}^{\infty }\sum_{x=0}^{n}\frac{x}{n}._{x}^{n}\textrm{C}.\theta ^{x}(1-\theta )^{n-x}.p_{i}$

$\fn_cm \small E[\frac{X}{N}]=(\sum_{i=1}^{\infty }p_{i})(\sum_{x=0}^{n}\frac{x}{n}._{x}^{n}\textrm{C}.\theta ^{x}(1-\theta )^{n-x})$

$\fn_cm \small E[\frac{X}{N}]=1.\frac{n\theta }{n}$

$\fn_cm \small E[\frac{X}{N}]=\theta$ ,for all values of $\theta$

Hence X/N is an Unbiased Estimator of $\theta$.