Chapter 9.5, Problem 67E

Refer to Exercise 9.66. Suppose that a sample of size n is taken from a normal population with mean μ and variance σ². Show that ${\sum }_{i=1}^n{Y}_i$, and ${\sum }_{i=1}^n{Y}_i^2$ jointly form minimal sufficient statistics for μ and σ².

*9.66 The likelihood function L (y₁, y₂, …, y_n|θ) takes on different values depending on the arguments (y₁, y₂, …, y_n). A method for deriving a minimal sufficient statistic developed by Lehmann and Scheffé uses the ratio of the likelihoods evaluated at two points, (x₁, x₂, …, x_n) and (y₁, y₂, …, y_n):

$\frac{L\left(x_1,x_2,\dots ,x_n|\theta \right)}{L\left(y_1,y_2,\dots ,y_n|\theta \right)}.$

Many times it is possible to find a function g (x₁, x₂, …, x_n) such that this ratio is free of the unknown parameter θ if and only if g (x₁, x₂, …, x_n) = g (y₁, y₂, …, y_n). If such a function g can be found, then g (Y₁, Y₂, …, Y_n) is a minimal sufficient statistic for θ.

1. a Let Y₁, Y₂, …, Y_n be a random sample from a Bernoulli distribution (see Example 9.6 and Exercise 9.65) with p unknown.
   1. i Show that

$\frac{L\left(x_1,x_2,\dots ,x_n|p\right)}{L\left(y_1,y_2,\dots ,y_n|p\right)}={\left(\frac{p}{1-p}\right)}^{\sum x_i-\sum y_i}.$

1. ii Argue that for this ratio to be independent of p, we must have

${\displaystyle \underset{i=1}{\overset{n}{\sum }}{x}_i}-{\displaystyle \underset{i=1}{\overset{n}{\sum }}{y}_i=0}\text{or}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{x}_i}={\displaystyle \underset{i=1}{\overset{n}{\sum }}{y}_i}.$

1. iii Using the method of Lehmann and Scheffé, what is a minimal sufficient statistic for p? How does this sufficient statistic compare to the sufficient statistic derived in Example 9.6 by using the factorization criterion?
2. b Consider the Weibull density discussed in Example 9.7.
3. i Show that

$\frac{L\left(x_1,x_2,\dots ,x_n|\theta \right)}{L\left(y_1,y_2,\dots ,y_n|\theta \right)}=\left(\frac{x_1{x}_2\cdots x_n}{y_1{y}_2\cdots y_n}\right)\exp \left[-\frac{1}{\theta }\left({\displaystyle \underset{i=1}{\overset{n}{\sum }}{x}_i^2}-{\displaystyle \underset{i=1}{\overset{n}{\sum }}{y}_i^2}\right)\right].$

1. ii Argue that ${\sum }_{i=1}^n{Y}_i^2$ minimal sufficient statistic for θ.