The likelihood function L(y1, y2, . . . , yn | θ) takes on different values depending on the arguments (y1, y2 , . . . , yn ). A method for deriving a minimal sufficient statistic developed by Lehmann and Scheff´e uses the ratio of the likelihoods evaluated at two points, (x1, x2, . . . , xn) and (y1, y2, . . . , yn ):
Many times it is possible to find a function g(x1, x2, . . . , xn ) such that this ratio is free of the unknown parameter θ if and only if g(x1, x2, . . . , xn ) = g(y1, y2, . . . , yn ). If such a function g can be found, then g(Y1, Y2, . . . , Yn ) is a minimal sufficient statistic for θ .
a Let Y1, Y2, . . . , Yn be a random sample from a Bernoulli distribution (see Example 9.6 and Exercise 9.65) with p unknown.
i Show that
ii Argue that for this ratio to be independent of p, we must have
iii Using the method of Lehmann and Scheff´e, 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?
b Consider the Weibull density discussed in Example 9.7.
i Show that
In this exercise, we illustrate the direct use of the Rao–Blackwell theorem. Let Y1, Y2, . . . , Ynbe independent Bernoulli random variables with
Chapter 4: General Features of Cells Overview Cell biology: the study of individual cells and their interactions with each other Cell theory: o All living organisms are composed of one or more cells o Cells are the smallest units of life o New cells come only from preexisting cells by division 4.1 Microscopy Overview o Microscope: a magnification tool that enables researchers to study the structure and function of cells Compound microscope: a microscope with more than one lens o Micrograph: an image taken with the aid of a microscope o Resolution: a measure of the clarity of an image Ability to observe two objects as d