Monotonicity of a Power Function: An Elementary Probabilistic Proof

Abstract

It is pointed out that in many one-sided testing situations for a real-valued parameter θ, the monotonicity of the power function hinges on the stochastic order of the underlying family of distributions [F_θ] rather than on the stronger property of monotone likelihood ratio of the family. An elementary proof, accessible to students of introductory probability and statistics, is presented.

Key Words:

• Power function
• Sums of independent random variables
• Stochastic order
• Distributions with given marginals