On the monotone likelihood ratio property for the convolution of independent binomial random variables

Abstract

Given that r and s are natural numbers and X ∼ Binomial ( r , q ) and Y ∼ Binomial ( s , p ) are independent random variables where q , p ∈ ( 0 , 1 ) , we prove that the likelihood ratio of the convolution Z = X + Y is decreasing, increasing, or constant when q < p , q > p or q = p , respectively.