I stumbled upon this term in McFadden - Analysis of qualitative choice behavior (page 111).
It is said that
"A random Variable $X$ is translation complete if for a function h of bounded absolute variation with $h(\pm\infty)=0$, the condition $E(h(X+a))=0$ for all real $a$ implies $h\equiv 0$ (except possibly on a set of measure zero)."
Further it is said that most common distributions have this porperty; in particular, the Gumble distribution is translation complete. Another qoute says that
"A distribution whose characteristic function is nonzero for real arguments is translation complete [apply Feller, 1966, An Introduction to Probability Theory and Its Applications, Vol. 2, Wiley, New York, p. 479]..."
if I check there I cannot find any reference.
My question are:
If I google translation complete in connection with random variable I get no feasible results. Is translation completeness connected to sufficient statistics: What are complete sufficient statistics?
Though I understand the mathematical meaning of translation completeness I dont get the intention of this definition.
Why does translation completeness imply $G(X - logK) ) = G(X)^K$ if $G$ is a cdf and $X$ a random variable?