Local score tests in mixture exponential family

For one-parameter natural exponential and power series families we give some general characterizations from an affine relation connecting the mean of member distributions and their length biased versions. Examples subsume many known cases. Other characterizations are explored using random variable relations involving the length biased version of partial sums. Finally, characterizations of the law of X admitting more general weights are obtained by equating the laws of a function H(X) and the weighted version of X.