The concept of "functional sufficiency" or "f-sufficiency" for a class of density functions is introduced and conditions given under which it corresponds to sufficiency as defined by Halmos and Savage . A minimal f-sufficient statistic is defined and shown to exist, and its construction is given. The minimal f-sufficient statistic for a class of densities for which the region of positive density varies with the parameter is shown to be equivalent to the combination of a "statistic of selection" and the minimal f-sufficient statistic for a class of densities for which the region of positive density is fixed. The construction of sufficient statistics in this latter case subject to certain conditions of regularity has been treated by Koopman . If the parameter is a parameter of selection from a fixed distribution, then the statistic of selection is the minimal f-sufficient statistic. If in addition the regions of positive density are monotone and indexed monotonely by a real parameter, then the statistic of selection is sufficient according to the Halmos and Savage definition. Three examples are given to illustrate the results.
"Sufficient Statistics and Selection Depending on the Parameter." Ann. Math. Statist. 23 (3) 417 - 425, September, 1952. https://doi.org/10.1214/aoms/1177729386