ABSTRACT
PROOF Asslrnle XL, < z l . Thcri on ( X L , , :cl) d;(r) is decreasing and f l (X) is increasing, so that - f2(X) is tiecreasirig 0x1 (:i2, :cl ) which ~oiltradicts
fl (X) (27.2.4).
I,ct riow ,Si, i - 1,2 , be inclcpenclent st,atistics such that ST?/a, has dmsit,y gi(z) T(a > 0) : wherc 01 ancl a:! are unknom~rl positive paritmeters with crl 5 aa. We further assume t , h t g,(z) satisfies (27.2.1). ViJe will study two classes of estimators of n l : rlarriely C = { a s , : [L > 0) and D - (4(V)S1 : 4 is a positive f~mctiorl ), where V = Sz/Sl . The first one is tlhe class of scale equivariant estirrlators of nl based only on S I . This class tlocs not make llsc of the full data and, in effect. ignores thc iriformation a1 5 m. Tllc second one is thc class of eqnivariarit estirrlators under the group of tlmsforrnations (Ss ,Sq) 4 ( a s l , US?), a > 0. Clearly, C c I).
