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## Generalized Distance Measures

The quantity  appearing in the unified expression (3.8) or in the entropy of order  and degree , i.e., in the expression (3.7) plays an important role. Let us write it in the simplified form:
 (3.22)

for all . The quantity (3.49) is famous as generalized distance measure (Boekee and Van der Lubbe, 1979 [15]; Capocelli et al., 1985 [24]) or the generalized certainty measure (Van der Lubbe et al., 1984 [116]).

Another distance measure arising from the entropy of order  is given by

 (3.23)

for all . This measure has been considered by Capocelli et al. (1985) [24].

The quantities (3.49) and (3.50) contain as a particular case the measures studied by Trouborst et al., (1974) [112], Györfi and Nemetz (1975) [42], Devijver (1974) [34], Vajda (1968) [113] etc..

The measures (3.22) and (3.23) satisfy some properties. These are given as follows:

Property 3.25. For all , we have

(i)  is a convex function of P for  or ,.
(ii)  is a concave function of P for ,.
(iii)  is a pseudoconvex/quasiconvex/Schur-convex function of P for , or .
(iv)  is a pseudoconcave/quasiconcave/Schur-concave function of P for  or .
Property 3.26. For all , we have
(i)  is a decreasing function of r ( fixed and ).
(ii)  is an increasing function of r ( fixed and ).
(iii)  is a decreasing function of  (r fixed and ).
(iv)  is an increasing function of  (r fixed and ).
Property 3.27. For all , and , we have
(i) (

(
or .
(ii) (

(
or .
Property 3.28. For all , we have
(i)( is an increasing function of r ( fixed).
( is an increasing function of  (r fixed).
(ii) (.
(.
(.

21-06-2001
Inder Jeet Taneja
Departamento de Matemática - UFSC
88.040-900 Florianópolis, SC - Brazil