Go to:Table of Contents

Generalized Distance Measures

The quantity $\Big(\sum_{i-1}^n{p^r_i}\Big)^{s-1\overr-1}$, $r>0$ appearing in the unified expression (3.8) or in the entropy of order $r$ and degree $s$, i.e., in the expression (3.7) plays an important role. Let us write it in the simplified form:

$$G^{\rho}_r(P)=\Big(\sum_{i=1}^n{p^r_i}\Big)^{\rho},\ r>0,\\rho\neq 0$$ (3.22)

for all $P=(p_1,p_2,...,p_n)\ \in\ \Delta_n$. 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 $(r,s)$ is given by

$$T^{\rho}_r(P)=\Big(\sum_{i=1}^n{p^r_i}\Big/\sum_{i=1}^n{p^{\rho}_i}\Big)^{1\overr-\rho}, \ r\neq \rho,\ r\geq 0,\ \rho \geq 0$$ (3.23)

for all $P=(p_1,p_2,...,p_n)\ \in\ \Delta_n$. 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 $P=(p_1,p_2,...,p_n)\ \in\ \Delta_n$, we have

(i) $G^{\rho}_r(P)$ is a convex function of P for $r>1, \ r\rho\geq 1$ or $0<r<1$,$\rho >0,\ r\rho\leq 1$.

(ii) $G^{\rho}_r(P)$ is a concave function of P for $0<r<1$,$\rho >0,\ r\rho\leq 1$.

(iii) $G^{\rho}_r(P)$ is a pseudoconvex/quasiconvex/Schur-convex function of P for $r>1$, $\rho >0$, or $0<r<1$, $\rho <0$.

(iv) $G^{\rho}_r(P)$ is a pseudoconcave/quasiconcave/Schur-concave function of P for $0<r<1$, $\rho >0$ or $r>1$, $\rho <0$.

Property 3.26. For all $P=(p_1,p_2,...,p_n)\ \in\ \Delta_n$, we have

(i) $G^{\rho}_r(P)$ is a decreasing function of r ($\rho$ fixed and $\rho >0$).

(ii) $G^{\rho}_r(P)$ is an increasing function of r ($\rho$ fixed and $\rho <0$).

(iii) $G^{\rho}_r(P)$ is a decreasing function of $\rho$ (r fixed and $r>1$).

(iv) $G^{\rho}_r(P)$ is an increasing function of $\rho$ (r fixed and $0<r<1$).

Property 3.27. For all $P=(p_1,p_2,...,p_n)\ \in\ \Delta_n$,$p_{max}=max\{p_1,p_2,...,p_n\}$ and $1\leq\sigma \leq n$, we have

(i) ($a_1$) $G^{\rho}_r\Big(1-\sum_{i=1}^\sigma{p_i},\ \sum_{i=1}^\sigma{p_i}\Big)\leqG^{......gma{p_i}}_{\sigma- \mbox{times}}, 1-{1\over\sigma}\sum_{i=1}^\sigma{p_i}\Big)$

($a_2$) $G^{\rho}_r(1-p_{max},\ p_{max})\leqG^{\rho}_r(P) \leq G^{\rho}_r\Big(\underb......_{max}\overn-1},...,{1-p_{max}\over n-1}}_{(n-1)-\mbox{times}},p_{max}\Big)$
$(0<r<1,\ \rho > 0)$ or $(r>1,\\rho < 0)$.

(ii) ($a_1$) $G^{\rho}_r\Big(1-\sum_{i=1}^\sigma{p_i},\ \sum_{i=1}^\sigma{p_i}\Big)\geqG^{......igma{p_i}}_{\sigma-\mbox{times}}, 1-{1\over\sigma}\sum_{i=1}^\sigma{p_i}\Big)$

($a_2$) $G^{\rho}_r(1-p_{max},\ p_{max})\geqG^{\rho}_r(P) \geq G^{\rho}_r\Big(\underb......_{max}\overn-1},...,{1-p_{max}\over n-1}}_{(n-1)-\mbox{times}},p_{max}\Big)$
$(r>1,\ \rho > 0)$ or $(0<r<1,\\rho < 0)$.

Property 3.28. For all $P=(p_1,p_2,...,p_n)\ \in\ \Delta_n$, we have

(i)($a_1$) $T^{\rho}_r(P)$ is an increasing function of r ($\rho$ fixed).

($a_2$) $T^{\rho}_r(P)$ is an increasing function of $\rho$ (r fixed).

(ii) ($a_1$) $T^{\rho}_r(P)\leq p_{max}$.

($a_2$) $T^{\rho}_r(P)\geq\Big(p^\rho_{max}/\sum_{i=1}^n{p^\rho_i}\Big) ^{1\over r-\rho}p_{max},\ r>\rho$.
($a_3$) $T^{\rho}_r(P)\geq\Big(p^r_{max}/\sum_{i=1}^n{p^r_i}\Big) ^{1\over \rho -r}p_{max},\ \rho>r$.