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Distance Measures

Let us consider the following three measures

$$V(P\vert\vert Q)=\sum_{i=1}^n{\vert p_i-q_i\vert},$$ (2.4)

$$B(P\vert\vert Q)=\sum_{i=1}^n{\sqrt{p_iq_i}}$$ (2.5)

and

$$M(P\vert\vert Q)=\sum_{i=1}^n{{2p_iq_i\over p_i+q_i}}.$$ (2.6)

for all $(P,Q) \in \Delta_n \times \delta_n.$

The measures (2.4), (2.5) and (2.6) are famous in the literarture as the ``Variational distance", the `Bhattacharyya distance" and the ``Harmonic mean" respectively. These measures satisfy the following properties

Property 2.15. We have

(i) $D(P\vert\vert Q)\geq V(P\vert\vert Q)-\ell n\big(1+V(P\vert\vert Q)\big),$

(ii) $D(P\vert\vert Q)\geq {V(P\vert\vert Q)^2\over 2},$

(iii) $D(P\vert\vert Q)\geq {V(P\vert\vert Q)^2\over 2}+{V(P\vert\vert Q)^4\over36},$

(iv) $D(P\vert\vert Q)\geq {V(P\vert\vert Q)^2\over 2}+{V(P\vert\vert Q)^4\over36}+{V(P\vert\vert Q)^6\over 288},$

(v) $D(P\vert\vert Q)\geq { V(P\vert\vert Q)\over 2}\ell n\Big({2+V(P\vert\vert Q)\over 2-V(P\vert\vert Q) }\Big),$

(vi) $2\big[1-B(P\vert\vert Q)\big]\leq V(P\vert\vert Q)\leq \,2\big[2\big(1-B(P\vert\vert Q)\big)\big]^{1/2},$

(vii) $V(P\vert\vert Q)\leq \,2\big(1-B(P\vert\vert Q)^2\big)^{1/2}\leq \,2\big[2\big(1-B(P\vert\vert Q)\big)\big]^{1/2},$

The measure $D(P\vert\vert Q)$ considered here is with natural logarithm base.

Property 2.16. We have

(i) $B(P\vert\vert Q)\geq T(P\vert\vert Q) \geq B(P\vert\vert Q)^2,$

(ii) $1-{V(P\vert\vert Q)\over 2} \leq T(P\vert\vert Q) \leq 1 - {1\over 4}V(P\vert\vert Q)^2,$

For more details on these measures refer to Mathai and Rathie (1975) [71] and references therein.