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Unified (r,s)-Inaccuracies
Similar to expression (4.1), we shall give below the three different forms
of writing directly the unified
inaccuracy
measures in three different forms:
|
|
|
(4.5) |
and 3, where
![$ \displaystyle ^1M_r(P\vert\vert Q)=\sum_{i=1}^n{p_iq^{r-1}_i},$](img989.gif)
![$ \displaystyle ^2M_r(P\vert\vert Q)=\Big(\sum_{i=1}^n{p_iq^{(r-1)/r}_i}\Big)^r$](img990.gif)
and
![$ \displaystyle^3M_r(P\vert\vert Q)={\sum_{i=1}^n{p^r_i}\over\sum_{i=1}^n{p^r_iq^{1-r}_i}}.$](img991.gif)
The expressions
,
and
are due to Nath (1968) [73], Van der Lubbe
(1978) [115] and Nath (1975) [74]
respectively. The expressions
and
are due to Sharma and Gupta (1976) [89].
We can write
and
where
is as given in (4.4).
From the unified espression (4.5) it is clear that the measures
and
are continuous with respect to the parameters
and
.
This allows us to write
and![$\displaystyle \ 3, $](img1003.gif)
where "CE" stands for "continuous extension".
In particular, when
,
we have
![$\displaystyle ^1{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)=\,\,......athscr{H}}}}^s_r(P\vert\vert Q)={\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P),$](img1004.gif)
and when
,
we have
where
21-06-2001
Inder Jeet Taneja
Departamento de Matemática - UFSC
88.040-900 Florianópolis, SC - Brazil