<s>
The	O
ratio	B-General_Concept
estimator	I-General_Concept
is	O
a	O
statistical	O
parameter	O
and	O
is	O
defined	O
to	O
be	O
the	O
ratio	O
of	O
means	O
of	O
two	O
random	O
variables	O
.	O
</s>
<s>
The	O
ratio	O
estimates	O
are	O
asymmetrical	O
and	O
symmetrical	O
tests	O
such	O
as	O
the	O
t	B-General_Concept
test	I-General_Concept
should	O
not	O
be	O
used	O
to	O
generate	O
confidence	O
intervals	O
.	O
</s>
<s>
An	O
upper	O
bound	O
on	O
the	O
relative	O
bias	O
of	O
the	O
estimate	O
is	O
provided	O
by	O
the	O
coefficient	O
of	O
variation	O
(	O
the	O
ratio	O
of	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
to	O
the	O
mean	O
)	O
.	O
</s>
<s>
Other	O
methods	O
of	O
estimating	O
a	O
ratio	B-General_Concept
estimator	I-General_Concept
include	O
maximum	O
likelihood	O
and	O
bootstrapping	B-Application
.	O
</s>
<s>
The	O
skewness	B-General_Concept
and	O
the	O
kurtosis	B-Error_Name
of	O
the	O
ratio	O
depend	O
on	O
the	O
distributions	O
of	O
the	O
x	O
and	O
y	O
variates	O
.	O
</s>
<s>
It	O
has	O
been	O
found	O
that	O
in	O
general	O
ratio	O
variables	O
are	O
skewed	B-General_Concept
to	O
the	O
right	O
,	O
are	O
leptokurtic	O
and	O
their	O
nonnormality	O
is	O
increased	O
when	O
magnitude	O
of	O
the	O
denominator	O
's	O
coefficient	O
of	O
variation	O
is	O
increased	O
.	O
</s>
<s>
Because	O
the	O
ratio	O
estimate	O
is	O
generally	O
skewed	B-General_Concept
confidence	O
intervals	O
created	O
with	O
the	O
variance	O
and	O
symmetrical	O
tests	O
such	O
as	O
the	O
t	B-General_Concept
test	I-General_Concept
are	O
incorrect	O
.	O
</s>
<s>
If	O
the	O
ratio	B-General_Concept
estimator	I-General_Concept
is	O
unimodal	B-Algorithm
(	O
which	O
is	O
frequently	O
the	O
case	O
)	O
then	O
a	O
conservative	O
estimate	O
of	O
the	O
95%	O
confidence	O
intervals	O
can	O
be	O
made	O
with	O
the	O
Vysochanskiï	O
–	O
Petunin	O
inequality	O
.	O
</s>
<s>
An	O
alternative	O
method	O
of	O
reducing	O
or	O
eliminating	O
the	O
bias	O
in	O
the	O
ratio	B-General_Concept
estimator	I-General_Concept
is	O
to	O
alter	O
the	O
method	O
of	O
sampling	O
.	O
</s>
<s>
Then	O
the	O
ratio	O
of	O
the	O
sum	O
of	O
the	O
y	O
variates	O
and	O
the	O
sum	O
of	O
the	O
x	O
variates	O
chosen	O
in	O
this	O
fashion	O
is	O
an	O
unbiased	O
estimate	O
of	O
the	O
ratio	B-General_Concept
estimator	I-General_Concept
.	O
</s>
<s>
The	O
ratio	B-General_Concept
estimator	I-General_Concept
given	O
by	O
this	O
scheme	O
is	O
unbiased	O
.	O
</s>
<s>
Tin	O
(	O
1965	O
)	O
described	O
and	O
compared	O
ratio	B-General_Concept
estimators	I-General_Concept
proposed	O
by	O
Beale	O
(	O
1962	O
)	O
and	O
Quenouille	O
(	O
1956	O
)	O
and	O
proposed	O
a	O
modified	O
approach	O
(	O
now	O
referred	O
to	O
as	O
Tin	O
's	O
method	O
)	O
.	O
</s>
<s>
These	O
ratio	B-General_Concept
estimators	I-General_Concept
are	O
commonly	O
used	O
to	O
calculate	O
pollutant	O
loads	O
from	O
sampling	O
of	O
waterways	O
,	O
particularly	O
where	O
flow	O
is	O
measured	O
more	O
frequently	O
than	O
water	O
quality	O
.	O
</s>
<s>
If	O
a	O
linear	O
relationship	O
between	O
the	O
x	O
and	O
y	O
variates	O
exists	O
and	O
the	O
regression	B-General_Concept
equation	O
passes	O
through	O
the	O
origin	O
then	O
the	O
estimated	O
variance	O
of	O
the	O
regression	B-General_Concept
equation	O
is	O
always	O
less	O
than	O
that	O
of	O
the	O
ratio	B-General_Concept
estimator	I-General_Concept
.	O
</s>
<s>
The	O
precise	O
relationship	O
between	O
the	O
variances	O
depends	O
on	O
the	O
linearity	O
of	O
the	O
relationship	O
between	O
the	O
x	O
and	O
y	O
variates	O
:	O
when	O
the	O
relationship	O
is	O
other	O
than	O
linear	O
the	O
ratio	O
estimate	O
may	O
have	O
a	O
lower	O
variance	O
than	O
that	O
estimated	O
by	O
regression	B-General_Concept
.	O
</s>
<s>
Although	O
the	O
ratio	B-General_Concept
estimator	I-General_Concept
may	O
be	O
of	O
use	O
in	O
a	O
number	O
of	O
settings	O
it	O
is	O
of	O
particular	O
use	O
in	O
two	O
cases	O
:	O
</s>
<s>
The	O
first	O
known	O
use	O
of	O
the	O
ratio	B-General_Concept
estimator	I-General_Concept
was	O
by	O
John	O
Graunt	O
in	O
England	O
who	O
in	O
1662	O
was	O
the	O
first	O
to	O
estimate	O
the	O
ratio	O
y/x	O
where	O
y	O
represented	O
the	O
total	O
population	O
and	O
x	O
the	O
known	O
total	O
number	O
of	O
registered	O
births	O
in	O
the	O
same	O
areas	O
during	O
the	O
preceding	O
year	O
.	O
</s>
