An element of choice arises when setting confidence limits with asymmetric probability densities. The requisite area, a, usually a = 0.05 or 0.01, can in principle be distributed over the two tails of the density in an infinity of ways. A common choice is to place half of the given area in each tail and then proceed to calculate the appropriate confidence interval on this basis with the aid of statistical tables or computer software. One drawback of this method is that it doesn’t usually provide the smallest confidence interval, indeed intervals of slightly smaller width can sometimes be obtained by distributing the tail areas in different proportions by trial and error [Reference 1]. This situation does not arise with symmetrical distributions, where placing half of the area in each tail automatically achieves the confidence interval of minimum width.

In the first part of this
article a simple procedure is described to obtain the minimum-width confidence
interval using a skewed density passing through the origin. It is shown that the minimum width occurs
when the two tails have equal heights.
The procedure is then applied to a c^{2} statistic for
the common cases a
= 0.05 and a
= 0.01.

The c^{2} distribution
is also widely used in the calculation for setting confidence limits on the
unknown population variance and standard deviation using a sample standard
deviation statistic. It is generally
desirable that the widths of these confidence intervals also be as small as
possible. The problem of minimising
these widths is considered in the latter part of this article.

** **

Figure 1

Figure 1 shows a skewed density
function for the positive random variable X over an infinite range. As usual the probability that X takes on a
value less than or equal to *u* is
given by

_{}

In Figure 1 the given constant
area a
has been arbitrarily divided into two parts; one part a_{L} has been
inserted in the left-hand tail and the remaining part a_{U} = a - a_{L}
placed in the right-hand tail. The
quantity D
= *x _{U} - x_{L}* is
therefore the width of the

(1- a)*100% confidence interval for this particular choice of the partitioning of the given area a and it is required to find the smallest value of D amongst all such possible partitions of a.

The independent variable *x _{L}* is allowed to vary between
zero (when all the area is in the right-hand tail) and some value

The requirement that the tail areas sum to a is expressed by

_{} [1]

Equation [1] defines D
implicitly as a function of the independent variable *x _{L}*

_{} [2]

Let _{} [3]

Clearly *F(0) = 0*, _{}* *and, since f* (x)* is a density function, it follows that

_{} [4]

Using [1], [3] and [4] the condition that the two tail areas sum to a may now be expressed by

_{}

_{} [5]

Differentiating both sides of [5] with respect to *x _{L}*

_{} [6]

so that

_{} [7]

We observe that _{} when

_{} [8]

By differentiating [7] and using [8] we find that at the turning point

_{} _{} [9]

where _{}.

For the type of situation
illustrated in Figure 1 we see from [9] that the turning point will be a
minimum since g(*x _{L}*) is
positive and g(

The condition [8] is a
necessary condition for the width to have a minimum value; it is automatically
satisfied for symmetrical distributions when equal areas are placed in each
tail. Satisfying the condition [8]
simultaneously with the area requirement expressed by equation [1] enables the
determination of the minimum width for a skew distribution with the given value
of a. This is carried out in the next section
using the c^{2
}density.

If X is a chi-squared distributed random variable the
appropriate density function is given by the c^{2} density with n
degrees of freedom [Reference 1]

_{} [10]

We shall restrict our attention in this
section to the cases where n>2. The
densities for n
= 1 and n
=2 decrease monotonically so that _{}and condition [8] cannot be satisfied. It can be shown that the smallest values of D for
these cases are simply obtained by placing all of the given area a in
the right hand tail.

The condition that _{}when applied to [10] leads to the condition that for the c^{2} density

_{} [11]

The required values of *x _{L}* and

_{} [12]

The simultaneous equation [11] and [12] may be solved numerically. A convenient procedure is to numerically
solve equation [12], treated as a function of the single variable *x _{L}*, repeatedly using [11]
where necessary to obtain and substitute the appropriate value of

Figure 2 shows the result of applying this procedure
to the c^{2}
density with n = 5 degrees of freedom and a=0.05.
For comparison the widths of all the possible 'trial' confidence
intervals consistent with the given value of a are also shown as well as
the minimum width of the confidence interval, D, calculated as described
above. This provides a reasonably
convincing demonstration that the simultaneous solution of Equations [11] and
[12] does indeed give the minimum width without the need for trial and
error.

** Figure 2**

Figure 2 is a graph showing the width, W, of the 95%
confidence intervals for

a = 0.05 and n = 5 as a function of the area, a_{L}, placed in the left hand
tail. The arrow marks the situation for
the minimum width and is obtained by placing most of the area in the right hand
tail. The right vertical line drawn at a_{L} = 0.025 shows the situation
when equal areas are placed in each tail. At the minimum width a_{L} = 0.0023, x_{L} =
0.296 and x_{U} =11.191

Denoting the minimum width by D* _{min}* and the corresponding
‘equal tail areas’ width by d the quantity

.

**Figure 3**

Figure 3
shows the percentage difference between the minimum width confidence interval
and the corresponding width obtained when the areas in each tail are equal,
plotted as a function of the number of degrees of freedom. The solid line is for

a = 0.05 and the dotted line
for a = 0.01.

The standard procedure to estimate the confidence
limits on the population variance, s^{2},^{ }and the
standard deviation, s, relies upon the fact that
the quantity _{}is chi-squared distributed with n = n-1 degrees of freedom. Here S is the given sample standard deviation
calculated from the n experimental observations.

This
implies that with a_{L} + a_{U} = a we can be (1-a)*100% certain that

_{}

_{} [13]

_{}

In practice Equation [13] is often rearranged to
provide the appropriate confidence limits on the variance or on the standard
deviation and used either in the form of Equation [14] or [15] respectively:

_{} [14]

_{} [15]

The appropriate values of *x _{L} *and

The width of the appropriate confidence interval for
[13] is given by D = x_{U} - x_{L}
and may thus be investigated by the method already described. The problem of finding the minimum widths of
the confidence intervals for Equations [14] and [15] is considered below using
Lagrange undetermined multipliers.
Although this method only identifies the turning points it is found that
in the cases of interest these do correspond to minimum values.

_{} [16]

and

_{} [17]

[16] and [17] serve
as new objective functions to be separately minimised subject to the constraint
[5] that *x _{L}* and

Consider first the
problem of minimising _{}. With l as
an undetermined multiplier we define the function

_{} [18]

The second term on the right of Equation [18]
represents the fixed-area constraint on *x _{L}*
and

_{} [19]

Using the definition of f*(x)*
for the chi-squared density in [9] we can now obtain the necessary condition
for minimising the width of the confidence interval containing the variance as

_{} _{} [20]

It is interesting to note the similarity between [20] and [11]. In this context it may be observed that [11] could also have been derived in the same way as [20] but by starting with

_{}

instead of the *V *used
in [18]

[20] plays the same role in the minimisation of the
variance interval _{}as
is played by [11] in the minimisation of D.

We can also use this same multiplier technique to
minimise D_{s}, the width of the
confidence interval for s, by repeating the steps
from [18] to [20] but this time starting with a new definition of *V* incorporating the expression for D_{s}_{ as} given in [17]

_{} [21]

The
results analogous to [19] and [20] are then found to be

_{} [22]

and

_{} [23]

We are now in a position to use [20] and [23] together with [12] to investigate the possible reductions in the widths of the confidence intervals for the variance and the standard deviation as compared with the practice of dividing the area a equally between the two tails. Some results are illustrated in Figures 4, 5 and 6.

**Figure 4**

Figure 4 is a graph showing the width, W, of the 95%
confidence intervals for the standard deviation with a = 0.05 and n = 5 as a function of the
area, a_{L}, placed in the left hand
tail. The sample standard deviation has
been taken as unity. The arrow marks
the situation for the minimum width found by simultaneously solving [23] and
[12] and this time it is obtained by placing most of the area in the left-hand
tail. The right vertical line drawn at
a_{L} = 0.025 shows the situation
when equal areas are placed in each tail.
At the minimum width a_{L} = 0.0467, *x _{L}* = 1.109 and

**Figure 5**

Figure 5 is a graph showing the width, W, of the 95%
confidence interval for the variance with n = 5 as a function of the
area, a_{L}, placed in the left hand
tail. The sample variance has been taken
as unity. Again the minimum width is
obtained by placing most of the area in the left-hand tail and is found by
simultaneously solving [20] and [12].
The right vertical line drawn at a_{L} = 0.025 shows the situation
when equal areas are placed in each tail.
At the minimum width a_{L} = 0.0494, *x _{L} *= 1.139 and

**Figure 6**

Figure 6 shows the percentage difference between the
minimum width confidence interval for the standard deviation and the
corresponding width obtained when the areas in each tail are equal, plotted as
a function of the number of degrees of freedom.
The solid line for a = 0.05 is very close to the
dotted line for a = 0.01.

Confidence intervals involving c^{2
}can be modestly reduced in width and in some extreme cases by as much as
50% over the ‘equal-area’ approach by using the methods described here. The scope for reduction decreases, as the
number of degrees of freedom increases and the density becomes more
symmetric. It is a matter of choice as
to which confidence intervals are reported but it can sometimes be useful to be
able to report narrower intervals for the population standard deviation if
required particularly when only small sample sizes are available.

To avoid unnecessary calculations a set of
reference tables has been drawn up and may be used to obtain the appropriate
critical values of c^{2}
in terms of n
and a
that are required to obtain the minimum width confidence intervals for s and s^{2}
.

‘Theory and Problems of Probability and Statistics’ by Murray. R. Spiegel. Schaum Outline Series, 1982.