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 c2 statistic for the common cases a = 0.05 and a = 0.01.
The c2 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 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 aL has been inserted in the left-hand tail and the remaining part aU = a - aL placed in the right-hand tail. The quantity D = xU - xL 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 xL is allowed to vary between zero (when all the area is in the right-hand tail) and some value xLmax (when all the area is in the left-hand tail). It is instructive to consider qualitatively the anticipated behaviour of D as xL varies over this range. It is clear from Figure 1 that when xL is zero D has some finite value which may be found by solving . It is also clear that D® ¥ as xL ® xLmax. It is possible that as D varies continuously between these two extremes there could be a value of D smaller than the starting value, if so there will be a minimum turning point on the graph of D v xL
The requirement that the tail areas sum to a is expressed by
Equation  defines D implicitly as a function of the independent variable xL for any given constant value of a and with 0£ xL £ xLmax. In the usual way D will have a minimum value when
Clearly F(0) = 0, and, since f (x) is a density function, it follows that
Using ,  and  the condition that the two tail areas sum to a may now be expressed by
Differentiating both sides of  with respect to xL gives
We observe that when
By differentiating  and using  we find that at the turning point
For the type of situation illustrated in Figure 1 we see from  that the turning point will be a minimum since g(xL) is positive and g(xL +D) is negative.
The condition  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  simultaneously with the area requirement expressed by equation  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 c2 density.
If X is a chi-squared distributed random variable the appropriate density function is given by the c2 density with n degrees of freedom [Reference 1]
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  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  leads to the condition that for the c2 density
The required values of xL and xU associated with the stationary-width confidence interval for the given a may be found by using  together with the requirement that the areas in the two tails must sum to the given value of a as expressed by equation 
The simultaneous equation  and  may be solved numerically. A convenient procedure is to numerically solve equation , treated as a function of the single variable xL, repeatedly using  where necessary to obtain and substitute the appropriate value of xU corresponding to the current value of xL. When the appropriate value of xL has finally been found the corresponding value of xU may be obtained using  and the stationary-width confidence interval for that value of a is then obtained from D = xU - xL
Figure 2 shows the result of applying this procedure to the c2 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  and  does indeed give the minimum width without the need for trial and error.
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, aL, 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 aL = 0.025 shows the situation when equal areas are placed in each tail. At the minimum width aL = 0.0023, xL = 0.296 and xU =11.191
Denoting the minimum width by Dmin and the corresponding ‘equal tail areas’ width by d the quantity is plotted as a function of n > 2 in Figure 3. This shows the modest reductions in widths that may be obtained.
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, s2, 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 aL + aU = a we can be (1-a)*100% certain that
In practice Equation  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  or  respectively:
The appropriate values of xL and xU are found from tables or software, usually by placing half of the area, a, into each tail of the appropriatec2 density.
The width of the appropriate confidence interval for  is given by D = xU - xL and may thus be investigated by the method already described. The problem of finding the minimum widths of the confidence intervals for Equations  and  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.
 and  serve as new objective functions to be separately minimised subject to the constraint  that xL and xU must always define two tail areas with a fixed sum equal to the given a. It is important to note that the particular choice of xL and xU required to minimise D will not generally serve to also minimise either . Indeed for the cases where n =1 and n = 2, the smallest value of D is achieved by placing all the area in the right hand tail thus setting xL to zero. This choice leads to infinite widths for both when the reciprocal of xL is formed in  and  and is clearly an unsuitable choice. In this section we shall also implicitly include consideration of the cases where n =1 and n = 2
Consider first the problem of minimising . With l as an undetermined multiplier we define the function
The second term on the right of Equation  represents the fixed-area constraint on xL and xU corresponding to Equation  and the first term is the constrained function corresponding to the variance width to be minimised. Differentiating V partially with respect to xL and xU, setting the results to zero and eliminating l yields the result
Using the definition of f(x) for the chi-squared density in  we can now obtain the necessary condition for minimising the width of the confidence interval containing the variance as
It is interesting to note the similarity between  and . In this context it may be observed that  could also have been derived in the same way as  but by starting with
instead of the V used in 
 plays the same role in the minimisation of the variance interval as is played by  in the minimisation of D.
We can also use this same multiplier technique to minimise Ds, the width of the confidence interval for s, by repeating the steps from  to  but this time starting with a new definition of V incorporating the expression for Ds as given in 
The results analogous to  and  are then found to be
We are now in a position to use  and  together with  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 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, aL, 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  and  and this time it is obtained by placing most of the area in the left-hand tail. The right vertical line drawn at aL = 0.025 shows the situation when equal areas are placed in each tail. At the minimum width aL = 0.0467, xL = 1.109 and xU = 17.743
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, aL, 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  and . The right vertical line drawn at aL = 0.025 shows the situation when equal areas are placed in each tail. At the minimum width aL = 0.0494, xL = 1.139 and xU = 21.8.
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 c2 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 c2 in terms of n and a that are required to obtain the minimum width confidence intervals for s and s2 .
‘Theory and Problems of Probability and Statistics’ by Murray. R. Spiegel. Schaum Outline Series, 1982.