For large sample sizes, the search for the confidence limits using Equation (2) can be very time consuming. When
sample sizes were over 300, an approximation to the binomial probabilities was used (see Johnson and Kotz, 1969).

There are two problems with this method which are evident in the tables produced for this project. First, when
calculated on small sample sizes, N<50, and when there are a number of ties in the data, confidence intervals for the
estimated percentile may have upper and lower bounds equal to each other and to the estimate. This is a direct result
of having ties and implies that the true precision of the estimate is below the scale of measurement inherent in the data.
In similar situations either the upper or lower bound may be equal to the estimator. Because of this, the confidence
intervals should only be used to indicate a general level of confidence in the estimator within the limits of measurement
resolution.

The second problem encountered, again with small samples and large number of ties, is that the distribution free method
may not be able to supply an acceptable point estimate of the percentile. In some cases, the confidence interval is
provided, indicating that the percentile, usually for percentage values close to one, is between the second largest and
largest observations. In other cases, neither the estimator nor the confidence limits are provided.

Finally, with large sample sizes, there are a large number of possible confidence intervals which have similar coverage
probabilities. Since, in theory at least, all of these intervals are acceptable, we choose one at random to report.


Smoothed Density Approach

The third approach involves approximating the density function underlying the data using density smoothing techniques
(Silverman, 1986). This smoothed density can then be integrated to construct an estimated distribution function from
which the percentile can be obtained. A kernel estimator was used to estimate the underlying density of the data. The
kernel estimator used was:


 1 x-X, (3)
 At) ,- K( )
 nh El h



The kernel function, K(.), is usually a standard parametric density function which defines the weights to be given to
observations around the target value, x. The window width (smoothing parameter or bandwidth), h, defines how far,
either side of x, the kernel function will apply. A larger h results in a very smooth density estimate, a very small h results
in a very rough, multimodal estimate. The smoother the kernel function, the smoother will be the estimated density
A x)"

The kernel density chosen was the standard normal distribution. The window width was chosen to be h 2o- n -s
This value of h has been shown to provide relatively smooth approximation of multimodal distributions when used with
the normal density.


Implementation

Smoothed and distribution-free estimates of designated percentiles using the methods described in the previous sections
were computed using the capabilities of the S-Plus statistical analysis package running on a UNIX workstation. S-Plus
is a product of the StatSci division of MathSoft.