Estimating a quantile by Wilks’ method¶
Let us denote , where is a random vector, and a deterministic vector. We seek here to evaluate, using the probability distribution of the random vector , the quantile of , where :
If we have a sample of independent samples of the random vector , can be estimated as follows:
the sample of vector is first transformed to a sample of the variable , using ,
the sample is then placed in ascending order, which gives the sample ,
this empirical estimation of the quantile is then calculated by the formula:
where denotes the integral part of .
For example, if and , is equal to , which is the largest value of the sample . We note that this estimation has no meaning unless . For example, if , one can only consider values of a to be between 1% and 99%.
It is also possible to calculate an upper limit for the quantile with a confidence level chosen by the user; one can then be sure with a level of confidence that the real value of is less than or equal to :
The most robust method for calculating this upper limit consists of taking where is an integer between 2 and found by solving the equation:
A solution to this does not necessarily exist, i.e. there may be no integer value for satisfying this equality; one can in this case choose the smallest integer such that:
which ensures that ; in other words, the level of confidence of the quantile estimation is greater than that initially required.
This formula of the confidence interval can be used in two ways:
either directly to determine for the values chosen by the user,
or in reverse to determine the number of simulations to be carried out for the values and chosen by the user; this is known as Wilks’ formula.
For example for , we take with simulations (that is the maximum value out of 59 samples) or else with simulations (that is the second largest result out of the 93 selections). For values of between and , the upper limit is the maximum value of the sample. The following tabular presents the whole results for , still for .
Rank of the upper bound of the quantile 
Rank of the empirical quantile 


59 
59 
57 
93 
92 
89 
124 
122 
118 
153 
150 
146 
181 
177 
172 
208 
203 
198 
234 
228 
223 
260 
253 
248 
286 
278 
272 
311 
302 
296 
336 
326 
320 
361 
350 
343 
386 
374 
367 
410 
397 
390 
434 
420 
413 
458 
443 
436 
482 
466 
458 
506 
489 
481 
530 
512 
504 
554 
535 
527 
577 
557 
549 
601 
580 
571 
624 
602 
593 
647 
624 
615 
671 
647 
638 
694 
669 
660 
717 
691 
682 
740 
713 
704 
763 
735 
725 
786 
757 
747 
809 
779 
769 
832 
801 
791 
855 
823 
813 
877 
844 
834 
900 
866 
856 
923 
888 
877 
945 
909 
898 
968 
931 
920 
991 
953 
942 
is often called the “empirical quantile” for the variable .
API:
See
Wilks
Examples:
References:
Wilks, S.S. (1962). “Mathematical Statistics”, New YorkLondon
Robert C.P., Casella G. (2004). MonteCarlo Statistical Methods, Springer, ISBN 0387212396, 2nd ed.
Rubinstein R.Y. (1981). Simulation and The MonteCarlo methods, John Wiley & Sons