naginterfaces.library.univar.robust_1var_ci(method, x, clevel)[source]

robust_1var_ci computes a rank based (nonparametric) estimate and confidence interval for the location parameter of a single population.

For full information please refer to the NAG Library document for g07ea

methodstr, length 1

Specifies the method to be used.

The exact algorithm is used.

The iterative algorithm is used.

xfloat, array-like, shape

The sample observations, , for .


The confidence interval desired.

For example, for a confidence interval set .


The estimate of the location, .


The estimate of the lower limit of the confidence interval, .


The estimate of the upper limit of the confidence interval, .


An estimate of the actual percentage confidence of the interval found, as a proportion between .


The upper value of the Wilcoxon test statistic, , corresponding to the lower limit of the confidence interval.


The lower value of the Wilcoxon test statistic, , corresponding to the upper limit of the confidence interval.

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: or .

(errno )

Not enough information to compute an interval estimate since the whole sample is identical. The common value is returned in , and .

(errno )

The iterative procedure used to estimate has not converged.

(errno )

The iterative procedure used to estimate, , the upper confidence limit has not converged.

(errno )

The iterative procedure used to estimate, , the lower confidence limit has not converged.


Consider a vector of independent observations, with unknown common symmetric density . robust_1var_ci computes the Hodges–Lehmann location estimator (see Lehmann (1975)) of the centre of symmetry , together with an associated confidence interval. The Hodges–Lehmann estimate is defined as

Let and let , for denote the ordered averages for . Then

if is odd, where ;

if is even, where .

This estimator arises from inverting the one-sample Wilcoxon signed-rank test statistic, , for testing the hypothesis that . Effectively is a monotonically decreasing step function of with

The estimate is the solution to the equation ; two methods are available for solving this equation. These methods avoid the computation of all the ordered averages ; this is because for large both the storage requirements and the computation time would be excessive.

The first is an exact method based on a set partitioning procedure on the set of all ordered averages for . This is based on the algorithm proposed by Monahan (1984).

The second is an iterative algorithm, based on the Illinois method which is a modification of the regula falsi method, see McKean and Ryan (1977). This algorithm has proved suitable for the function which is asymptotically linear as a function of .

The confidence interval limits are also based on the inversion of the Wilcoxon test statistic.

Given a desired percentage for the confidence interval, , expressed as a proportion between and , initial estimates for the lower and upper confidence limits of the Wilcoxon statistic are found from


where is the inverse cumulative Normal distribution function.

and are rounded to the nearest integer values. These estimates are then refined using an exact method if , and a Normal approximation otherwise, to find and satisfying


Let ; then . This is the largest value such that .

Let ; then . This is the smallest value such that .

As in the case of , these equations may be solved using either the exact or the iterative methods to find the values and .

Then is the confidence interval for . The confidence interval is thus defined by those values of such that the null hypothesis, , is not rejected by the Wilcoxon signed-rank test at the level.


Lehmann, E L, 1975, Nonparametrics: Statistical Methods Based on Ranks, Holden–Day

Marazzi, A, 1987, Subroutines for robust estimation of location and scale in ROBETH, Cah. Rech. Doc. IUMSP, No. 3 ROB 1, Institut Universitaire de Médecine Sociale et Préventive, Lausanne

McKean, J W and Ryan, T A, 1977, Algorithm 516: An algorithm for obtaining confidence intervals and point estimates based on ranks in the two-sample location problem, ACM Trans. Math. Software (10), 183–185

Monahan, J F, 1984, Algorithm 616: Fast computation of the Hodges–Lehman location estimator, ACM Trans. Math. Software (10), 265–270