NAG Library Routine Document
G07EAF
1 Purpose
G07EAF computes a rank based (nonparametric) estimate and confidence interval for the location parameter of a single population.
2 Specification
| SUBROUTINE G07EAF ( | METHOD, N, X, CLEVEL, THETA, THETAL, THETAU, ESTCL, WLOWER, WUPPER, WRK, IWRK, IFAIL) |
| INTEGER | N, IWRK(3*N), IFAIL |
| double precision | X(N), CLEVEL, THETA, THETAL, THETAU, ESTCL, WLOWER, WUPPER, WRK(4*N) |
| CHARACTER*1 | METHOD |
3 Description
Consider a vector of independent observations,
x=x1,x2,…,xnT with unknown common symmetric density
fxi-θ. G07EAF computes the Hodges–Lehmann location estimator (see
Lehmann (1975)) of the centre of symmetry
θ, together with an associated confidence interval. The Hodges–Lehmann estimate is defined as
Let
m=nn+1/2 and let
ak, for
k=1,2,…,m denote the
m ordered averages
xi+xj/2 for
1≤i≤j≤n. Then
- if m is odd, θ^=ak where k=m+1/2;
- if m is even, θ^=ak+ak+1/2 where k=m/2.
This estimator arises from inverting the one-sample Wilcoxon signed-rank test statistic,
Wx-θ0, for testing the hypothesis that
θ=θ0. Effectively
Wx-θ0 is a monotonically decreasing step function of
θ0 with
The estimate
θ^ is the solution to the equation
Wx-θ^=μ; two methods are available for solving this equation. These methods avoid the computation of all the ordered averages
ak; this is because for large
n 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
xi+xj/2 for
i≤j. 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
Wx-θ0 which is asymptotically linear as a function of
θ0.
The confidence interval limits are also based on the inversion of the Wilcoxon test statistic.
Given a desired percentage for the confidence interval,
1-α, expressed as a proportion between
0 and
1, initial estimates for the lower and upper confidence limits of the Wilcoxon statistic are found from
and
where
Φ-1 is the inverse cumulative Normal distribution function.
Wl and
Wu are rounded to the nearest integer values. These estimates are then refined using an exact method if
n≤80, and a Normal approximation otherwise, to find
Wl and
Wu satisfying
and
Let
Wu=m-k; then
θl=ak+1. This is the largest value
θl such that
Wx-θl=Wu.
Let Wl=k; then θu=am-k. This is the smallest value θu such that Wx-θu=Wl.
As in the case of θ^, these equations may be solved using either the exact or the iterative methods to find the values θl and θu.
Then θl,θu is the confidence interval for θ. The confidence interval is thus defined by those values of θ0 such that the null hypothesis, θ=θ0, is not rejected by the Wilcoxon signed-rank test at the 100×α% level.
4 References
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
5 Parameters
- 1: METHOD – CHARACTER*1Input
On entry: specifies the method to be used.
- If METHOD='E', the exact algorithm is used.
- If METHOD='A', the iterative algorithm is used.
Constraint:
METHOD='E' or 'A'.
- 2: N – INTEGERInput
On entry:
n, the sample size.
Constraint:
N≥2.
- 3: X(N) – double precision arrayInput
On entry: the sample observations, xi for i=1,2,…,n.
- 4: CLEVEL – double precisionInput
On entry: the confidence interval desired.
For example, for a 95% confidence interval set CLEVEL=0.95.
Constraint:
0.0<CLEVEL<1.0.
- 5: THETA – double precisionOutput
On exit: the estimate of the location, θ^.
- 6: THETAL – double precisionOutput
-
On exit: the estimate of the lower limit of the confidence interval, θl.
- 7: THETAU – double precisionOutput
-
On exit: the estimate of the upper limit of the confidence interval, θu.
- 8: ESTCL – double precisionOutput
-
On exit: an estimate of the actual percentage confidence of the interval found, as a proportion between 0.0,1.0.
- 9: WLOWER – double precisionOutput
On exit: the upper value of the Wilcoxon test statistic, Wu, corresponding to the lower limit of the confidence interval.
- 10: WUPPER – double precisionOutput
On exit: the lower value of the Wilcoxon test statistic, Wl, corresponding to the upper limit of the confidence interval.
- 11: WRK(4×N) – double precision arrayWorkspace
- 12: IWRK(3×N) – INTEGER arrayWorkspace
- 13: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
0,
-1 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
On exit:
IFAIL=0 unless the routine detects an error (see
Section 6).
For environments where it might be inappropriate to halt program execution when an error is detected, the value
-1 or 1 is recommended. If the output of error messages is undesirable, then the value
1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
0.
When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.
6 Error Indicators and Warnings
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
- IFAIL=1
-
| On entry, | METHOD≠'E' or 'A', |
| or | N<2, |
| or | CLEVEL≤0.0, |
| or | CLEVEL≥1.0. |
- IFAIL=2
-
There is not enough information to compute a confidence interval since the whole sample consists of identical values.
- IFAIL=3
-
For at least one of the estimates θ^, θl and θu, the underlying iterative algorithm (when METHOD='A') failed to converge. This is an unlikely exit but the estimate should still be a reasonable approximation.
7 Accuracy
G07EAF should produce results accurate to five significant figures in the width of the confidence interval; that is the error for any one of the three estimates should be less than 0.00001×THETAU-THETAL.
8 Further Comments
The time taken increases with the sample size n.
9 Example
The following program calculates a 95% confidence interval for θ, a measure of symmetry of the sample of 50 observations.
9.1 Program Text
Program Text (g07eafe.f)
9.2 Program Data
Program Data (g07eafe.d)
9.3 Program Results
Program Results (g07eafe.r)