NAG Library Routine Document

G07EAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

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
REAL (KIND=nag_wp)	X(N), CLEVEL, THETA, THETAL, THETAU, ESTCL, WLOWER, WUPPER, WRK(4*N)
CHARACTER(1)	METHOD

3 Description

Consider a vector of independent observations,

x = {(x_{1}, x_{2}, \dots, x_{n})}^{T}

with unknown common symmetric density

f (x_{i} - 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

\hat{θ} = median \{\frac{x_{i} + x_{j}}{2}, 1 \leq i \leq j \leq n\} .

Let

m = (n (n + 1)) / 2

and let

a_{k}

, for

k = 1, 2, \dots, m

denote the

m

ordered averages

(x_{i} + x_{j}) / 2

for

1 \leq i \leq j \leq n

. Then

if $m$ is odd, $\hat{θ} = a_{k}$ where $k = (m + 1) / 2$ ;
if $m$ is even, $\hat{θ} = (a_{k} + a_{k + 1}) / 2$ where $k = m / 2$ .

This estimator arises from inverting the one-sample Wilcoxon signed-rank test statistic,

W (x - θ_{0})

, for testing the hypothesis that

θ = θ_{0}

. Effectively

W (x - θ_{0})

is a monotonically decreasing step function of

θ_{0}

with

\begin{matrix} mean ​ (W) = μ = \frac{n (n + 1)}{4}, \\ var (W) = σ^{2} = \frac{n (n + 1) (2 n + 1)}{24} . \end{matrix}

The estimate

\hat{θ}

is the solution to the equation

W (x - \hat{θ}) = μ

; two methods are available for solving this equation. These methods avoid the computation of all the ordered averages

a_{k}

; 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

(x_{i} + x_{j}) / 2

for

i \leq 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

W (x - θ_{0})

which is asymptotically linear as a function of

θ_{0}

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

W_{l} = μ - 0.5 + (σ Φ^{- 1} (α / 2))

and

W_{u} = μ + 0.5 + (σ Φ^{- 1} (1 - α / 2)),

where

Φ^{- 1}

is the inverse cumulative Normal distribution function.

W_{l}

and

W_{u}

are rounded to the nearest integer values. These estimates are then refined using an exact method if

n \leq 80

, and a Normal approximation otherwise, to find

W_{l}

and

W_{u}

satisfying

\begin{array}{l} P (W \leq W_{l}) \leq α / 2 \\ P (W \leq W_{l} + 1) > α / 2 \end{array}

and

\begin{array}{l} P (W \geq W_{u}) \leq α / 2 \\ P (W \geq W_{u} - 1) > α / 2 . \end{array}

Let

W_{u} = m - k

; then

θ_{l} = a_{k + 1}

. This is the largest value

θ_{l}

such that

W (x - θ_{l}) = W_{u}

Let

W_{l} = k

; then

θ_{u} = a_{m - k}

. This is the smallest value

θ_{u}

such that

W (x - θ_{u}) = W_{l}

As in the case of

\hat{θ}

, these equations may be solved using either the exact or the iterative methods to find the values

θ_{l}

and

θ_{u}

Then

(θ_{l}, θ_{u})

is the confidence interval for

θ

. The confidence interval is thus defined by those values of

θ_{0}

such that the null hypothesis,

θ = θ_{0}

, is not rejected by the Wilcoxon signed-rank test at the

(100 \times 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(1)Input

On entry: specifies the method to be used.

$METHOD ='E'$: The exact algorithm is used.
$METHOD ='A'$: The iterative algorithm is used.

Constraint:

METHOD ='E'

'A'

2: N – INTEGERInput

On entry:

n

, the sample size.

Constraint:

N \geq 2

3: X(N) – REAL (KIND=nag_wp) arrayInput

On entry: the sample observations,

x_{i}

, for

i = 1, 2, \dots, n

4: CLEVEL – REAL (KIND=nag_wp)Input

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 – REAL (KIND=nag_wp)Output

On exit: the estimate of the location,

\hat{θ}

6: THETAL – REAL (KIND=nag_wp)Output

On exit: the estimate of the lower limit of the confidence interval,

θ_{l}

7: THETAU – REAL (KIND=nag_wp)Output

On exit: the estimate of the upper limit of the confidence interval,

θ_{u}

8: ESTCL – REAL (KIND=nag_wp)Output

On exit: an estimate of the actual percentage confidence of the interval found, as a proportion between

(0.0, 1.0)

9: WLOWER – REAL (KIND=nag_wp)Output

On exit: the upper value of the Wilcoxon test statistic,

W_{u}

, corresponding to the lower limit of the confidence interval.

10: WUPPER – REAL (KIND=nag_wp)Output

On exit: the lower value of the Wilcoxon test statistic,

W_{l}

, corresponding to the upper limit of the confidence interval.

11: WRK( $4 \times N$ ) – REAL (KIND=nag_wp) arrayWorkspace

12: IWRK( $3 \times N$ ) – 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.

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.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 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,	$METHOD \neq'E'$ or $'A'$ ,
or	$N < 2$ ,
or	$CLEVEL \leq 0.0$ ,
or	$CLEVEL \geq 1.0$ .

$IFAIL = 2$: 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 $\hat{θ}$ , $θ_{l}$ and $θ_{u}$ , 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 \times (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.

NAG Library Routine DocumentG07EAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G07EAF