void	nag_rank_ci_1var (Nag_RCIMethod method, Integer n, const double x[], double clevel, double theta, double thetal, double thetau, double estcl, double wlower, double wupper, NagError *fail)

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)

. nag_rank_ci_1var (g07eac) 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 Arguments

1: $method$ – Nag_RCIMethodInput

On entry: specifies the method to be used.

$method = Nag_RCI_Exact$: The exact algorithm is used.
$method = Nag_RCI_Approx$: The iterative algorithm is used.

Constraint:

method = Nag_RCI_Exact

Nag_RCI_Approx

2: $n$ – IntegerInput

On entry:

n

, the sample size.

Constraint:

n \geq 2

3: $x [n]$ – const doubleInput

On entry: the sample observations,

x_{i}

, for

i = 1, 2, \dots, n

4: $clevel$ – doubleInput

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 *Output

On exit: the estimate of the location,

\hat{θ}

6: $thetal$ – double *Output

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

θ_{l}

7: $thetau$ – double *Output

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

θ_{u}

8: $estcl$ – double *Output

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

(0.0, 1.0)

9: $wlower$ – double *Output

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

W_{u}

, corresponding to the lower limit of the confidence interval.

10: $wupper$ – double *Output

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

W_{l}

, corresponding to the upper limit of the confidence interval.

11: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_CONVERGENCE: Warning. The iterative procedure to find an estimate of the lower confidence point had not converged in $100$ iterations.

Warning. The iterative procedure to find an estimate of Theta had not converged in $100$ iterations.

Warning. The iterative procedure to find an estimate of the upper confidence point had not converged in $100$ iterations.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 2$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL: On entry, clevel is out of range: $clevel = 〈value〉$ .
NE_SAMPLE_IDEN: Not enough information to compute an interval estimate since the whole sample is identical. The common value is returned in theta, thetal and thetau.

7 Accuracy

nag_rank_ci_1var (g07eac) 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 Parallelism and Performance

nag_rank_ci_1var (g07eac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken increases with the sample size

n

10 Example

The following program calculates a 95% confidence interval for

θ

, a measure of symmetry of the sample of

50

observations.

NAG Library Function Documentnag_rank_ci_1var (g07eac)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_rank_ci_1var (g07eac)