The function may be called by the names: g07eac, nag_univar_robust_1var_ci or nag_rank_ci_1var.
3Description
Consider a vector of independent observations, $x={({x}_{1},{x}_{2},\dots ,{x}_{n})}^{\mathrm{T}}$ with unknown common symmetric density $f({x}_{i}-\theta )$. g07eac computes the Hodges–Lehmann location estimator (see Lehmann (1975)) of the centre of symmetry $\theta $, together with an associated confidence interval. The Hodges–Lehmann estimate is defined as
Let $m=\left(n(n+1)\right)/2$ and let ${a}_{\mathit{k}}$, for $\mathit{k}=1,2,\dots ,m$ denote the $m$ ordered averages $({x}_{i}+{x}_{j})/2$ for $1\le i\le j\le n$. Then
if $m$ is odd, $\hat{\theta}={a}_{k}$ where $k=(m+1)/2$;
if $m$ is even, $\hat{\theta}=({a}_{k}+{a}_{k+1})/2$ where $k=m/2$.
This estimator arises from inverting the one-sample Wilcoxon signed-rank test statistic, $W(x-{\theta}_{0})$, for testing the hypothesis that $\theta ={\theta}_{0}$. Effectively $W(x-{\theta}_{0})$ is a monotonically decreasing step function of ${\theta}_{0}$ with
The estimate $\hat{\theta}$ is the solution to the equation $W(x-\hat{\theta})=\mu $; 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 $i\le j$. 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 $W(x-{\theta}_{0})$ which is asymptotically linear as a function of ${\theta}_{0}$.
The confidence interval limits are also based on the inversion of the Wilcoxon test statistic.
Given a desired percentage for the confidence interval, $1-\alpha $, expressed as a proportion between $0$ and $1$, initial estimates for the lower and upper confidence limits of the Wilcoxon statistic are found from
where ${\Phi}^{-1}$ 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 $n\le 80$, and a Normal approximation otherwise, to find ${W}_{l}$ and ${W}_{u}$ satisfying
Let ${W}_{u}=m-k$; then ${\theta}_{l}={a}_{k+1}$. This is the largest value ${\theta}_{l}$ such that $W(x-{\theta}_{l})={W}_{u}$.
Let ${W}_{l}=k$; then ${\theta}_{u}={a}_{m-k}$. This is the smallest value ${\theta}_{u}$ such that $W(x-{\theta}_{u})={W}_{l}$.
As in the case of $\hat{\theta}$, these equations may be solved using either the exact or the iterative methods to find the values ${\theta}_{l}$ and ${\theta}_{u}$.
Then $({\theta}_{l},{\theta}_{u})$ is the confidence interval for $\theta $. The confidence interval is thus defined by those values of ${\theta}_{0}$ such that the null hypothesis, $\theta ={\theta}_{0}$, is not rejected by the Wilcoxon signed-rank test at the $(100\times \alpha )\%$ level.
4References
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. Software10 183–185
Monahan J F (1984) Algorithm 616: Fast computation of the Hodges–Lehman location estimator ACM Trans. Math. Software10 265–270
5Arguments
1: $\mathbf{method}$ – Nag_RCIMethodInput
On entry: specifies the method to be used.
${\mathbf{method}}=\mathrm{Nag\_RCI\_Exact}$
The exact algorithm is used.
${\mathbf{method}}=\mathrm{Nag\_RCI\_Approx}$
The iterative algorithm is used.
Constraint:
${\mathbf{method}}=\mathrm{Nag\_RCI\_Exact}$ or $\mathrm{Nag\_RCI\_Approx}$.
On entry: the sample observations,
${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
4: $\mathbf{clevel}$ – doubleInput
On entry: the confidence interval desired.
For example, for a $95\%$ confidence interval set ${\mathbf{clevel}}=0.95$.
Constraint:
$0.0<{\mathbf{clevel}}<1.0$.
5: $\mathbf{theta}$ – double *Output
On exit: the estimate of the location, $\hat{\theta}$.
6: $\mathbf{thetal}$ – double *Output
On exit: the estimate of the lower limit of the confidence interval, ${\theta}_{l}$.
7: $\mathbf{thetau}$ – double *Output
On exit: the estimate of the upper limit of the confidence interval, ${\theta}_{u}$.
8: $\mathbf{estcl}$ – double *Output
On exit: an estimate of the actual percentage confidence of the interval found, as a proportion between $(0.0,1.0)$.
9: $\mathbf{wlower}$ – double *Output
On exit: the upper value of the Wilcoxon test statistic, ${W}_{u}$, corresponding to the lower limit of the confidence interval.
10: $\mathbf{wupper}$ – double *Output
On exit: the lower value of the Wilcoxon test statistic, ${W}_{l}$, corresponding to the upper limit of the confidence interval.
11: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_CONVERGENCE
The iterative procedure used to estimate $\theta $ has not converged. This is an unlikely exit but the estimate should still be a reasonable approximation.
The iterative procedure used to estimate, ${\theta}_{l}$, the lower confidence limit has not converged. This is an unlikely exit but the estimate should still be a reasonable approximation.
The iterative procedure used to estimate, ${\theta}_{u}$, the upper confidence limit has not converged. This is an unlikely exit but the estimate should still be a reasonable approximation.
NE_INT
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 2$.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL
On entry, ${\mathbf{clevel}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: $0.0<{\mathbf{clevel}}<1.0$.
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.
7Accuracy
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 $0.00001\times ({\mathbf{thetau}}-{\mathbf{thetal}})$.
8Parallelism and Performance
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.
9Further Comments
The time taken increases with the sample size $n$.
10Example
The following program calculates a 95% confidence interval for $\theta $, a measure of symmetry of the sample of $50$ observations.