naginterfaces.library.univar.robust_​1var_​ci

naginterfaces.library.univar.robust_1var_ci(method, x, clevel)

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

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g07/g07eaf.html

Parameters

methodstr, length 1

Specifies the method to be used.

$method=\text{'E'}$

The exact algorithm is used.

$method=\text{'A'}$

The iterative algorithm is used.

xfloat, array-like, shape $\left(n\right)$

The sample observations, $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

clevelfloat

The confidence interval desired.

For example, for a $95\%$ confidence interval set $clevel=0.95$.

Returns

thetafloat

The estimate of the location, $\hat{\theta }$.

thetalfloat

The estimate of the lower limit of the confidence interval, ${\theta }_l$.

thetaufloat

The estimate of the upper limit of the confidence interval, ${\theta }_u$.

estclfloat

An estimate of the actual percentage confidence of the interval found, as a proportion between $(0.0,1.0)$.

wlowerfloat

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

wupperfloat

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

Raises

NagValueError

(errno $1$)

On entry, $clevel=⟨value⟩$.

Constraint: $0.0<clevel<1.0$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 2$.

(errno $1$)

On entry, $method=⟨value⟩$.

Constraint: $method=\text{'E'}$ or $\text{'A'}$.

(errno $2$)

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

(errno $3$)

The iterative procedure used to estimate $\theta$ has not converged.

(errno $3$)

The iterative procedure used to estimate, ${\theta }_u$, the upper confidence limit has not converged.

(errno $3$)

The iterative procedure used to estimate, ${\theta }_l$, the lower confidence limit has not converged.

Notes

Consider a vector of independent observations, $x={(x_1,x_2,\dots ,x_n)}^T$ with unknown common symmetric density $f(x_i-\theta )$. robust_1var_ci 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

$$\hat{\theta }=median\{\frac{x_i+x_j}{2},1\le i\le j\le n\}\text{.}$$

Let $m=\left(n(n+1)\right)/2$ and let $a_{\text{k}}$, for $\text{k}=1,2,\dots ,m$ 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

$$\begin{array}{c}\begin{array}{c}\text{mean}\left(W\right)=\mu =\frac{n(n+1)}{4}\text{,}\\ \\ \\ var\left(W\right)={\sigma }^2=\frac{n(n+1)(2n+1)}{24}\text{.}\end{array}\end{array}$$

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

$$W_l=\mu -0.5+\left(\sigma {\Phi }^{-1}\left(\alpha /2\right)\right)$$

and

$$W_u=\mu +0.5+\left(\sigma {\Phi }^{-1}(1-\alpha /2)\right)\text{,}$$

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

$$\begin{array}{c}\begin{array}{c}P(W\le W_l)\le \alpha /2\\ P(W\le W_l+1)>\alpha /2\end{array}\end{array}$$

and

$$\begin{array}{c}\begin{array}{c}P(W\ge W_u)\le \alpha /2\\ P(W\ge W_u-1)>\alpha /2\text{.}\end{array}\end{array}$$

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.

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