tieStatistics¶

Compute tie statistics for a sample of observations.

Synopsis¶

tieStatistics (x)

Required Arguments¶

float x[] (Input): Array of length nObservations containing the observations. x must be ordered monotonically increasing with all missing values removed.

Return Value¶

Array of length 4 containing the tie statistics.

\[\begin{split}\begin{aligned} \mathit{ties}[0] =& \sum_{j=1}^\tau\left[t_j\left(t_j - 1\right)\right] / 2 \\ \mathit{ties}[1] =& \sum_{j=1}^\tau\left[t_j\left(t_j - 1\right)\left(t_j + 1\right)\right] / 12 \\ \mathit{ties}[2] =& \sum_{j=1}^\tau\left[t_j\left(t_j - 1\right)\left(2t_j + 5\right)\right] \\ \mathit{ties}[3] =& \sum_{j=1}^\tau\left[t_j\left(t_j - 1\right)\left(t_j - 2\right)\right] \\ \end{aligned}\end{split}\]

where \(t_j\) is the number of ties in the j-th group (rank) of ties, and \(\tau\) is the number of tie groups in the sample.

Optional Arguments¶

fuzz, float (Input)

Value used to determine ties. Observations i and j are tied if the successive differences

x[k + 1] – x[k] between observations i and j, inclusive, are all less than fuzz. fuzz must be nonnegative.

Default: fuzz = 0.0

Description¶

Function tieStatistics computes tie statistics for a monotonically increasing sample of observations. “Tie statistics” are statistics that may be used to correct a continuous distribution theory nonparametric test for tied observations in the data. Observations i and j are tied if the successive differences X(k + 1) - X(k), inclusive, are all less than fuzz. Note that if each of the monotonically increasing observations is equal to its predecessor plus a constant, if that constant is less than fuzz, then all observations are contained in one tie group. For example, if fuzz = 0.11, then the following observations are all in one tie group.

0.0, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 1.00

Example¶

We want to compute tie statistics for a sample of length 7.

from numpy import *
from pyimsl.stat.tieStatistics import tieStatistics
from pyimsl.stat.writeMatrix import writeMatrix

ties = []
fuzz = .001
x = array([1.0, 1.0001, 1.0002, 2., 3., 3., 4.])

ties = tieStatistics(x, fuzz=fuzz)

writeMatrix("TIES", ties, writeFormat="%5.2f")

Output¶

 
           TIES
    1      2      3      4
 4.00   2.50  84.00   6.00