bosfr (version 0.1.0)

boundsKendall: Bounds of Kendall's tau in the Presence of Missing Data

Description

Computes bounds of Kendall's tau in the presence of missing data. Suitable only for univariate distinct data where no ties is allowed.

Usage

boundsKendall(X, Y)

Value

bounds

bounds of Kendall's tau.

bounds.scaled

bounds of scaled Kendall's tau.

Arguments

X, Y

Numeric vectors of data values with potential missing data. No ties in the data is allowed. Inf and -Inf values will be omitted.

Details

boundsKendall() computes bounds of Kendall's tau for partially observed univariate, distinct data. The bounds are computed by first calculating the bounds of Spearman's footrule (Zeng et al., 2025), and then applying the combinatorial inequality between Kendall's tau and Spearman's footrule (Kendall, 1948). See Zeng et al., 2025 for more details.

Let \(X = (x_1, \ldots, x_n)\) and \(Y = (y_1, \ldots, y_n)\) be two vectors of univariate, distinct data. Kendall's tau is defined as the number of discordant pairs between \(X\) and \(Y\): $$\tau(X,Y) = \sum\limits_{i < j} \{I(x_i < x_j)I(y_i > y_j) + I(x_i > x_j)I(y_i < y_j)\}.$$ Scaled Kendall's tau \(\tau_{Scale}(X,Y) \in [0,1]\) is defined as (Kendall, 1948): $$\tau_{Scale}(X,Y) = 1 - 4\tau(X,Y)/(n(n-1)).$$

References

• Zeng Y., Adams N.M., Bodenham D.A. Exact Bounds of Spearman's footrule in the Presence of Missing Data with Applications to Independence Testing. arXiv preprint arXiv:2501.11696. 2025 Jan 20.

• Kendall, M.G. (1948) Rank Correlation Methods. Charles Griffin, London.

• Diaconis, P. and Graham, R.L., 1977. Spearman's footrule as a measure of disarray. Journal of the Royal Statistical Society Series B: Statistical Methodology, 39(2), pp.262-268.

Examples

### compute bounds of Kendall's tau between incomplete ranked lists
X <- c(1, 2, NA, 4, 3)
Y <- c(3, NA, 4, 2, 1)
boundsKendall(X, Y)

### compute bounds of Kendall's tau between incomplete vectors of distinct data
X <- c(1.3, 2.6, NA, 4.2, 3.5)
Y <- c(5.5, NA, 6.5, 2.6, 1.1)
boundsKendall(X, Y)