Pairwise Comparisons in the Form of Difference of Ranks - Testing of Estimates of the Preference Relation

Abstract

The paper presents tests for verification of estimates of the preference relation, obtained of the basis of multiple independent pairwise comparisons, in the form of difference of ranks, with random errors. The relation can have strict or weak form, while an estimate is obtained with the use of the idea of the nearest adjoining order (NAO). The approach to verification is the original concept of the author – it develops the ideas applied to binary comparisons. Some of the proposed tests are non-parametric, i.e. do not require any parameters of distribution of comparisons errors; remaining tests are based on exact or limiting distributions. Estimates verified with the use of the proposed tests are highly reliable and require acceptable computational costs.

Appendix

The test for a mode equal zero in multinomial distribution (see Domański 1990).

The values $z_{i} , \ldots ,z_{\nu } \left( {\nu = 2\mu + 1} \right)$ are independent sample from multinomial distribution with the set of values (integers) $\left\{ { - \mu , - \mu + 1, \ldots ,0,\mu - 1,\mu } \right\} \left( {\mu \ge 1} \right)$; any value $z_{i} \left( {i = 1, \ldots ,\nu } \right)$ expresses number $k_{i}$ of realizations of i-th element of the set of values. The test for verification of the null hypothesis stating that the probability of the mode of the distribution (zero) is equal $p_{0}$, i.e. H₀:  $p_{m} = p_{0}$, under alternative H₁:  $p_{m} < p_{0}$  (or $p_{m} \ne p_{0}$), is based on the statistics (Domański 1990, point 3.5.2):

$$ U = \left| {\frac{{k_{m} }}{\nu } - p_{0} } \right|/\sqrt {\frac{{k_{m} }}{\nu }\left( {1 - \frac{{k_{m} }}{\nu }} \right)/\nu } , $$

(A1)

where: $k_{m} = max\left\{ {z_{i} , \ldots ,z_{\nu } } \right\}$.

The probability $p_{0}$ has to be determined on the basis of parameters of the distribution under consideration.

The statistics U can be applied for $0,2 \le \frac{{k_{m} }}{\nu } \le 0,8$ and has (under H₀) limiting Gaussian standard distribution. For remaining values of ratio $\frac{{k_{m} }}{\nu }$ the statistics U assumes the form:

$$ U = \left| {y - y_{0} } \right|/\nu^{ - 0,5} , $$

where:

$$ y = 2{ \arcsin }(\frac{{k_{m} }}{\nu })^{0,5} ,\quad y = 2{ \arcsin }(p_{0} )^{0,5} . $$