The Sign Test for Interval-Valued Data

The sign test and the signed‐rank test for interval‐valued data

“…The projection-based methodology also appears useful for adopting the signed-rank test for random intervals. In particular, to verify hypothesis (26) that the population median of a sample of random intervals X X , …, n 1 equals M M M = [mid ± spr ] 0 0 0 versus the one-sided or two-sided alternative, we may transform the interval-valued observations by operator (29) as it is shown in Section 5.2 and then apply the following test statistic…”

“…One way to escape this problem is to rewrite the null hypothesis as a conjunction of two sub‐hypotheses: one for the mid‐point and the second for spreads, and then apply two classical sign tests for each sub‐hypothesis separately. By this strategy, we obtain two separate P ‐values that should be somehow combined to obtain a final unique P ‐value (a short discussion can be found, eg, in Grzegorzewski and Śpiewak). However, both conceptual and technical problems connected with P ‐value aggregation are not insuperable.…”

“…Indeed, testing assumptions on distributions may cause difficulties even in the case of crisp data, and known goodness‐of‐fit techniques for interval‐valued data are not very powerful. This is also the reason that nonparametric tests seem to be attractive to the authors who generalize classical statistical tools for interval‐valued data …”

“…3,4 Interval-valued data that appear in different contexts drew the attention of many researchers. Various problems concerning interval-valued data in regression analysis, [5][6][7][8][9] time series, 10 principal component analysis, 11,12 correlation analysis, 13,14 classification, 9,[15][16][17][18][19][20][21] clustering, 22,23 analysis of variance, 24 and hypothesis testing [25][26][27][28][29][30][31][32][33] have been deeply studied in the literature.…”

“…This is also the reason that nonparametric tests seem to be attractive to the authors who generalize classical statistical tools for interval-valued data. [29][30][31] Of course, one may apply some other tools for modeling uncertainty in data analysis and statistical inference, like fuzzy sets (a generalization of some nonparametric tests for fuzzy data was proposed by Denoeux et al 34 or Grzegorzewski 35,36 ). However, it seems that interval-valued data constitute actually a noteworthy construction which allows grasping uncertainty in a sound mathematical model which is flexible enough and effective at the same time.…”