Although there are many methods to test of homogeneity under umbrella ordering alternatives, most of these tests are aimed at the case of multinomial distributions or parametric families of continuous distributions. In this paper, we focus on testing homogeneity for umbrella ordering alternatives under nonparametric continuous distributions, considering both scenarios in which the peak of the umbrella ordering is known and unknown. The test statistics are constructed using isotonic estimates of the distribution functions, and the asymptotic distributions of the test statistics are derived. Critical values are obtained via a bootstrap procedure, and simulation results are provided to illustrate the performance of the proposed testing method.
Citation: Jianling Zhang. A non-parametric test of homogeneity for umbrella ordering alternative based on an isotonic estimators[J]. AIMS Mathematics, 2026, 11(2): 3425-3440. doi: 10.3934/math.2026139
Although there are many methods to test of homogeneity under umbrella ordering alternatives, most of these tests are aimed at the case of multinomial distributions or parametric families of continuous distributions. In this paper, we focus on testing homogeneity for umbrella ordering alternatives under nonparametric continuous distributions, considering both scenarios in which the peak of the umbrella ordering is known and unknown. The test statistics are constructed using isotonic estimates of the distribution functions, and the asymptotic distributions of the test statistics are derived. Critical values are obtained via a bootstrap procedure, and simulation results are provided to illustrate the performance of the proposed testing method.
| [1] |
E. L. Lehmann, Order families of distributions, Ann. Math. Statist., 26 (1955), 399–419. https://doi.org/10.1214/aoms/1177728487 doi: 10.1214/aoms/1177728487
|
| [2] |
M. L. Puri, Some distribution-free k-Sample rank tests of homogeneity against ordered alternatives, Commun. Pure Appl. Math., 18 (1965), 51–63. https://doi.org/10.1002/cpa.3160180108 doi: 10.1002/cpa.3160180108
|
| [3] |
P. V. Tryon, T. P. Hettmansperger, A class of non-parametric tests for homogeneity against ordered alternative, Ann. Statist., 1 (1973), 1061–1070. https://doi.org/10.1214/aos/1176342557 doi: 10.1214/aos/1176342557
|
| [4] |
G. A. Mack, D. A. Wolfe, $k$-sample rank tests for umbrella alternatives, J. Amer. Stat. Assoc., 76 (1981), 175–181. https://doi.org/10.1080/01621459.1981.10477625 doi: 10.1080/01621459.1981.10477625
|
| [5] |
T. P. Hettmansperger, R. M. Norton, Tests for patterned alternatives in k-sample problems, J. Amer. Stat. Assoc., 82 (1987), 292–299. https://doi.org/10.1080/01621459.1987.10478432 doi: 10.1080/01621459.1987.10478432
|
| [6] |
H. Büning, Adaptive Jonckheere-type tests for ordered alternatives, J. Appl. Stat., 26 (1999), 541–551. https://doi.org/10.1080/02664769922214 doi: 10.1080/02664769922214
|
| [7] |
W. Kössler, H. Büning, The asymptotic power and relative efficiency of some c-sample rank tests of homogeneity against umbrella alternatives, Statistics, 34 (2000), 1–26. https://doi.org/10.1080/02331880008802703 doi: 10.1080/02331880008802703
|
| [8] |
Y. I. Chen, D. A. Wolfe, Nonparametric procedures for comparing umbrella pattern treatment effects with a control in a one-way layout, Biometrics, 49 (1993), 455–465. https://doi.org/10.2307/2532558 doi: 10.2307/2532558
|
| [9] |
N.-Z. Shi, A test of homogeneity for umbrella alternatives and tables of the level probabilities, Commun. Stat.-Theor. Method., 17 (1988), 657–670. https://doi.org/10.1080/03610928808829647 doi: 10.1080/03610928808829647
|
| [10] |
Y. I. Chen, D. A. Wolfe, A study of distribution-free tests for umbrella alternatives, Biometrical J., 32 (1990), 47–57. https://doi.org/10.1002/bimj.4710320109 doi: 10.1002/bimj.4710320109
|
| [11] |
Y. I. Chen, Notes on the Mack-Wolfe and Chen-Wolfe Tests for Umbrella Alternatives, Biometrical J., 33 (1991), 281–290. https://doi.org/10.1002/bimj.4710330305 doi: 10.1002/bimj.4710330305
|
| [12] |
R. Magel, L. Qin, A non-parametric test for umbrella alternatives based on ranked-set sampling, J. Appl. Stat., 30 (2003), 925–937. https://doi.org/10.1080/0266476032000075994 doi: 10.1080/0266476032000075994
|
| [13] |
W. Kössler, Some c-sample rank tests of homogeneity against umbrella alternatives with unknown peak, J. Stat. Comput. Sim., 76 (2006), 57–74. https://doi.org/10.1080/00949650412331320882 doi: 10.1080/00949650412331320882
|
| [14] |
M. Alvo, Nonparametric tests of hypotheses for umbrella alternatives, Can. J. Stat., 36 (2008), 143–156. https://doi.org/10.1002/cjs.5550360113 doi: 10.1002/cjs.5550360113
|
| [15] |
C.-H. Chang, C.-H. Yen, A new nonparametric test for the umbrella alternative problem, Commun. Stat.-Simulat. Comput., 41 (2012), 1478–1488. https://doi.org/10.1080/03610918.2011.606946 doi: 10.1080/03610918.2011.606946
|
| [16] |
E. Gökpinar, F. Gökpinar, A modified Mack–Wolfe test for the umbrella alternative problem, Commun. Stat.-Theor. Method., 45 (2016), 7226–7241. https://doi.org/10.1080/03610926.2014.975821 doi: 10.1080/03610926.2014.975821
|
| [17] |
P. Singh, W. Liu, A test against an umbrella ordered alternative, Comput. Stat. Data Anal., 51 (2006), 1957–1964. https://doi.org/10.1016/j.csda.2005.12.010 doi: 10.1016/j.csda.2005.12.010
|
| [18] |
M. Carpenter, P. Singh, On tests based on sample quasi ranges for ordered alternative (scale case of exponential distribution), Commun. Stat.-Simulat. Comput., 38 (2009), 846–855. https://doi.org/10.1080/03610910802664215 doi: 10.1080/03610910802664215
|
| [19] |
A. Gaur, K. K. Mahajan, S. Arora, New nonparametric tests for testing homogeneity of scale parameters against umbrella alternative, Stat. Probabil. Lett., 82 (2012), 1681–1689. https://doi.org/10.1016/j.spl.2012.05.009 doi: 10.1016/j.spl.2012.05.009
|
| [20] | A. Gaur, A nonparametric test for umbrella alternate scale problem using ranked-set data, International Journal of Probability and Statistics, 1 (2012), 95–100. |
| [21] |
A. Gaur, Test of homogeneity against umbrella scale alternative based on sample spacings, J. Stat. Manag. Syst., 18 (2015), 1–31. https://doi.org/10.1080/09720510.2013.842058 doi: 10.1080/09720510.2013.842058
|
| [22] | N.-Z. Shi, Testing for umbrella order restrictions on multinomial parameters, Sankhya, 51 (1989), 13–23. https://www.jstor.org/stable/25052567 |
| [23] |
D. Basso, L. Salmaso, A permutation test for umbrella alternatives, Stat. Comput., 21 (2011), 45–54. https://doi.org/10.1007/s11222-009-9145-8 doi: 10.1007/s11222-009-9145-8
|
| [24] |
W. Xiong, J. Ding, A flexible nonparametric test for ordered and umbrella alternatives, Commun. Stat.-Simulat. Comput., 44 (2015), 1257–1263. https://doi.org/10.1080/03610918.2013.815774 doi: 10.1080/03610918.2013.815774
|
| [25] | R. E. Barlow, D. J. Bartholomew, J. M. Bremner, H. D. Brunk, Statistical inference under order restrictions, New York: John Wiley & Sons, 1972. |
| [26] |
H. El Barmi, H. Mukerjee, Inferences under a stochastic ordering constraint: the k-sample case, J. Amer. Stat. Assoc., 100 (2005), 252–261. https://doi.org/10.1198/016214504000000764 doi: 10.1198/016214504000000764
|
| [27] | T. Robertson, T. Wright, R. L. Dykstra, Order-restricted inferences, New York: John Wiley & Sons, 1988. |
| [28] | P. Billingsley, Convergence of probability measure, New York: John Wiley & Sons, 1999. https://doi.org/10.1002/9780470316962 |
| [29] |
L. Baringhaus, R. Grübel, Nonparametric two-sample tests for increasing convex order, Bernoulli, 15 (2009), 99–123. https://doi.org/10.3150/08-BEJ151 doi: 10.3150/08-BEJ151
|
| [30] | A. W. Vaart, J. A. Wellner, Weak convergence and empirical processes, New York: Springer, 1996. https://doi.org/10.1007/978-1-4757-2545-2 |
Figures(1) / Tables(5)
Jianling Zhang. A non-parametric test of homogeneity for umbrella ordering alternative based on an isotonic estimators[J]. AIMS Mathematics, 2026, 11(2): 3425-3440. doi: 10.3934/math.2026139
DownLoad: