The restrictiveness of the hazard rate order and the moments of the maximal coordinate of a random vector uniformly distributed on the probability \(n\)-simplex. (English) Zbl 1489.60032
Summary: Continuing the work of S. Fried [Stat. Probab. Lett. 170, Article ID 109012, 8 p. (2021; Zbl 1457.60036)] who defined the restrictiveness of stochastic orders and calculated it for the usual stochastic order and for the likelihood ratio order, we calculate the restrictiveness of the hazard rate order. Inspired by the works of S. Onn and I. Weissman [Ann. Oper. Res. 189, 331–342 (2011; Zbl 1268.65005)] and I. Weissman [“Testing for serial correlation by means of extreme values”, RT&A 6, No. 4(23), 79–93 (2011)], we propose an application of the restrictiveness of stochastic orders in randomness testing. We then apply the inductive dimension reduction technique, that proved useful in obtaining the restrictiveness results, and provide an alternative proof for Whitworth’s formula, which we then use to derive the moments of the maximal coordinate of a random vector that is uniformly distributed on the probability \(n\)-simplex.
Keywords:
hazard rate order; probability simplex; Dilcher’s formula; Whitworth’s formula; randomness testingSoftware:
TestU01References:
| [1] | Andreescu, T.; Feng, Z., A Path to Combinatorics for Undergraduates: counting Strategies (2003), Springer Science & Business Media |
| [2] | Boland, P. J.; El-Neweihi, E.; Proschan, F., Applications of the hazard rate ordering in reliability and order statistics, J. Appl. Probab., 180-192 (1994) · Zbl 0793.62053 |
| [3] | Buchta, C., On the average number of maxima in a set of vectors, Inform. Process. Lett., 33, 2, 63-65 (1989) · Zbl 0682.68041 |
| [4] | Candelpergher, B.; Coppo, M.-A., A new class of identities involving Cauchy numbers, harmonic numbers and zeta values, Ramanujan J., 27, 3, 305-328 (2012) · Zbl 1276.11148 |
| [5] | Darling, D. A., On a class of problems related to the random division of an interval, Ann. Math. Stat., 239-253 (1953) · Zbl 0053.09902 |
| [6] | Devroye, L., Non-uniform random variate generation, Handbooks in Operations Research and Management Science, 13, 83-121 (2006) · Zbl 1170.90300 |
| [7] | Dilcher, K., Some q-series identities related to divisor functions, Discrete Math., 145, 1-3, 83-93 (1995) · Zbl 0834.05005 |
| [8] | Edwards, C. H., Advanced Calculus of Several Variables (2012), Courier Corporation · Zbl 0308.26002 |
| [9] | Fisher, R. A., Tests of significance in harmonic analysis, Proc. Royal Soc. Lond. Ser. A, 125, 796, 54-59 (1929) · JFM 55.0950.16 |
| [10] | Fried, S., On the restrictiveness of the usual stochastic order and the likelihood ratio order, Statist. Probab. Lett., 170, Article 109012 pp. (2021) · Zbl 1457.60036 |
| [11] | Garwood, F., An application of the theory of probability to the operation of vehicular-controlled traffic signals, Suppl. J. Royal Stat. Soc., 7, 1, 65-77 (1940) |
| [12] | Gentle, J. E., Random Number Generation and Monte Carlo Methods, Vol. 381 (2003), Springer · Zbl 1028.65004 |
| [13] | Graham, R. L.; Knuth, D. E.; Patashnik, O., Concrete Mathematics: A Foundation for Computer Science (1989), Addison-Wesley: Addison-Wesley Reading · Zbl 0668.00003 |
| [14] | L’Ecuyer, P.; Simard, R., TestU01: A C library for empirical testing of random number generators, ACM Trans. Math. Softw. (TOMS), 33, 4, 1-40 (2007) · Zbl 1365.65008 |
| [15] | Marichal, J.-L.; Mossinghoff, M., Slices, slabs, and sections of the unit hypercube, Online J. Anal. Comb., 3, 1-11 (2008) · Zbl 1189.52011 |
| [16] | Munkres, J. R., Analysis on Manifolds (2018), CRC Press |
| [17] | Onn, S.; Weissman, I., Generating uniform random vectors over a simplex with implications to the volume of a certain polytope and to multivariate extremes, Ann. Oper. Res., 189, 1, 331-342 (2011) · Zbl 1268.65005 |
| [18] | Pinelis, I., Order statistics on the spacings between order statistics for the uniform distribution (2019), arXiv Preprint arXiv:1909.06406 |
| [19] | Riordan, J., (Combinatorial Identities. Combinatorial Identities, Wiley series in probability and mathematical statistics (1968), Wiley), URL https://books.google.co.il/books?id=X6ccBWwECP8C |
| [20] | Rubinstein, R. Y.; Kroese, D. P., Simulation and the Monte Carlo Method, Vol. 10 (2016), John Wiley & Sons · Zbl 1352.68002 |
| [21] | Shaked, M.; Shanthikumar, J. G., Stochastic Orders (2007), Springer Science & Business Media · Zbl 1111.62016 |
| [22] | Weissman, I., Testing for serial correlation by means of extreme values, Reliab. Theory Appl., 6, 4 (23) (2011) |
| [23] | Whitworth, W. A., DCC Exercises: Including Hints for the Solution of All the Questions in Choice and Chance (1897), D. Bell · JFM 28.0212.14 |
| [24] | Zar, J. H., Biostatistical Analysis (1999), Pearson Education India |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
