Cramer, E.; Herle, K.; Balakrishnan, N. Permanent expansions and distributions of order statistics in the inid case. (English) Zbl 1173.62006 Commun. Stat., Theory Methods 38, No. 12, 2078-2088 (2009). Summary: Expansions of permanents are applied to derive expressions for the distribution functions of order statistics arising from \(n\) independent, nonidentically distributed (inid) random variables. For this purpose, we apply two types of expansions. The first one is based on H. J. Ryser’s method [Combinatorial mathematics. 4th ed., Carus Math. Monogr. 14 (1973; Zbl 0302.05001)] and leads to a representation of the distribution function as a generalized mixture of distributions of order statistics from independent, identically distributed (iid) random variables. The second approach, which is based on the calculation of the permanents of a sum of two matrices, yields an expression in terms of products of both distributions of minima and maxima. It turns out that the most efficient way of calculating the distribution function is the one based on Ryser’s expansion. Cited in 3 Documents MSC: 62E15 Exact distribution theory in statistics 62G30 Order statistics; empirical distribution functions 15A15 Determinants, permanents, traces, other special matrix functions Keywords:generalized mixtures; INID variables; order statistics; permanents; Ryser’s expansion of permanents Citations:Zbl 0302.05001 PDFBibTeX XMLCite \textit{E. Cramer} et al., Commun. Stat., Theory Methods 38, No. 12, 2078--2088 (2009; Zbl 1173.62006) Full Text: DOI References: [1] DOI: 10.1007/BF00052344 · Zbl 0668.62029 · doi:10.1007/BF00052344 [2] DOI: 10.1007/BF00049399 · Zbl 0721.62049 · doi:10.1007/BF00049399 [3] Balakrishnan N., Rev. Mat. Complut. 20 pp 7– (2007) [4] DOI: 10.1007/s11749-007-0061-y · Zbl 1121.62052 · doi:10.1007/s11749-007-0061-y [5] DOI: 10.1007/s10463-006-0070-8 · Zbl 1184.62071 · doi:10.1007/s10463-006-0070-8 [6] Balasubramanian K., Sankhy Ser. A pp 375– (1991) [7] DOI: 10.1016/0378-3758(95)00138-7 · Zbl 0859.62051 · doi:10.1016/0378-3758(95)00138-7 [8] Bapat R., Sankhy Ser. A pp 79– (1989) [9] Bebiano N., Pac. J. Math. 101 pp 1– (1982) [10] Fischer , T. ( 2006 ). Ordnungsstatistiken aus nicht identisch verteilten Zufallsvariablen (in German) . Master’s thesis, RWTH Aachen University , Germany . [11] DOI: 10.1016/j.jmva.2008.02.007 · Zbl 1151.60306 · doi:10.1016/j.jmva.2008.02.007 [12] Guilbaud O., Scand. J. Statist. Theor. Appl. 9 pp 229– (1982) [13] DOI: 10.1007/s11749-007-0068-4 · doi:10.1007/s11749-007-0068-4 [14] Kräuter A. R., Sémin. Lothar. Comb. 9 pp 1– (1983) [15] DOI: 10.1080/03081088708817770 · Zbl 0633.15004 · doi:10.1080/03081088708817770 [16] DOI: 10.1016/j.amc.2004.11.020 · Zbl 1137.65343 · doi:10.1016/j.amc.2004.11.020 [17] Minc H., Permanents (1978) [18] Reiss R.-D., Approximate Distributions of Order Statistics (1989) · Zbl 0682.62009 [19] Ryser H. J., Combinatorial Mathematics., 4. ed. (1973) · Zbl 0302.05001 [20] Vaughan R. J., J. Roy. Statist. Soc. Ser. B 34 pp 308– (1972) 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.