×

Characterization of stochastic orders by \(L\)-functionals. (English) Zbl 1114.60021

Summary: Random variables may be compared with respect to their location by comparing certain functionals ad hoc, such as the mean or median, or by means of stochastic ordering based directly on the properties of the corresponding distribution functions. These alternative approaches are brought together in this paper. We focus on the class of L-functionals discussed by P. J. Bickel and E. L. Lehmann [Ann. Stat. 3, 1038–1044 (1975; Zbl 0321.62054)] and characterize the comparison of random variables in terms of these measures by means of several stochastic orders based on iterated integrals, including the increasing convex order.

MSC:

60E15 Inequalities; stochastic orderings

Citations:

Zbl 0321.62054
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andrews DF, Bickel PJ, Hampel FR, Hubert PJ, Rogers WH, Tukey JW (1972) Robust estimation of location. Princeton Univ. Press, Princeton NJ
[2] Apostol TH (1973) Mathematical Analysis (6th pr.). Reading: Addison-Wesley
[3] Arnold BC (1987) Majorization and the Lorenz order: a brief introduction. Springer-Verlag, New York, NY · Zbl 0649.62041
[4] Baccelli F, Makowski AM (1989) Multi-dimensional stochastic ordering and associated random variables. Oper. Res. 37, 478–487 · Zbl 0671.60013 · doi:10.1287/opre.37.3.478
[5] Bhattacharjee MC, Sethuraman J (1990) Families of life distributions characterizated by two moments. J. Appl. Probab. 27, 720–725 · Zbl 0719.60095 · doi:10.2307/3214556
[6] Bhattacharjee MC (1991) Some generalized variability orderings among life distributions with reliability applications. J. Appl. Prob. 28, 374–383 · Zbl 0729.62097 · doi:10.2307/3214873
[7] Bhattacharjee MC, Bhattacharya RN (2000) Stochastic equivalence of convex ordered distributions and applications. Probab. Engrg. Inform. Sci. 14, 33–48 · Zbl 0955.60019 · doi:10.1017/S026996480014104X
[8] Bickel PJ, Lehmann EL (1975) Descriptive statistics for nonparametric models. I. Introduction. Ann. Statist. 3, 1038–1044. II. Location. Ann. Statist. 3, 1045–1069 · Zbl 0321.62054 · doi:10.1214/aos/1176343239
[9] Bickel PJ, Lehmann EL (1976) Descriptive statistics for nonparametric models. III. Dispersion. Ann. Statist. 4, 1139–1158 · Zbl 0351.62031 · doi:10.1214/aos/1176343648
[10] Cai J, Wu Y (1997) Characterization of life distributions under some generalized stochastic orderings. J. Appl. Probab. 34, 711–719 · Zbl 0881.62013 · doi:10.2307/3215096
[11] Chong KM (1974) Some extensions of a theorem of Hardy, Littlewood and Pólya and their applications. Canad. J. Math. 26, 1321–1340 · Zbl 0295.28006 · doi:10.4153/CJM-1974-126-1
[12] Denuit M, Lefèvre C, Shaked M (2000) On the theory of high convexity stochastic orders. Statist. Probab. Lett. 47, 287–293 · Zbl 0958.60011 · doi:10.1016/S0167-7152(99)00166-2
[13] Fagiuoli E, Pellerey F, Shaked M (1999) A characterization of the dilation order and its applications. Statist. Papers 40, 393–406 · Zbl 0938.62008 · doi:10.1007/BF02934633
[14] Giovagnoli A, Regoli G (1993) Some results on the representation of measures of location and spread as L-functionals. Statist. Probab. Lett. 16, 269–278 · Zbl 0764.62002 · doi:10.1016/0167-7152(93)90130-B
[15] Hardy GH, Littlewood JE, Pólya G (1929) Some simple inequalities satisfied by convex functions. Mess. Math. 58, 145–152
[16] Hickey RJ (1986) Concepts of dispersion in distributions: a comparative note. J. Appl. Probab. 23, 914–921 · Zbl 0607.60016 · doi:10.2307/3214465
[17] Huber PJ (1972) Robust statistics: a review. Ann. Math. Statist. 43, 1041–1067 · Zbl 0254.62023 · doi:10.1214/aoms/1177692459
[18] Jun C (1994) Characterizations of life distributions by moments of extremes and sample mean. J. Appl. Prob. 31, 148–155 · Zbl 0803.62009 · doi:10.2307/3215242
[19] Li H, Zhu H (1994) Stochastic equivalence of ordered random variables with applications in reliability theory. Statist. Probab. Lett. 20, 383–393 · Zbl 0804.62015 · doi:10.1016/0167-7152(94)90130-9
[20] Muliere P, Scarsini M (1989) A note on stochastic dominance and inequality measures. J. Econ. Theory 49, 314–323 · Zbl 0682.90029 · doi:10.1016/0022-0531(89)90084-7
[21] Ogryczak W, Ruszczynski A (2002) Dual stochastic dominance and related mean-risk models. SIAM J. Optim. 13, 60–78 · Zbl 1022.91017 · doi:10.1137/S1052623400375075
[22] Ramos HM, Sordo MA (2002) Characterizations of inequality orderings by means of dispersive orderings. Qüestiió 26, 15–28 · Zbl 1042.60007
[23] Ramos HM, Sordo MA (2003) Dispersion measures and dispersive orderings. Statist. Probab. Lett. 61, 123–131 · Zbl 1043.60013 · doi:10.1016/S0167-7152(02)00341-3
[24] Röel, A (1987) Risk aversion in Quiggin and Yaari’s rank-order model of choice under uncertainty. Economic J. 97, 143–159. · doi:10.2307/3038236
[25] Ross SM (1983) Stochastic Processes, Wiley, New York.
[26] Ryff JV (1963) On the representation of doubly stochastic operators. Pacific J. Math. 13, 1379–1386 · Zbl 0125.08405
[27] Scarsini M, Shaked M (1990) Some conditions for stochastic equality. Naval Res. Logist. 37, 617–625 · Zbl 0718.60010 · doi:10.1002/1520-6750(199010)37:5<617::AID-NAV3220370503>3.0.CO;2-L
[28] Scarsini M (1998) Multivariate convex orderings, dependence, and stochastic equality. J. Appl. Probab. 35, 93–103 · Zbl 0906.60020 · doi:10.1239/jap/1032192554
[29] Sendler W (1979) On statistical inference in concentration measurement. Metrika 26, 109–122 · Zbl 0416.62038 · doi:10.1007/BF01893478
[30] Serfling RJ (1980) Approximation Theorems of Mathematical Statistics. Wiley, New York · Zbl 0538.62002
[31] Shaked M, Shanthikumar JG (1994) Stochastic Orders and their Applications. Academic Press, San Diego, CA · Zbl 0806.62009
[32] Shorack GR (1972) Functions of Order Statistics. Ann. Math. Statist. 43, 412–427 · Zbl 0239.62037 · doi:10.1214/aoms/1177692622
[33] Shorack GR, Wellner JA (1986) Empirical Processes with Applications to Statitics. Wiley, New York
[34] Stoyan D (1983) Comparison Methods for Queues and Other Stochastic Models. John Wiley, New York · Zbl 0536.60085
[35] Wang S, Young VR (1998) Ordering risks: Expected utility theory versus Yaari’s dual theory of risk. Insurance: Math. and Econ. 22, 145–161 · Zbl 0907.90102 · doi:10.1016/S0167-6687(97)00036-X
[36] Yaari, ME (1987) The dual theory of choice under risk. Econometrica 55, 95–115 · Zbl 0616.90005 · doi:10.2307/1911158
[37] Zygmund A (1959) Trigonometric Series, Vol. I, Cambridge · Zbl 0085.05601
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.