Abstract
The paper gives the joint distribution of maxima of contralateral observations taken from the same individual at several occasions, when the data are normal with given constraints on the parameters. Different constraints lead to different well-known generalizations of the normal distribution. As an immediate consequence, evaluation of several features of these maxima is greatly simplified. The results of the paper are also useful in modeling other phenomena, such as measures of physical fitness based on the best of two trials. Further applications deal with statistical inference, for example, testing whether an identified treatment is best.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aldous DJ (1985) Exchangeability and related topics. In: Hennequin PL (ed) Ecole d’été de probabilités de Saint-Flour, XIII. Springer series in statistics. Springer, Berlin
Arellano-Valle RB, Genton MG (2005) On fundamental skew distributions. J Multivar Anal 96:93–116
Arnold BC, Beaver RJ (2000) Hidden truncation models. Sankhya Ser A 62:22–35
Arnold BC, Beaver RJ (2002) Skewed multivariate models related to hidden truncation and/or selective reporting (with discussion). Test 11:7–54
Arnold BC, Beaver RJ (2004) Elliptical models subject to hidden truncation or selective sampling. In: Genton MG (ed) Skew-elliptical distributions and their applications: a journey beyond normality. Chapman & Hall/CRC, Boca Raton, pp 101–112
Azzalini A (1985) A class of distributions which includes the normal ones. Scand J Stat 12:171–178
Azzalini A (2002) Discussion of Arnold and Beaver’s paper “Skewed multivariate models related to hidden truncation and/or selective reporting”. Test 11:36
Azzalini A (2005) The skew-normal distribution and related multivariate families (with discussion). Scand J Stat 32:159–200
Azzalini A, Capitanio A (2003) Distributions generated by perturbation of symmetry with emphasis on a multivariate skew-t distribution. J Roy Stat Soc Ser B 65:367–389
Azzalini A, Cox DR (1984) Two new tests associated with analysis of variance. J Roy Stat Soc Ser B 46:335–343
Azzalini A, Dalla Valle A (1996) The multivariate skew-normal distribution. Biometrika 83:715–726
Cain M (1994) The moment-generating function of the minimum of bivariate normal random variables. Am Stat 48:124–125
Cain M, Pan M (1995) Moments of the minimum of bivariate normal random variables. Math Sci 20:119–122
Capitanio A, Azzalini A, Stanghellini E (2003) Graphical models for skew-normal variates. Scand J Stat 30:129–144
Cox DR, Wermuth N (1991) A simple approximation for bivariate and trivariate normal integrals. Int Stat Rev 59:263–269
Dalla Valle A (2004) The skew-normal distribution. In: Genton MG (ed) Skew-elliptical distributions and their applications: a journey beyond normality. Chapman & Hall/CRC, Boca Raton, pp 3–24
Fishman GA, Baca W, Alexander KR, Derlacky DJ, Glenn AM, Viana MAG (1993) Visual acuity in patients with best vitelliform macular distrophy. Ophthalmology 100:1665–1670
Frenkel R, Shin D (1986) Prevention and management of delayed suprachoroidal hemorrhage after filtration surgery. Arch Ophthalmol 104:1459–1463
Gupta PL, Gupta RC (2001) Failure rate of the minimum and maximum of a multivariate normal distribution. Metrika 53:39–49
Gupta SS, Pillai S (1965) On linear functions of ordered correlated normal random variables. Biometrika 52:367–379
Jiang J (1997) Sharp upper and lower bounds for asymptotic levels of some statistical tests. Stat Probab Lett 35:395–400
Kella O (1986) On the distribution of maximum of bivariate normal random variables with general means and variances. Commun Stat Theory Methods 15:3265–3276
Kotz S, Balakrishnan N, Johnson NL (2000) Continuous multivariate distributions, vol 1. Wiley, New York
Laska EM, Meisner MJ (1989) Testing whether an identified treatment is best. Biometrics 45:1139–1151
Loperfido N (2002) Statistical implications of selectively reported inferential results. Stat Probab Lett 56:13–22
Nagarajah HN (1982) A note on linear functions of ordered correlated normal random variables. Biometrika 69:284–285
Olkin I, Viana M (1995) Correlation analysis of extreme observations from a multivariate normal distribution. J Am Stat Assoc 90:1373–1379
Parving A, Christiansen B (1990) Hearing of the mentally retarded living at home. Ugeskr Læ152:3161–3164
Roberts C (1966) A correlation model useful in the study of twins. J Am Stat Assoc 61:1184–1190
Sonty SP, Sonty S, Viana MAG (1996) The additive ocular hypotensive effect of topical 2% dorzolamide on glaucomatous eyes on topical beta blocker therapy. Investig Ophthalmol Vis Sci 37:1100
Tomohiro O, Yoshio N, Kiyoji T (2003) Effects of exercise intensity on physical fitness and risk factors for coronary heart disease. Obes Res 11:1131–1139
Tong YL (1990) The multivariate normal distribution. Springer series in statistics. Springer, Berlin
Viana M (1998) Linear combinations of ordered symmetric observations with application to visual acuity. In: Balakrishnan N, Rao CR (eds) Order statistics: applications. Handbook of statistics, vol 17. North Holland, Amsterdam, pp 513–524. Chap 19
Viana M, Olkin I (2000) Symmetrically dependent models arising in visual assessment data. Biometrics 56:1188–1191
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Loperfido, N. Modeling maxima of longitudinal contralateral observations. TEST 17, 370–380 (2008). https://doi.org/10.1007/s11749-006-0037-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11749-006-0037-3
Keywords
- Canonical fundamental skew-normal distribution
- Exchangeability
- Extended skew-normal distribution
- Multivariate skew-normal distribution
- Nonlinear prediction