On connections between skewed, weighted and distorted distributions: applications to model extreme value distributions. (English) Zbl 1537.60022
Given identically distributed continuous random variables \(X_1,\dots,X_n\) with copula \(C\), the authors consider representations of the density functions of their minimum and maximum (as well as of other order statistics). These representations connect the notions of skewed, weighted and distorted distributions. The authors also explore connections to the likelihood ratio ordering. Several examples with explicit copulas are considered, and results are illustrated with numerical applications.
Reviewer: Fraser Daly (Edinburgh)
MSC:
| 60E05 | Probability distributions: general theory |
| 60E15 | Inequalities; stochastic orderings |
| 62N05 | Reliability and life testing |
| 62G30 | Order statistics; empirical distribution functions |
References:
| [1] | Arellano-Valle, RB; Genton, MG, On the exact distribution of the maximum of absolutely continuous dependent random variables, Stat Probab Lett, 78, 27-35 (2008) · Zbl 1134.60033 |
| [2] | Arevalillo, JM; Navarro, H., A study of the effect of kurtosis on discriminant analysis under elliptical populations, J Multivar Anal, 107, 53-63 (2012) · Zbl 1236.62066 |
| [3] | Arevalillo, JM; Navarro, H., Data projections by skewness maximization under scale mixtures of skew-normal vectors, Adv Data Anal Classif, 14, 435-461 (2020) · Zbl 1459.62075 |
| [4] | Arevalillo, JM; Navarro, H., New insights on the multivariate skew exponential power distribution, Math Slovaca, 73, 2, 529-544 (2023) · Zbl 1508.60014 |
| [5] | Arevalillo, JM; Navarro, H., A stochastic ordering based on the canonical transformation of skew-normal vectors, TEST, 28, 475-498 (2019) · Zbl 1456.60046 |
| [6] | Arnold BC, Groeneveld RA (1993) Skewness and kurtosis orderings: an introduction. Stochastic Inequalities IMS Lecture Notes - Monograph Series Volume 22 |
| [7] | Arnold, BC; Groeneveld, RA, Measuring skewness with respect to the mode, Am Stat, 49, 34-38 (1995) |
| [8] | Arnold BC, Castillo E, Sarabia JM (1999) Conditional specification of statistical models. Springer Series in Statistics. Springer-Verlag, New York · Zbl 0932.62001 |
| [9] | Arnold, BC; Beaver, RJ, Hidden truncation models, Sankhya Ser A, 62, 22-35 (2000) · Zbl 0973.62041 |
| [10] | Arnold, BC; Balakrishnan, N.; Nagaraja, HN, A first course in order statistics (2008), Philadelphia: SIAM, Philadelphia |
| [11] | Arriaza, A.; Di Crescenzo, A.; Sordo, MA; Suáirez-Llorens, A., Shape measures based on the convex transform order, Metrika, 82, 99-124 (2019) · Zbl 1416.62261 |
| [12] | Azzalini, A., A class of distributions which includes the normal ones, Scand J Stat, 12, 171-178 (1985) · Zbl 0581.62014 |
| [13] | Azzalini, A., The skew-normal distribution and related multivariate families, Scand J Stat, 32, 159-188 (2005) · Zbl 1091.62046 |
| [14] | Azzalini, A.; Capitanio, A., Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution, J R Stat Soc B, 65, 367-389 (2003) · Zbl 1065.62094 |
| [15] | Azzalini, A.; Capitanio, A., The skew-normal and related families (2014), Cambridge: Cambridge University Press, Cambridge · Zbl 1338.62007 |
| [16] | Azzalini, A.; Regoli, G., Some properties of skew-symmetric distributions, Ann Inst Stat Math, 64, 857-879 (2012) · Zbl 1253.62038 |
| [17] | Azzalini, A.; Regoli, G., On symmetry-modulated distributions: revisiting an old result and a step further, Stat, 7 (2018) · Zbl 07851059 |
| [18] | Blenkinsop, S.; Lewis, E.; Chan, SC; Fowler, HJ, Quality-control of an hourly rainfall dataset and climatology of extremes for the UK, Int J Climatol, 37, 722-740 (2017) |
| [19] | Buishand, TA, Statistics of extremes in climatology, Stat Neerl, 43, 1-30 (1989) · Zbl 0668.62085 |
| [20] | Chan W, Proschan F, Sethuraman J (1990) Convex-ordering among functions,with applications to reliability and mathematical statistics. In Block HW, Sampson AR, Savits TH (eds) Topics in statistical dependence. IMS Lecture Notes-Monograph Series 16. Hayward, California, pp 121-134 · Zbl 0770.62086 |
| [21] | Cox, DR, Regression models and life-tables, J R Stat Soc B, 34, 187-220 (1972) · Zbl 0243.62041 |
| [22] | David, HA; Nagaraja, HN, Order statistics (2003), Hoboken: Wiley, Hoboken · Zbl 1053.62060 |
| [23] | Domma, F.; Popoviś, BV; Nadarajah, S., An extension of Azzalini’s method, J Comput Appl Math, 278, 37-47 (2015) · Zbl 1301.60014 |
| [24] | Durante, F.; Sempi, C., Principles of copula theory (2016), London: CRC/Chapman & Hall, London · Zbl 1380.62008 |
| [25] | De Luca G, Loperfido N (2004) A skew-in-mean GARCH model for financial returns. In: Skew-elliptical distributions and their applications: a journey beyond normality. CRC/Chapman & Hall, pp 205-222 |
| [26] | Fisher, RA, The effect of methods of ascertainment upon the estimation of frequencies, Ann Eugen, 6, 13-25 (1934) · JFM 62.1373.02 |
| [27] | Ferreira, JTAS; Steel, MFJ, A constructive representation of univariate skewed distributions, J Am Stat Assoc, 101, 823-829 (2006) · Zbl 1119.62311 |
| [28] | Ghosh, I.; Ng, HKT, A class of skewed distributions with applications in environmental data, Commun Stat Case Stud Data Anal Appl, 5, 346-365 (2019) |
| [29] | Groeneveld, RA; Meeden, G., Measuring skewness and kurtosis, The Statistician, 33, 391-399 (1984) |
| [30] | Huber, PJ, Projection pursuit (with discussion), Ann Stat, 13, 435-475 (1985) · Zbl 0595.62059 |
| [31] | Jones, MC, On families of distributions with shape parameters, Int Stat Rev, 83, 175-192 (2015) · Zbl 07763428 |
| [32] | Jupp PE, Regoli G, Azzalini A (2016) A general setting for symmetric distributions and their relationship to general distributions. J Multivar Anal 148:107-119 · Zbl 1338.62047 |
| [33] | Ley, C., Flexible modelling in statistics: past, present and future, Journal de la Société Française de Statistique, 156, 76-96 (2015) · Zbl 1316.62023 |
| [34] | Loperfido, N., A note on skew-elliptical distributions and linear functions of order statistics, Stat Probab Lett, 78, 3184-3186 (2008) · Zbl 1489.62152 |
| [35] | Loperfido, N.; Navarro, J.; Ruiz, JM; Sandoval, CJ, Some relationships between skew-normal distributions and order statistics from exchangeable normal random vectors, Commun Stat Theory Methods, 36, 1719-1733 (2007) · Zbl 1128.62014 |
| [36] | Loperfido, N., Skewness-based projection pursuit: a computational approach, Comput Stat Data Anal, 120, 42-57 (2018) · Zbl 1469.62111 |
| [37] | Loperfido, N., Kurtosis-based projection pursuit for outlier detection in financial time series, Eur J Finance, 26, 142-164 (2020) |
| [38] | Navarro, J., Introduction to system reliability theory (2022), Berlin: Springer, Berlin |
| [39] | Navarro, J.; Calì, C.; Longobardi, M.; Durante, F., Distortion representations of multivariate distributions, Stat Methods Appl, 31, 925-954 (2022) · Zbl 1495.62035 |
| [40] | Navarro, J.; Ruiz, JM; del Aguila, Y., Multivariate weighted distributions: a review and some extensions, Statistics, 40, 51-64 (2006) · Zbl 1098.62068 |
| [41] | Navarro, J.; Sordo, MA, Stochastic comparisons and bounds for conditional distributions by using copula properties, Depend Model, 6, 156-177 (2018) · Zbl 1404.62057 · doi:10.1515/demo-2018-0010 |
| [42] | Navarro, J.; Torrado, N.; del Águila, Y., Comparisons between largest order statistics from multiple-outlier models with dependence, Methodol Comput Appl Probab, 20, 411-433 (2018) · Zbl 1458.62102 |
| [43] | Nelsen, RB, An introduction to copulas (2006), New York: Springer, New York · Zbl 1152.62030 |
| [44] | Patil, GP; Rao, CR, Weighted distributions and size biased sampling with applications to wildlife populations and human families, Biometrics, 34, 179-189 (1978) · Zbl 0384.62014 |
| [45] | Rao, CR, On discrete distributions arising out of methods of ascertainment, Sankhya Ser A, 27, 311-324 (1965) · Zbl 0212.21903 |
| [46] | Roberts, C., A correlation model useful in the study of twins, J Am Stat Assoc, 61, 1184-1190 (1966) · Zbl 0147.38001 |
| [47] | Schweizer, B.; Sklar, A., Operations on distribution functions not derivable from operations on random variables, Stud Math, 52, 43-52 (1974) · Zbl 0292.60035 |
| [48] | Shaked, M.; Shanthikumar, JG, Stochastic orders (2007), New York: Springer, New York · Zbl 1111.62016 |
| [49] | Serrano-Notivoli R (2022) agroclim: climatic indices for agriculture. R package version 0.2.0. https://CRAN.R-project.org/package=agroclim |
| [50] | Thas, O., Comparing distributions (2010), New York: Springer, New York · Zbl 1234.62014 |
| [51] | Van Zwet, WR, Convex transformations of random variables (1964), Amsterdan: Mathematish Centrum, Amsterdan · Zbl 0125.37102 |
| [52] | Wang, S., Premium calculation by transforming the layer premium density, Astin Bull, 26, 71-92 (1996) |
| [53] | Wang, J., A family of kurtosis orderings for multivariate distributions, J Multivar Anal, 100, 509-517 (2009) · Zbl 1154.62043 |
| [54] | Yaari, ME, The dual theory of choice under risk, Econometrica, 55, 95-115 (1987) · Zbl 0616.90005 |
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