Box-Cox symmetric distributions and applications to nutritional data. (English) Zbl 1443.62051
Summary: We introduce and study the Box-Cox symmetric class of distributions, which is useful for modeling positively skewed, possibly heavy-tailed, data. The new class of distributions includes the Box-Cox \(t\), Box-Cox Cole-Green (or Box-Cox normal), Box-Cox power exponential distributions, and the class of the log-symmetric distributions as special cases. It provides easy parameter interpretation, which makes it convenient for regression modeling purposes. Additionally, it provides enough flexibility to handle outliers. The usefulness of the Box-Cox symmetric models is illustrated in a series of applications to nutritional data.
MSC:
| 62E15 | Exact distribution theory in statistics |
| 60E05 | Probability distributions: general theory |
| 62P10 | Applications of statistics to biology and medical sciences; meta analysis |
Keywords:
Box-Cox transformation; symmetric distributions; Box-Cox power exponential distribution; Box-Cox slash distribution; Box-Cox \(t\) distribution; log-symmetric distributions; nutrients intakeReferences:
| [1] | Azzalini, A, The skew-normal distribution and related multivariate families, Scand. J. Stat., 32, 159-188, (2005) · Zbl 1091.62046 · doi:10.1111/j.1467-9469.2005.00426.x |
| [2] | Box, GEP; Cox, DR, An analysis of transformations, J. R. Stat. Soc. Ser. B, 26, 211-252, (1964) · Zbl 0156.40104 |
| [3] | Cole, T; Green, PJ, Smoothing reference centile curves: the LMS method and penalized likelihood, Stat. Med., 11, 1305-1319, (1992) · doi:10.1002/sim.4780111005 |
| [4] | Cordeiro, GM; Andrade, MG, Transformed symmetric models, Stat. Model., 11, 371-388, (2011) · Zbl 1420.62314 · doi:10.1177/1471082X1001100405 |
| [5] | de Haan, L.: On Regular Variation and Its Application to the Weak Convergence of Sample Extremes, Mathematical Centre Tracts, vol. 32. Mathematics Centre, Amsterdam (1970) · Zbl 0226.60039 |
| [6] | Dunn, PK; Smyth, GK, Randomized quantile residuals, J. Comput. Graph. Stat., 5, 236-244, (1996) |
| [7] | Fang, K.T., Kotz, S., NG, K.W.: Symmetric Multivariate and Related Distributions. Chapman and Hall, London (1990) · Zbl 0699.62048 · doi:10.1007/978-1-4899-2937-2 |
| [8] | Hubert, M; Vandervieren, E, An adjusted boxplot for skewed distributions, Comput. Stat. Data Anal., 52, 5186-5201, (2008) · Zbl 1452.62074 · doi:10.1016/j.csda.2007.11.008 |
| [9] | Jondeau, E; Rockinger, M, Conditional volatility, skewness, and kurtosis: existence, persistence, and comovements, J. Econ. Dyn. Control, 27, 1699-1737, (2003) · Zbl 1178.91226 · doi:10.1016/S0165-1889(02)00079-9 |
| [10] | Kelker, D, Distribution theory of spherical distributions and a location-scale parameter generalization, Sankhya A, 32, 419-430, (1970) · Zbl 0223.60008 |
| [11] | Luceño, A, Fitting the generalized Pareto distribution to data using maximum goodness-of-fit estimators, Comput. Stat. Data Anal., 51, 904-917, (2005) · Zbl 1157.62399 · doi:10.1016/j.csda.2005.09.011 |
| [12] | Poirier, DJ, The use of box-Cox transformation in limited dependent variable models, J. Am. Stat. Assoc., 73, 284-287, (1978) · Zbl 0399.62117 · doi:10.1080/01621459.1978.10481570 |
| [13] | Resnick, S.I.: Heavy-Tail Phenomena Probabilistic and Statistical Modeling. Springer, New York (2007) · Zbl 1152.62029 |
| [14] | Rigby, RA; Stasinopoulos, DM, Smooth centile curves for skew and kurtotic data modelled using the box-Cox power exponential distribution, Stat. Med., 23, 3053-3076, (2004) · doi:10.1002/sim.1861 |
| [15] | Rigby, RA; Stasinopoulos, DM, Generalized additive models for location, scale and shape, J. R. Stat. Soc. Ser. C Appl. Stat., 54, 507-554, (2005) · Zbl 1490.62201 · doi:10.1111/j.1467-9876.2005.00510.x |
| [16] | Rigby, RA; Stasinopoulos, DM, Using the box-Cox t distribution in GAMLSS to model skewness and kurtosis, Stat. Model., 6, 209-229, (2006) · Zbl 07257135 · doi:10.1191/1471082X06st122oa |
| [17] | Rigby, R.A., Stasinopoulos, D.M., Heller, G. Voudouris, V.: The Distribution Toolbox of GAMLSS. London (2014). http://www.gamlss.org/wp-content/uploads/2014/10/distributions.pdf |
| [18] | Rogers, WH; Tukey, JW, Understanding some long-tailed symmetrical distributions, Stat. Neerl., 26, 211-226, (1972) · Zbl 0245.62021 · doi:10.1111/j.1467-9574.1972.tb00191.x |
| [19] | Stasinopoulos, D.M., Rigby, R.A. Akantziliotou, C.: Instructions on how to use the GAMLSS package in R. London (2008). http://www.gamlss.org · Zbl 1231.62019 |
| [20] | Vanegas, LH; Paula, GA, A semiparametric approach for joint modeling of Median and skewness, Test, 24, 110-135, (2015) · Zbl 1315.62058 · doi:10.1007/s11749-014-0401-7 |
| [21] | Vanegas, LH; Paula, GA, Log-symmetric distributions: statistical properties and parameter estimation, Braz. J. Probab. Stat., 30, 196-220, (2016) · Zbl 1381.60047 · doi:10.1214/14-BJPS272 |
| [22] | Voudouris, V; Gilchrist, R; Rigby, RA; Sedgwick, J; Stasinopoulos, DM, Modelling skewness and kurtosis with the BCPE density in GAMLSS, J. Appl. Stat., 39, 1279-1293, (2012) · Zbl 1514.62333 · doi:10.1080/02664763.2011.644530 |
| [23] | Yang, Z, A modified family of power transformations, Econ. Lett., 92, 14-19, (2006) · Zbl 1255.62031 · doi:10.1016/j.econlet.2006.01.011 |
| [24] | Yang, ZL, Some asymptotic results on box-Cox transformation methodology, Commun. Stat. Theory Methods, 25, 403-415, (1996) · Zbl 0875.62114 · doi:10.1080/03610929608831703 |
| [25] | Yeo, IK; Johnson, RA, A new family of power transformation to improve normality or symmetry, Biometrika, 87, 954-959, (2000) · Zbl 1028.62010 · doi:10.1093/biomet/87.4.954 |
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