This function performs test for symmetry about an unknown median. Users can choose between the Cabilio-Masaro test (Cabilio and Masaro, 1996), the Mira test (Mira, 1999), or the MGG test (Miao, Gel, and Gastwirth, 2006); and using asymptotic distribution of respective statistics or a distribution from \(m\)-out-of-\(n\) bootstrap. Additionally to the general distribution asymmetry, the function allows to test for negative or positive skeweness (see the argument side). NAs from the data are omitted.
symmetry.test(x, option = c("MGG", "CM", "M"), side = c("both", "left", "right"),
boot = TRUE, B = 1000, q = 8/9)data to be tested for symmetry.
test statistic to be applied. Options include statistic by Miao, Gel, and Gastwirth (2006) (default), Cabilio and Masaro (1996), and by Mira (1999).
choice from the three possible alternative hypotheses: general distribution asymmetry (side="both", default), left skewness (side="left"), or right skewness (side="right").
logical value indicates whether \(m\)-out-of-\(n\) bootstrap will be used to obtain critical values (default), or asymptotic distribution of the chosen statistic.
number of bootstrap replications to perform (default is 1000).
scalar from 0 to 1 to define a set of possible \(m\) for the \(m\)-out-of-\(n\) bootstrap. Default q = 8/9. Possible \(m\) are then set as the values unique(round(n*(q^j)) greater than 4, where n = length(x) and j = c(0:20).
A list of class htest containing the following components:
name of the method.
name of the data.
value of the test statistic.
\(p\)-value of the test.
alternative hypothesis.
bootstrap optimal \(m\) (given in the output only if bootstrap was used, i.e., boot = TRUE).
If the bootstrap option is used (boot = TRUE), a bootstrap distribution is obtained for each candidate subsample size \(m\). Then, a heuristic method (Bickel et al., 1997; Bickel and Sakov, 2008) is used for the choice of optimal \(m\). Particularly, we use the Wasserstein metric (Ruschendorf, 2001) to calculate distances between different bootstrap distributions and select \(m\), which corresponds to the minimal distance.
Bickel, P. J., Gotze, F., and van Zwet, W. R. (1997). Resampling fewer than n observations: gains, losses, and remedies for losses. Statistica Sinica 7: 1--31.
Bickel, P. J. and Sakov, A. (2008). On the choice of \(m\) in the \(m\) out of \(n\) bootstrap and confidence bounds for extrema. Statistica Sinica 18: 967--985.
Cabilio, P. and Masaro, J. (1996). A simple test of symmetry about an unknown median. The Canadian Journal of Statistics, 24(3): 349--361. DOI: 10.2307/3315744
Lyubchich, V., Wang, X., Heyes, A., and Gel, Y. R. (2016). A distribution-free m-out-of-n bootstrap approach to testing symmetry about an unknown median. Computational Statistics and Data Analysis 104: 1--9. DOI: 10.1016/j.csda.2016.05.004
Miao, W., Gel, Y. R., and Gastwirth, J. L. (2006). A new test of symmetry about an unknown median. In: A. Hsiung, C.-H. Zhang, and Z. Ying (Eds.) Random Walk, Sequential Analysis and Related Topics --- A Festschrift in Honor of Yuan-Shih Chow. World Scientific Publisher, Singapore, pp. 199--214. DOI: 10.1142/9789812772558_0013
Mira, A. (1999). Distribution-free test for symmetry based on Bonferroni's measure. Journal of Applied Statistics, 26(8): 959--972. DOI: 10.1080/02664769921963
Ruschendorf, L. (2001). Wasserstein metric. In: M. Hazewinkel (Ed.) Encyclopedia of Mathematics. Springer, Berlin.
# NOT RUN {
data(zuni) #run ?zuni to see the data description
symmetry.test(zuni[,"Revenue"], boot = FALSE)
## Symmetry test by Miao, Gel, and Gastwirth (2006)
##
## data: zuni[, "Revenue"]
## Test statistic = 5.0321, p-value = 4.851e-07
## alternative hypothesis: the distribution is asymmetric.
# }
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