TestStatistics {symmetry}

Available test statistics for symmetry tests

Description

The list of implemented test statistics and their functions

Usage

B1(X)

BH2(X)

BHC1(X, k)

BHC2(X, k)

BHI(X)

BHK(X)

CM(X)

FM(X)

HM(X, k)

K2U(X)

K2(X)

KS(X)

SGN(X)

WCX(X)

MGG(X)

MI(X)

MOI(X, k)

MOK(X, k)

NAC1(X, k)

NAC2(X, k)

NAI(X, k)

NAK(X, k)

RW(X)

Arguments

X

the numeric vector for which to calculate the test statistic

k

the 'k' parameter in the formula (if applicable)

Details

Below is a list of the implemented test statistics in the package. Each statistic is listed by it's name, a code string (e.g. 'B1', CM','MOI') and the formula of the statistic which is evaluated. The code string is used as an argument to the symmetry_test function. Some statistics depend on a parameter 'k' which can be seen from the formulas and is also passed as an argument.

Each statistic is implemented as a function with the same name as the code string, so the name of the function is passed as the argument "stat" to the symmetry_test function

Value

The value of the test statistic.

Test statistics

The list of available statitics in the format "code(s) : name (reference)"

• MI : The Mira test statistic (see (Mira 1999))

• CM : The Cabilio–Masaro test statistic (see (Cabilio and Masaro 1996))

• MGG : The Miao, Gel and Gastwirth test statistic (see (Miao et al. 2006))

• B1 : The \sqrt{b_1} test statistic (see (Milošević and Obradović 2019))

• KS : The Kolmogorov–Smirnov test statistic (see (Milošević and Obradović 2019))

• SGN : The Sign test statistic (see (Milošević and Obradović 2019))

• KS : The Wilcoxon test statistic (see (Milošević and Obradović 2019))

• FM : The characterization based test statistic (see (Feuerverger et al. 1977))

• RW : The Rothman-Woodroofe test statistic (see (Gaigall 2019))

• BHI : The Litvinova test statistic (see (Litvinova 2001))

• BHK : The Baringhaus and Henze supremum-type test statistic (see (Baringhaus and Henze 1992))

• BH2 : The Baringhaus-Henze test statistic (see (Baringhaus and Henze 1992))

• MOI and MOK : The Milošević and Obradović test statistics (see (Milošević and Obradović 2016))

• NAI and NAK : The Nikitin and Ahsanullah test statistics (see (Nikitin and Ahsanullah 2015))

• K2 and K2U : The Božin, Milošević, Nikitin and Obradović Kolmogorov type statistics based on V- and U- statistics respectively (see (Božin et al. 2018))

• NAC1, NAC2, BHC1 and BHC2 : The Allison and Pretorius test statistics (see (Allison and Pretorius 2017))

References

Allison JS, Pretorius C (2017). “A Monte Carlo evaluation of the performance of two new tests for symmetry.” Computational Statistics, 32(4), 1323–1338. doi:10.1007/s00180-016-0680-4.

Baringhaus L, Henze N (1992). “A characterization of and new consistent tests for symmetry.” Communications in statistics-theory and methods, 21(6), 1555–1566. doi:10.1080/03610929208830863.

Božin V, Milošević B, Nikitin Y, Obradović M (2018). “New Characterization-Based Symmetry Tests.” Bulletin of the Malaysian Mathematical Sciences Society, 10–1007. doi:10.1007/s40840-018-0680-3.

Cabilio P, Masaro J (1996). “A simple test of symmetry about an unknown median.” Canadian Journal of Statistics, 24(3), 349–361. doi:10.2307/3315744.

Feuerverger A, Mureika RA, others (1977). “The empirical characteristic function and its applications.” The Annals of Statistics, 5(1), 88–97. doi:10.1214/aos/1176343742.

Gaigall D (2019). “Rothman-Woodroofe symmetry test statistic revisited.” Computational Statistics & Data Analysis, 106837.

Litvinova VV (2001). “New nonparametric test for symmetry and its asymptotic efficiency.” Vestnik St. Petersburg University Mathematics, 34(4), 12–14.

Miao W, Gel YR, Gastwirth JL (2006). “A new test of symmetry about an unknown median.” In Random Walk, Sequential Analysis And Related Topics: A Festschrift in Honor of Yuan-Shih Chow, 199–214. World Scientific.

Milošević B, Obradović M (2016). “Characterization based symmetry tests and their asymptotic efficiencies.” Statistics & Probability Letters, 119, 155–162.

Milošević B, Obradović M (2019). “Comparison of efficiencies of some symmetry tests around an unknown centre.” Statistics, 53(1), 43–57.

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.

Nikitin Y, Ahsanullah M (2015). “New U-empirical Tests of Symmetry Based on Extremal Order Statistics, and their Efficiencies.” In Mathematical Statistics and Limit Theorems, 231–248. Springer. doi:10.1007/978-3-319-12442-1_13.