Description Usage Arguments Value Author(s) References See Also Examples

This function performs highly intriguing test for normality using L-skew (*τ_3*) and L-kurtosis (*τ_4*) computed from an input vector of data. The test is simultaneously focused on L-skew and L-kurtosis. Harri and Coble (2011) presented two types of normality tests based on these two L-moment ratios. Their first test is dubbed the *τ_3τ_4* test. Those authors however conclude that a second test dubbed the *τ^2_{3,4}* test “in particular shows consistently high power against [sic] symmetric distributions and also against [sic] skewed distributions and is a powerful test that can be applied against a variety of distributions.”

A sample-size transformed quantity of the sample L-skew (*\hatτ_3*) is

*Z(τ_3) = \hatτ_3 \times \frac{1}{√{0.1866/n + 0.8/n^2}}\mathrm{,}*

which has an approximate Standard Normal distribution. A sample-sized transformation of the sample L-kurtosis (*\hatτ_4*) is

*Z(τ_4)' = \hatτ_4 \times \frac{1}{√{0.0883/n}}\mathrm{,}*

which also has an approximate Standard Normal distribution. A superior approximation for the variate of the Standard Normal distribution however is

*Z(τ_4) = \hatτ_4 \times \frac{1}{√{0.0883/n + 0.68/n^2 + 4.9/n^3}}\mathrm{,}*

and is highly preferred for the algorithms in `tau34sq.normtest`

.

The *τ_3τ_4* test (not implemented in `tau34sq.normtest`

) by Harri and Coble (2011) can be constructed from the *Z(τ_3)* and *Z(τ_4)* statistics as shown, and a square rejection region constructed on an L-moment ratio diagram of L-skew versus L-kurtosis. However, the preferred method is the “Tau34-squared” test *τ^2_{3,4}* that can be developed by expressing an ellipse on the L-moment ratio diagram of L-skew versus L-kurtosis. The *τ^2_{3,4}* test statistic is defined as

*τ^2_{3,4} = Z(τ_3)^2 + Z(τ_4)^2\mathrm{,}*

which is approximately distributed as a *χ^2* distribution with two degrees of freedom. The *τ^2_{3,4}* also is the expression of the ellipical region on the L-moment ratio diagram of L-skew versus L-kurtosis.

1 2 | ```
tau34sq.normtest(x, alpha=0.05, pvalue.only=FALSE, getlist=TRUE,
useHoskingZt4=TRUE, verbose=FALSE, digits=4)
``` |

`x` |
A vector of values. |

`alpha` |
The |

`pvalue.only` |
Only return the p-value of the test and superceeds the |

`getlist` |
Return a list of salient parts of the computations. |

`useHoskingZt4` |
J.R.M. Hosking provided a better approximation |

`verbose` |
Print a nice summary of the test. |

`digits` |
How many digits to report in the summary. |

An **R** `list`

is returned if `getlist`

argument is true. The list contents are

`SampleTau3` |
The sample L-skew. |

`SampleTau4` |
The sample L-kurtosis. |

`Ztau3` |
The Z-value of |

`Ztau4` |
The Z-value of |

`Tau34sq` |
The |

`ChiSq.2df` |
The Chi-squared distribution nonexceedance probability. |

`pvalue` |
The p-value of the test. |

`isSig` |
A logical on whether the p-value is “statistically significant” based on the |

`source` |
The source of the parameters: “tau34sq.normtest”. |

W.H. Asquith

Harri, A., and Coble, K.H., 2011, Normality testing—Two new tests using L-moments: Journal of Applied Statistics, v. 38, no. 7, pp. 1369–1379.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 | ```
HarriCoble <- tau34sq.normtest(rnorm(20), verbose=TRUE)
## Not run:
# If this basic algorithm is run repeatedly with different arguments,
# then the first three rows of table 1 in Harri and Coble (2011) can
# basically be repeated. Testing by WHA indicates that even better
# empirical alphas will be computed compared to those reported in that table 1.
# R --vanilla --silent --args n 20 s 100 < t34.R
# Below is file t34.R
library(batch) # for command line argument parsing
a <- 0.05; n <- 50; s <- 5E5 # defaults
parseCommandArgs() # it will echo out those arguments on command line
sims <- sapply(1:s, function(i) {
return(tau34sq.normtest(rnorm(n),
pvalue.only=TRUE)) })
p <- length(sims[sims <= a])
print("RESULTS(Alpha, SampleSize, EmpiricalAlpha)")
print(c(a, n, p/s))
## End(Not run)
``` |

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