Jackson's test of exponentiality

Usage

Jackson.test(x, method = c("num", "sim", "asymp"), nSamp = 15000)

Arguments

x: numeric vector or matrix.
method: Character: choice of the method used to compute the \(p\)-value. See the Details section.
nSamp: Number of samples used to compute the \(p\)-value if method is "sim".

Value

A list of results.

statistic, p.value

The statistic and \(p\)-value.

df

Number \(n\) of observations.

method

Description of the test implemented, regardless of how the \(p\)-value has been computed.

Details

Compute the Jackson's test of exponentiality. The test statistic is the ratio of weighted sums of the order statistics. Both sums can also be written as weighted sums of the scalings.

The Jackson's statistic for a sample of size \(n\) of the exponential distribution can be shown to be approximately normal. More precisely \(\sqrt{n}(J_n -2)\) has approximately a standard normal distribution. This distribution is used to compute the \(p\)-value when method is "asymp". When method is "num", a numerical approximation of the distribution is used. Finally, when method is "sim" the \(p\)-value is computed by simulating nSamp samples of size length(x) and estimating the probability to have a Jackson's statistic larger than that of the 'observed' x.

Note

Jackson's test of exponentiality works fine for a Lomax alternative (GPD with heavy tail). It then reaches nearly the same power as a Likelihood Ratio (LR) test, see Kozubowski et al. It can be implemented more easily than the LR test because simulated values of the test statistic can be obtained quickly enough to compute the \(p\)-value by simulation.

Author

Yves Deville

References

J. Beirlant and T. de Weit and Y. Goegebeur(2006) "A Goodness-of-fit Statistic for Pareto-Type Behaviour", J. Comp. Appl. Math., 186(1), pp. 99-116.

T.J. Kozubowski, A. K. Panorska, F. Qeadan, A. Gershunov and D. Rominger (2009) "Testing Exponentiality Versus Pareto Distribution via Likelihood Ratio" Comm. Statist. Simulation Comput. 38(1), pp. 118-139.

Examples

set.seed(1234)
x <- rGPD(n = 50, loc = 0, scale = 1, shape = 0.1)
Jackson.test(x, method = "num")$p.value
#> [1] 0.534
Jackson.test(x, method = "asymp")$p.value
#> [1] 0.775
Jackson.test(x, method = "sim")$p.value
#> [1] 0.539