HK1 function

The first Henze-Klar test statistic

This function computes the first test statistic of the goodness-of-fit test for the inverse Gaussian family due to Henze and Klar (2002).

HK1(data, a = 0)

Arguments

• data: a vector of positive numbers.
• a: positive tuning parameter.

Returns

value of the test statistic

Details

The representation of the first Henze-Klar test statistic used for computation is given by:

with $\hat{\varphi}_n = \frac{\hat{\lambda}_n}{\hat{\mu}_n}$, where $\hat{\mu}_n,\hat{\lambda}_n$ are the maximum likelihood estimators for $\mu$ and $\lambda$, respectively, the parameters of the inverse Gaussian distribution. Furthermore $\hat{Z}_{jk} = \hat{\varphi}_n(Y_j + Y_k +a)$, where $Y_i = \frac{X_i}{\hat{\mu}_n}$ for $(X_i)_{i = 1,...,n}$, a sequence of independent observations of a nonnegative random variable $X$. To ensure numerical stability of the implementation the exponentially scaled complementary error function $\text{erfce}(x)$ is used: $\text{erfce}(x) = \exp{(x^2)}\text{erfc}(x)$, with $\text{erfc}(x) = 2\int_x^\infty \exp{(-t^2)}dt/\pi$. The null hypothesis is rejected for large values of the test statistic $HK_{n,a}^{(1)}$.

Examples

HK1(rmutil::rinvgauss(20,2,1))

References

Henze, N. and Klar, B. (2002) "Goodness-of-fit tests for the inverse Gaussian distribution based on the empirical Laplace transform", Annals of the Institute of Statistical Mathematics, 54(2):425-444. tools:::Rd_expr_doi("https://doi.org/10.1023/A:1022442506681")