ABEV1 {gofIG} | R Documentation |
This function computes the first test statistic of the goodness-of-fit tests for the inverse Gaussian family due to Allison et al. (2022). Two different estimation procedures are implemented, namely the method of moment and the maximum likelihood method.
ABEV1(data, a = 10, meth = "MME")
data |
a vector of positive numbers. |
a |
positive tuning parameter. |
meth |
method of estimation used. Possible values are |
The numerically stable test statistic for the first Allison-Betsch-Ebner-Visagie test is defined as:
ABEV1_{n,a} = \frac{1}{4n} \sum_{j,k=1}^{n} \left( \hat{\varphi}_n + \frac{3}{Y_{n,j}} - \frac{\hat{\varphi}_n}{Y_{n,j}^2} \right) \left( \hat{\varphi}_n + \frac{3}{Y_{n,k}} - \frac{\hat{\varphi}_n}{Y_{n,k}^2} \right) h_{1,a}(Y_{n,j}, Y_{n,k})
- 2 \left( \hat{\varphi}_n + \frac{3}{Y_{n,j}} - \frac{\hat{\varphi}_n}{Y_{n,j}^2} \right) h_{2,a}(Y_{n,j}, Y_{n,k})
- 2 \left( \hat{\varphi}_n + \frac{3}{Y_{n,k}} - \frac{\hat{\varphi}_n}{Y_{n,k}^2} \right) h_{2,a}(Y_{n,k}, Y_{n,j})
+ \frac{4}{a} e^{-a \max(Y_{n,j}, Y_{n,k})},
with \hat{\varphi}_n = \frac{\hat{\lambda}_n}{\hat{\mu}_n}
, where \hat{\mu}_n,\hat{\lambda}_n
are consistent estimators of \mu, \lambda
, respectively, the parameters of the inverse Gaussian distribution. Furthermore Y_{n,j} = \frac{X_j}{\hat{\mu}_n}
, j = 1,...,n
, for (X_j)_{j = 1,...,n}
, a sequence of independent observations of a positive random variable X
. The functions h_{i,a}(s,t)
, i = 1,2
, are defined in Allison et al. (2022), section 5.1.
The null hypothesis is rejected for large values of the test statistic ABEV1_{n,a}
.
value of the test statistic.
Allison, J.S., Betsch, S., Ebner, B., Visagie, I.J.H. (2022) "On Testing the Adequacy of the Inverse Gaussian Distribution". LINK
ABEV1(rmutil::rinvgauss(20,2,1),a=10,meth='MLE')