KS {gofIG} | R Documentation |
The Kolmogorov-Smirnov test statistic
Description
This function computes the test statistic of the goodness-of-fit test for the inverse Gaussian family in the spirit of Kolmogorov and Smirnov. Note that this tests the composite hypothesis of fit to the family of inverse Gaussian distributions.
Usage
KS(data)
Arguments
data |
a vector of positive numbers. |
Details
Let X_{(j)}
denote the j
th order statistic of X_1, \ldots, X_n
, a sequence of independent observations of a positive random variable X
. Furthermore, let \hat{F}(x) = F(x; \hat{\mu}_n, \hat{\lambda}_n)
, where F
is the distribution function of the inverse Gaussian distribution.
Note that \hat{\mu}_n,\hat{\lambda}_n
are the maximum likelihood estimators for \mu
and \lambda
, respectively, the parameters of the inverse Gaussian distribution.
The null hypothesis is rejected for large values of the test statistic:
KS = \max(D^+, D^-),
where
D^+ = \max_{j=1,\ldots,n} \left( \frac{j}{n} - \hat{F}(X_{(j)}) \right)
and
D^- = \max_{j=1,\ldots,n} \left( \hat{F}(X_{(j)}) - \frac{j-1}{n} \right).
Value
value of the test statistic.
References
Allison, J.S., Betsch, S., Ebner, B., Visagie, I.J.H. (2022) "On Testing the Adequacy of the Inverse Gaussian Distribution". LINK
Examples
KS(rmutil::rinvgauss(20,2,1))