Mathematics Ph.D. Dissertations

The Energy Goodness-Of-fit Test for the Inverse Gaussian Distribution

Date of Award


Document Type


Degree Name

Doctor of Philosophy (Ph.D.)



First Advisor

Maria Rizzo (Advisor)

Second Advisor

Neil Baird (Other)

Third Advisor

Wei Ning (Committee Member)

Fourth Advisor

Junfeng Shang (Committee Member)


The inverse Gaussian distribution is one of the most widely used distributions for modelling positively skewed data. Areas of its application includes lifetime models, cardiology, linguistics, employment service, labor dispute, finance, electrical networks hydrology, demography, and meteorology (Folks and Chhikara, 1978; Seshadri, 1993; Norman, Kotz, and Balakrishnan, 1994;Bardsley, 1980). When making statistical inference with the inverse Gaussian distribution, it is animportant practice to determine if the data fits the inverse Gaussian family.

In this dissertation two new goodness-of-fit test based on energy distance are proposed. The first test called the energy goodness-of-fit test is consistent against general alternatives. Under the null hypothesis, the test statistic converges to a weighted sum of Chi-square random variable. The second proposed test is based on the independence characterization of the inverse Gaussian distribution. The proposed test utilize the distance correlation as a measure of dependence. The distance correlation test is implemented as non-parametric permutation test and requires the observations to be exchangeable under the null hypothesis.

Simulation results indicates that, the energy goodness-of-fit test is more powerful compared to other test considered. However, the distance correlation based test outperforms other tests for large shape parameter values of the alternative distributions. The distance correlation based test is extended to the Pareto distribution, which is characterized by the independence of of the rth order statistic X(r) and the ratio X(s)/X(r). Simulation results indicate that, the distance correlation based test has high power when the effective sample size is large.