In this article we derive two simple tests for two-way ANOVA under unequal variances requiring some constraints, which cannot be solved by classical regression formulas. We do so by taking the generalized p-value approach and provide explicit formulas to handle the constraints. The first test is an extension of fiducial one-way ANOVA test, which tends to assure the intended size of the test, but tends to be conservative, and the second test reduces that drawback, but may slightly exceed the intended size of the test. The second test is also a generalized test that is numerically equivalent to the parametric bootstrap (PB) test, which contains some unintended glitches in formulas, which we will rectify. Moreover, our approach does not require good estimators of parameters as PB approach does. Hence, one can take our approach dealing with such distributions as Weibull and Gamma that are used in analysis of lifetime data in engineering and analysis of Survival data in studies of public health. By taking similar approach, researchers are also encouraged to derive generalized tests in other ANOVA applications, such as higher-way ANOVA, ANCOVA and MANOVA under heteroscedasticity, especially in Mixed Effects Model applications such as repeated measures (RM) ANOVA, where the PB approach run into difficulties.

It is observed that, for some of the data in engineering and medical fields, the hazard rates increase to a high peak at the beginning and quickly decrease to a low level. In the context of survival analysis, such a hazard rate is called a upside-down bathtub hazard rate. In this paper, we investigated the properties of a model named exponentiated exponential-Pareto distribution. The model was recently proposed and applied to insurance data. We demonstrated that the model has upside-down bathtub-shaped hazard rates with specific choices of parameters. The theoretical properties such as moments, survival functions, and hazard functions were derived. The parameter estimation procedures were also introduced. We then briefly discussed the goodness-of-fit tests of the model with the simulations. Finally, we applied the model to a specific time-to-event data set along with a comparison of the performances with previous existing models. When compared to previous proposed models, the exponentiated exponential-Pareto model demonstrated good performance when fitting to such data sets.

Exponentiated models have been widely used in modeling various types of data such as survival data and insurance claims data. However, the exponentiated composite distribution models have not been explored yet. In this paper, we introduce an improvement of the one-parameter Inverse Gamma-Pareto composite model by exponentiating the random variable associated with the one-parameter Inverse Gamma-Pareto composite distribution function. The goodness-of-fit of the exponentiated Inverse Gamma-Pareto was assessed using three different insurance data sets. The two-parameter exponentiated Inverse Gamma-Pareto model outperforms the one-parameter Inverse Gamma-Pareto model in terms of goodness-of-fit measures for all datasets. In addition, the proposed exponentiated composite Inverse Gamma-Pareto model provides a very good fit with some well-known insurance datasets.

In this paper, we propose a new family of distributions, by exponentiating the random variables associated with the probability density functions of composite distributions. We also derive some mathematical properties of this new family of distributions, including the moments and the limited moments. Specifically, two special models in this family are discussed. Three real datasets were chosen, to assess the performance of these two special exponentiated-composite models. When fitting to these three datasets, these three special exponentiated-composite distributions demonstrate significantly better performance, compared to the original composite distributions.

This article presents two approaches to extend classical MANOVA tests to the unequal covariances case. The first approach is illustrated by extending the classical Wilks test, which is valid only when covariances are equal. Such tests will be based on exact probabilities of certain extreme regions. We will also show how tests numerically equivalent to the parametric bootstrap tests could be easily obtained without using any bootstrap sampling arguments, so that resulting p-values are also based on exact probabilities of well defined extreme regions. Being systematic approaches, by taking similar approaches, researchers should be able to derive generalized tests in MANCOVA, higher-way MANOVA, and in RM MANOVA under heteroscedasticity, in which the parametric bootstrap type approaches run into difficulties.