Analysis of covariance (ANCOVA) is the standard procedure for comparing several treatments when the response variable depends on one or more covariates. We consider the problem of testing the equality of treatment effects when the variances are not assumed to be equal. It is well known that classical F test is not robust with respect to the assumption of equal variances and may lead to misleading conclusions if the variances are not equal. Ananda ( 1998 Ananda , M. M. A. ( 1998 ). Bayesian and non-Bayesian solutions to analysis of covariance models under heteroscedasticity . J. Econometrics 86 : 177 – 192 . [Crossref], [Web of Science ®] [Google Scholar] ) developed a generalized F test for testing the equality of treatment effects. However, simulation studies show that the actual size of this test can be much higher than the nominal level when the sample sizes are small, particularly when the number of treatments is large. In this article, we develop a test using the parametric bootstrap approach of Krishnamoorthy et al. ( 2007 Krishnamoorthy , K. , Lu , F. , Mathew , T. ( 2007 ). A parametric bootstrap approach for ANOVA with unequal variances: Fixed and random models . Computat. Statist. Data Anal. 51 : 5731 – 5742 . [Crossref], [Web of Science ®] [Google Scholar] ). Our simulations show that the actual size of our proposed test is close to the nominal level, irrespective of the number of treatments and sample sizes. Our simulations also indicate that our proposed PB test is more robust, with respect to the assumption of normality, than the generalized F test. Therefore, our proposed PB test provides a satisfactory alternative to the generalized F test.

In this article, a new class of distributions is introduced, which generalizes the linear failure rate distribution and is obtained by compounding this distribution and power series class of distributions. This new class of distributions is called the linear failure rate-power series distributions and contains some new distributions such as linear failure rate-geometric, linear failure rate-Poisson, linear failure rate-logarithmic, linear failure rate-binomial distributions, and Rayleigh-power series class of distributions. Some former works such as exponential-power series class of distributions, exponential-geometric, exponential-Poisson, and exponential-logarithmic distributions are special cases of the new proposed model. The ability of the linear failure rate-power series class of distributions is in covering five possible hazard rate function, that is, increasing, decreasing, upside-down bathtub (unimodal), bathtub and increasing-decreasing-increasing shaped. Several properties of this class of distributions such as moments, maximum likelihood estimation procedure via an EM-algorithm and inference for a large sample, are discussed in this article. In order to show the flexibility and potentiality, the fitted results of the new class of distributions and some of its submodels are compared using two real datasets.

This article introduces a five-parameter lifetime model called the McDonald Gompertz (McG) distribution to extend the Gompertz, generalized Gompertz, generalized exponential, beta Gompertz, and Kumaraswamy Gompertz distributions among several other models. The hazard function of new distribution can be increasing, decreasing, upside-down bathtub, and bathtub shaped. We obtain several properties of the McG distribution including moments, entropies, quantile, and generating functions. We provide the density function of the order statistics and their moments. The parameter estimation is based on the usual maximum likelihood approach. We also provide the observed information matrix and discuss inferences issues. The flexibility and usefulness of the new distribution are illustrated by means of application to two real datasets.