Validation of Burr XII inverse Rayleigh model via a modified chi-squared goodness-of-fit test

In this work, we propose a new three parameter distribution called the Burr XII inverse Rayleigh model, this model is a generalization of the inverse Rayleigh distribution using the Burr XII family introduced by Cordeiro et al. [The burr XII system of densities: properties, regression model and applications. J. Stat. Comput. Simul. 88 (2018), pp. 432-456]. After studying the statistical characterization of this model, we construct a modified chi-squared goodness-of-fit test based on the Nikulin-Rao-Robson statistic in the presence of two cases: censored and complete data. We describe the theory and the mechanism of the $$ statistic test which can be used in survival and reliability data analysis. We use the maximum likelihood estimators based on initial non grouped data. Then, we conduct numerical simulations to reinforce the results. For showing the applicability of our model in various fields, we illustrate it and the proposed test by applications to two real data sets for complete data case and two other data sets in the presence of right censored.