The inverted Gompertz-Fréchet distribution with applications

Abstract

The probability distribution is an important aspect of probability theory because of its vast relevance in almost all human disciplines. Its applicability in finance, medicine, agriculture, actuarial science, demography, and econometrics, to mention but a few, is highly commendable. The use of probability distributions to model real-life data is an age-old practice, but most of the standard distributions are not flexible enough to model emergent real-life occurrences. This shortcoming gave birth to diverse extensions of the standard distribution. In this work, the Inverted Gompertz-Fréchet (IGoFre) distribution is developed by transforming the independent variable of Gompertz-Fréchet, and one great feature of this model is its capacity to model both positively and negatively skewed datasets with increasing and decreasing hazard rates. It is a distribution whose random variable follows the reciprocal of the Gompertz-Fréchet distribution. The proposed probability distribution is developed with the aim of modeling real data with non-monotonic failure rates. The proposed distribution does not involve the addition of extra parameters, thereby removing the difficulties encountered in deriving its properties. The statistical and mathematical properties of the new distribution, such as the survival function, hazard function, distributions of the minimum and maximum order statistics, moments, mean, median, variance, skewness, and kurtosis, and Renyi entropy are derived. The method of maximum likelihood was used to estimate the model's parameters. Tables of percentage points, which could be of immense benefit for the test of the hypothesis, were generated for different values of the parameters. The proposed distribution was applied to two real-life data sets. The results revealed that the IGoFre distribution performs better than the Gompertz-Weibull, Gompertz-Fréchet, Gompertz Bur XII, and Gompertz-Lomax distributions for the datasets. Also, it was discovered that the proposed model is suitable for modeling both positively skewed and negatively skewed datasets.