Abstract
Statistical distributions already in existence are not the most appropriate model that adequately describes real-life data such as those obtained from experimental investigations. Therefore, there are needs to come up with their extended forms to give substitutive adaptable models. By adopting the method of Transformed-Transformer family of distributions, an extension of Exponentiated Rayleigh distribution titled Gompertz- Exponentiated Rayleigh (GOM-ER) distribution was proposed and proved to be valid. Some properties of the new distribution including random number generator, quartiles, distribution of smallest and largest order statistics, reliability function, hazard rate function, cumulative or integrated hazard function, odds function, non-central moments, moment generating function, mean, variance and entropy measures were derived. Using the methods of maximum likelihood and maximum product of spacing, the four unknown parameters were estimated. Shapes of the hazard function depicts that GOM-ER is a distribution that is strictly increasing while those of the PDF depicts that GOM-ER can be skewed or symmetrical. Two datasets were fitted to determine the flexibility of GOM-ER. Simulation study evaluates the consistency, accuracy and unbiasedness of the GOM-ER parameter estimates obtained from the two frequentist estimation methods adopted.
Highlights
Long time ago, probability distributions are known to be used by researchers to fit any given data adequately
For maximizing the geometric mean of spacings between this reason, we propose a four parameter lifetime distribution cumulative distribution function in close observations are called Gompertz-Exponentiated Rayleigh (GOM-ER)
Using different parameters values and sample sizes (25-1000), the estimation methods were compared based on bias and root mean square error (RMSE) of the estimators
Summary
Probability distributions are known to be used by researchers to fit any given data adequately. Dating back to 19th century, several methods of defining probability distributions have been proposed , some of which include; ”Method of transformation” by (Johnson, 1949), ”Method of Generating Skewed Distributions” by (Azzalini, 1985), modified by (Azzalini, 1986), ”the Method of adding parameters” by (Mudholkar and Srivastava, 1993) and (Marshal and Olkin), ”Beta-Generated” by (Eugene et al, 2002) and (Jones, 2009), ”Kumaraswamy-Generated” by (Cordeiro and de Castro, 2011) and lastly the modern and most used method in the recent decade ”Transformed- Transformer(T-X)” by (Alzaatreh et al, 2013) modified to ”Exponentiated (T-X)” by (Alzaghal et al, 2013) These methods expand families of distributions for more flexibility and applications. Adopting the method of ”Transformed- Transformer(T-X)”, a handful families of distributions namely “Weibull-G” by (Bourgiugnon et al, 2014) “Kumaraswamy-G” by (Cordeiro and De Castro, 2011)” “The generalized transmuted-G” by (Nofal et al, 2017), “Gompertz-G” by (Alizadeh et al, 2017),
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