A multitude of real-world phenomena can be elegantly captured as proportions, ratios, or fractions nestled within the bounded interval (0, 1). These captivating representations are not merely abstract concepts; they reflect the intricate relationships found in various fields such as biology, where the delicate balance of ecosystems is analyzed; finance, where the flow of market ratios unfold; and mortality rates, which provide profound insights into human health and longevity. Additionally, recovery rates in medical science showcase the resilience of life, while economics delves into the nuanced distributions of wealth and resources. Engineering and hydrology further enrich this tapestry, modeling everything from structural integrity to the flow of water in our environments. The measurement sciences, too, rely on these continuous distributions, breathing life into data that inform our understanding of the world.

Some of these distributions include: the Johnson SB distribution [1], Beta distribution [2], Unit Johnson distribution [3], Topp-Leone distribution [4], Unit Gamma distribution [5–8], Unit Logistic distribution [9], Kumaraswamy distribution [10], Unit Burr-III distribution [11], Unit Modified Burr-III distribution [12], Unit Burr-XII distribution [13], Unit-Gompertz distribution [14], Unit-Lindley distribution [15], Unit-Weibull distribution [16], and Unit Muth distribution [17].

Generalizing a distribution by adding new parameters can undeniably enrich and amend the estimation process in numerous manners. First, adding parameters empowers the distribution to apprehend different data conducts such as skewness, kurtosis and different tail behaviors. For example, this is glorified when generalized gamma [18] distribution extends the gamma distribution to accommodate modeling wide range of data with different characteristics. The extra parameter in the generalized Pareto [19] distribution controlling the tail behavior facilitates it to better analyze the extreme values. Second, the newly added parameters amplify the goodness of fit to empirical data and reduce the systematic bias. An example for such an effect, when the data exhibit extreme values; the four parameter Beta distribution [20] extending the standard beta distribution can better model these heavy tail features data. Third, extending the distribution with new parameters augments it flexibility to align more properly with the basic data hence diminishing the bias in parameter estimation, obtaining more efficient estimators with minor variance and improve the implementation of MLE. The Generalized Weibull Distribution [21] enhances the estimates of failures rates in reliability studies. Fourth, the newly introduced parameters behave as regularizers that take control against over-fitting thereby improving stability in estimation. For example, the additional shape parameter in the Exponentiated Weibull Distribution [21] enables the modeling of both decreasing and increasing failure rates. Likewise, the skewness parameter in the Skew-Normal Distribution [22] promotes the distribution to better model the asymmetric data. Fifth, in real-world analysis like Generalized Logistic Distribution [23], implanting a shape parameter can regulate the rate of decay in growth models. The incorporated extra parameter in the Generalized Gamma [24] Distribution aids modeling diverse hazard rates.

The generalization of the unit distribution can be achieved through different mechanisms like power transformation to obtain the power Johnson B [25], power Generalized Johnson SB [26], and power unit inverse Lindley distribution [27]. The generalization can also be conducted using the T-X family method (transformed-transformer mechanism) like the transmuted power unit inverse Lindley distribution [28], Kumaraswamy generalized family of distribution [29], generalized distribution based on T-Topp -Leone family of distributions [30], and the generalized unit half logistic geometric distributions [31]. The author discusses in this paper a different method for adding a new parameter to the unit distribution using the general formula for the order statistics.

It's important to recognize the limitations of this generalization. While adding more parameters can enhance a model's complexity and capability, it also brings challenges that must be carefully considered. Increased parameters may complicate computations and require sophisticated optimization techniques to manage effectively. Additionally, a larger set of parameters can introduce identifiability issues, where the effects of certain parameters become unclear. Many parameters may even lack practical significance, questioning their value in real-world applications. Therefore, it's crucial to weigh the advantages of incorporating new parameters against the potential difficulties they could introduce, ensuring that we make informed decisions in our modeling approaches.

This paper is structured into several sections for clarity and coherence. Section 2 provides a comprehensive discussion of the methodology employed to derive the new distribution. Section 3 delves into its fundamental characteristics, including the probability density function (PDF), cumulative distribution function (CDF), survival function (S), hazard function (HF), reversed hazard function (RHF), and quantile function. Section 4 demonstrates the maximum likelihood estimation method. Section 5 offers an in-depth discussion that encompasses an analysis of real data as well as a detailed examination of the findings. In conclusion, Section 6 provides a comprehensive overview of our findings and offers valuable recommendations for future research, inviting further exploration and innovation in the field.

Theorem 2: the cumulative distribution function (CDF) of the generalized odd MBUR is:

P(Y < y) = I_w(n+1, n+1) for version 1 and P(Y<y)=Iw(n+12,n+12) for version 2.

Proof: for version 1:

Apply the transformation of Equation 12 and substitute in Equation 13 yields Equation 14

For version 2:

Apply the transformation of Equation 12 and substitute in equation 15 yields Equation 16

Lemma 1: the survival function (S) for version 1 is shown in Equation 17

Lemma 2: the survival function (S) for version 2 is shown in Equation 18

Lemma 3: The Hazard function or rate (HF or hr) for version 1 is shown in Equation 19

Lemma 4: The Hazard function or rate (HF or hr) for version 2 is shown in Equation 20

Lemma 5: the reversed hazard function (RHF) or reversed hazard rate (rhr) for version 1 is shown in Equation 21

Lemma 6: the reversed hazard function (RHF) or reversed hazard rate (rhr) for version 2 is shown in Equation 22

The quantile function of the distribution has no closed explicit form.

Theorem 3: the rth raw moment of the first version of the distribution is given by

Proof: the expectation of the r^th moment in Equation 23 is obtained with the help of the transformation mentioned in Equation 12

Theorem 4: the rth raw moment of the second version of the distribution is given by

Proof: the expectation of the rth moment in Equation 24 is obtained with the help of the transformation mentioned in Equation 12

Figures 1–4 show PDF, CDF, survival function, Hazard function for the distribution at different parameters values. See Supplementary material 1 for more figures.

The addition of a new parameter to the previously studied MBUR distribution enhances the capability of MBUR to fit the data. Two versions for generalizing the MBUR were discussed by the author. Both have a non-explicit closed form of the CDF, but they can be expressed using special function. Subsequently, the quantile functions for both versions are not expressed in closed form. Also the rth moments of both versions are expressed with the aid of special function. This is considered as a limitation for being used in applications like median based quantile regression or mean based regression in generalized linear model. The advantage of adding this new parameter is that it helps MBUR to fit near symmetric data. Moreover, the two versions of this Generalized Odd MBUR exhibit a new shape for the hazard rate in the form of an oscillating pattern at the end of the distribution before approaching infinity and at different values of the random variable depending on the level of the alpha and the n parameter (See Supplementary materials 1).

The analysis of the datasets shows that adding a new parameter to the MBUR model increases its flexibility, allowing it to accommodate diverse shapes of data with different characteristics, such as skewness, kurtosis, and various tail behaviors. This newly introduced parameter enhances the estimation process by improving validity indices like AIC, CAIC, BIC, and HQIC. Additionally, it enhances the goodness of fit by reducing test statistics such as the (AD), (CVM), and (KS) tests. Furthermore, it increases the value of the Log-Likelihood. The two versions of the generalization yield different values for the parameter (n), but they have equal values for the parameter (alpha). The variances of the estimated (alpha) obtained from the two versions are identical, and the covariance between the two parameters is minimal, which is significantly lower than the covariance observed when fitting distributions like the Beta and the Kumaraswamy. The determinant of the estimated variance-covariance matrix obtained from fitting GOMBUR-1 is minimal, almost the lowest compared to that achieved after fitting the Beta and Kumaraswamy distributions. In Supplementary material 3, the author discusses various new generalized unit distributions utilizing the general formula for the order statistics. These distributions can be candidates for various applications in future work.

In this paper, the Nelder-Mead optimizer was employed for Maximum Likelihood Estimation (MLE) of the parameters. Future work may explore Bayesian inference procedures. Methods like Maximum product of spacing, least square and weighted least square methods can be attempted but the encountered limitation is that the CDF does not have a well-closed form. This limitation will impose some cumbersome calculations to estimate the parameters. The generalized method of moments using moment based approach by equating the sample moments with the population moments can offer an alternative for parameter estimation. But this will also face some difficult calculations as the moments require special functions. Applying derivative based algorithms like Quasi-Newton methods could be another solution for the optimization procedure.