Abstract
In this article, we introduced a new extension of the binomial-exponential 2 distribution. We discussed some of its structural mathematical properties. A simple type Copula-based construction is also presented to construct the bivariate- and multivariate-type distributions. We estimated the model parameters via the maximum likelihood method. Finally, we illustrated the importance of the new model by the study of two real data applications to show the flexibility and potentiality of the new model in modeling skewed and symmetric data sets.
Highlights
Introduction and MotivationThe monotonicity of the hazard rate function (HRF) of a life model plays an important role in modeling failure time data
The model of [3] is named the binomial-exponential[2] (BE-2) model, which is constructed as the distribution of the random sum (RSum) of independent exponential random variables (IID RVs) when the sample size (n) has a zero truncated binomial (ZTB) model
We notice that when β = 0 we get the standard exponential distribution, and when β = 1 the BE-2 distribution reduces to the gamma distribution with shape parameter 2 and scale parameter α
Summary
The monotonicity of the hazard (failure) rate function (HRF) of a life model plays an important role in modeling failure time data. The probability density function (PDF) corresponding to (1) can be expressed as:. We notice that when β = 0 we get the standard exponential distribution, and when β = 1 the BE-2 distribution reduces to the gamma distribution with shape parameter 2 and scale parameter α. The BE-2 distribution has a PDF whose shape is like those of Ga, W, WhE, and EE distributions. [4] proposed a general family of distributions called the Marshall–Olkin (MO-G) family of distributions. The new PDF of the proposed lifetime model distribution can be right-skewed, symmetric, and left-skewed with many different useful shapes (see Figure 1), and this means that the new model will be suitable for modeling different real data sets, and the HRF of the new model exhibits many important HRF shapes such as the “increasing-constant”, “decreasing”, “increasing”, “constant”, and “bathtub” shapes (see Figure 2). The proposed lifetime model is much better than many competitive versions of the exponential model, such as the odd Lindley exponential, the Marshall–Olkin exponential, moment exponential, the logarithmic Burr–Hatke exponential, the generalized Marshall–Olkin exponential, beta exponential, the Marshall–Olkin–Kumaraswamy exponential, the Kumaraswamy exponential, and the Kumaraswamy–Marshall–Olkin exponential, so the new lifetime model may be a good alternative to these models in modeling relief times and survival times data sets
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