Abstract
In this paper, we introduce two families of general bivariate distributions. We refer to these families as general bivariate exponential family and general bivariate inverse exponential family. Many bivariate distributions in the literature are members of the proposed families. Some properties of the proposed families are discussed, as well as a characterization associated with the stress-strength reliability parameter, R, is presented. Concerning R, the maximum likelihood estimators and a simple estimator with an explicit form depending on some marginal distributions are obtained in case of complete sampling. When the stress is censored at the strength, an explicit estimator of R is also obtained. The results obtained can be applied to a variety of bivariate distributions in the literature. A numerical illustration is applied on some well-known distributions. Finally a real data example is presented to fit one of the proposed models.
Highlights
Mokhlis et al [1] presented two forms of survival functions, given byFðu; θ; cÞ 1⁄4 e−θg1ðu;cÞ; ð1ÞFðu; β; cÞ 1⁄4 1−e−βg2ðu;cÞ; ð2Þ where g1(u; c) does not contain θ, θ ∈ Θ, and g2(u; c) does not contain β ∈ β, c ∈ C, { Θ, β, and C} are the parametric spaces, where g1(u; c) is a continuous, monotone increasing, and differential function such that g1(u; c) → 0 as u → 0 and g1(u; c) → ∞ as u → ∞, while g2(u; c) is continuous, monotone decreasing and differential function such that g2(u; c) → 0 as u → ∞ and g2(u; c) → ∞ as u → 0
We introduce a simple estimator of R, depending on the marginal distributions of X and min{X, Y} for the General bivariate exponential distribution (BEF) and depending on the marginal distributions of Y and max{X, Y} for the General bivariate inverse exponential distribution (BIEF)
Some distributions in the literature belong to these families, such as the M-O bivariate exponential distribution, Marshall and Olkin [15], and bivariate Rayleigh distribution, Pak et al [9]
Summary
Fðu; β; cÞ 1⁄4 1−e−βg2ðu;cÞ; ð2Þ where g1(u; c) does not contain θ, θ ∈ Θ, and g2(u; c) does not contain β ∈ β, c ∈ C, { Θ, β, and C} are the parametric spaces, where g1(u; c) is a continuous, monotone increasing, and differential function such that g1(u; c) → 0 as u → 0 and g1(u; c) → ∞ as u → ∞, while g2(u; c) is continuous, monotone decreasing and differential function such that g2(u; c) → 0 as u → ∞ and g2(u; c) → ∞ as u → 0. Many bivariate distributions in the literature have forms of the proposed models, for example, M-O bivariate exponential distribution, Marshal and Olkin [15], and the bivariate Rayleigh distribution introduced by Pak et al [9] for the BEF and bivariate inverse Weibull and bivariate Burr type III for the BIEF. The above values are less than the critical value D0.05 ≅ 0.2099, for n = 42, so that each of exponential distribution and inverse exponential distribution is an appropriate fit for the given data This means that there may exist three independent random variables, say Ui, i = 1, 2, 3, with EF or IEF X = min {U0, U1} or max{U0, U1} and Y = min {U0, U2} or max{U0, U2}.
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