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Table 1 Special models and the corresponding functions H ( x ; ξ ) and h ( x ; ξ )

From: The Marshall-Olkin extended Weibull family of distributions

Distribution H( x; ξ) h( x; ξ) α ξ References
Exponential (x≥0) x 1 α Johnson et al. (1994)
Pareto (xk) log(x/k) 1/x α k Johnson et al. (1994)
Burr XII (x≥0) log(1+xc) c xc−1/(1+xc) α c Rodriguez (1977)
Lomax (x≥0) log(1+x) 1/(1+x) α Lomax (1954)
Log-logistic (x≥0) log(1+xc) c xc−1/(1+xc) 1 c Fisk (1961)
Rayleigh (x≥0) x 2 2x α Rayleigh (1880)
Weibull (x≥0) x γ γ x γ−1 α γ Johnson et al. (1994)
Fréchet (x≥0) x γ γ x−(γ+1) α γ Fréchet (1927)
Linear failure rate(x≥0) a x+b x2/2 a+b x 1 [a,b] Bain (1974)
Modified Weibull (x≥0) xγexp(λ x) xγ−1 exp(λ x)(γ+λ x) α [γ, λ] Lai et al. (2003)
Weibull extension (x≥0) λ[ exp(x/λ)β−1] β exp(x/λ)β(x/λ)β−1 α [γ, λ, β] Xie et al. (2002)
Phani (0<μ<x<σ<) [(xμ)/(σx)]β β[(xμ)/(σx)]β−1[(σμ)/(σt)2] α [μ, σ, β] Phani (1987)
Weibull Kies (0<μ<x<σ<) ( x μ ) β 1 / ( σ x ) β 2 ( x μ ) β 1 1 ( σ x ) β 2 1 [ β 1 (σx)+ β 2 (xμ)] α [μ, σ, β1, β2] Kies (1958)
Additive Weibull (x≥0) ( x / β 1 ) α 1 + ( x / β 2 ) α 2 ( α 1 / β 1 ) ( x / β 1 ) α 1 1 +( α 2 / β 2 ) ( x / β 2 ) α 2 1 1 [α1, α2, β1, β2] Xie and Lai (1995)
Traditional Weibull (x≥0) xb[ exp(c xd−1)] b xb−1[ exp(c xd)−1]+c d xb+d−1 exp(c xd) α [b, c, d] Nadarajah and Kotz (2005)
Gen. power Weibull (x≥0) [ 1 + ( x / β ) α 1 ] θ 1 (θ α 1 /β) [ 1 + ( x / β ) α 1 ] θ 1 ( x / β ) α 1 1 [α1, β, θ] Nikulin and Haghighi (2006)
Flexible Weibull extension(x≥0) exp(γ xβ/x) exp(γ xβ/x)(γ+β/x2) 1 [γ, β] Bebbington et al. (2007)
Gompertz (x≥0) β−1[ exp(β x)−1] exp(β x) α β Gompertz (1825)
Exponential power (x≥0) exp[(λ x)β]−1 β λ exp[(λ x)β](λ x)β−1 1 [λ, β] Smith and Bain (1975)
Chen (x≥0) exp(xb)−1 b xb−1 exp(xb) α b Chen (2000)
Pham (x≥0) (ax)β−1 β(ax)βlog(a) 1 [a, β] Pham (2002)