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

Exponential (x≥0)

x

1

α

∅

Johnson et al. (1994)

Pareto (x≥k)

log(x/k)

1/x

α

k

Johnson et al. (1994)

Burr XII (x≥0)

log(1+x^c)

c x^c−1/(1+x^c)

α

c

Rodriguez (1977)

Lomax (x≥0)

log(1+x)

1/(1+x)

α

∅

Lomax (1954)

Log-logistic (x≥0)

log(1+x^c)

c x^c−1/(1+x^c)

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)

x^γexp(λ x)

x^γ−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<∞)

[(x−μ)/(σ−x)]^β

β[(x−μ)/(σ−x)]^β−1[(σ−μ)/(σ−t)²]

α

[μ, σ, β]

Phani (1987)

Weibull Kies (0<μ<x<σ<∞)

${(x-\mu )}^{{\beta }_1}/{(\sigma -x)}^{{\beta }_2}$

${(x-\mu )}^{{\beta }_1-1}{(\sigma -x)}^{-{\beta }_2-1}\left[{\beta }_1\right(\sigma -x)+{\beta }_2(x-\mu \left)\right]$

α

[μ, σ, β₁, β₂]

Kies (1958)

Additive Weibull (x≥0)

${(x/{\beta }_1)}^{{\alpha }_1}+{(x/{\beta }_2)}^{{\alpha }_2}$

$({\alpha }_1/{\beta }_1){(x/{\beta }_1)}^{{\alpha }_1-1}+({\alpha }_2/{\beta }_2){(x/{\beta }_2)}^{{\alpha }_2-1}$

1

[α₁, α₂, β₁, β₂]

Xie and Lai (1995)

Traditional Weibull (x≥0)

x^b[ exp(c x^d−1)]

b x^b−1[ exp(c x^d)−1]+c d x^b+d−1 exp(c x^d)

α

[b, c, d]

Nadarajah and Kotz (2005)

Gen. power Weibull (x≥0)

${[1+{(x/\beta )}^{{\alpha }_1}]}^{\theta }-1$

$(\theta {\alpha }_1/\beta ){[1+{(x/\beta )}^{{\alpha }_1}]}^{\theta -1}{(x/\beta )}^{{\alpha }_1}$

1

[α₁, β, θ]

Nikulin and Haghighi (2006)

Flexible Weibull extension(x≥0)

exp(γ x−β/x)

exp(γ x−β/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(x^b)−1

b x^b−1 exp(x^b)

α

b

Chen (2000)

Pham (x≥0)

(a^x)^β−1

β(a^x)^βlog(a)

1

[a, β]

Pham (2002)