Reliability Modeling for Systems with Multiple Degradation Processes Using Inverse Gaussian Process and Copulas

Table 1

Summary of some one-parameter Archimedean copulas.

Copulas	𝐶(𝑢,𝑣)	Parameter
Gumbel copula	exp−(−ln𝑢)^𝛼+(−ln𝑣)^𝛼1/𝛼	𝛼∈[1,∞)
Frank copula	−1𝛼ln1+(𝑒^−𝛼𝑢−1)(𝑒^−𝛼𝑣−1)𝑒^−𝛼−1	𝛼∈(−∞,∞)⧵{0}
Clayton copula	max(𝑢^−𝛼+𝑣^−𝛼−1,0)^−1/𝛼	𝛼∈[−1,∞)⧵{0}
Joe copula	1−(1−𝑢)^𝛼+(1−𝑣)^𝛼−(1−𝑢)^𝛼(1−𝑣)^𝛼1/𝛼	𝛼∈[1,∞)
Amh copula	𝑢𝑣1−𝛼(1−𝑢)(1−𝑣)	𝛼∈[−1,1]