Journal of Probability and Statistics

Research Article

Computing the Moments of Order Statistics from Independent Nonidentically Distributed Exponentiated Frechet Variables

Table 1

𝐼 𝑗 ( 𝑘 ) using (3.3) when 𝑛 = 3 .
j 𝐼 𝑗 ( 𝑘 )
1 𝐼 1 ( 𝑘 ) = 𝑘 𝜎 𝑘 Γ ( − 𝑘 / 𝜆 ) 𝜆 𝛼 1 ∑ 𝑚 1 = 0 ( − 1 ) 𝑚 1 Γ ( 𝛼 1 + 1 ) 𝑚 1 ! Γ ( 𝛼 1 − 𝑚 1 + 1 ) ( 𝑚 1 ) 𝑘 / 𝜆 + 𝛼 2 ∑ 𝑚 1 = 0 ( − 1 ) 𝑚 1 Γ ( 𝛼 2 + 1 ) 𝑚 1 ! Γ ( 𝛼 2 − 𝑚 1 + 1 ) ( 𝑚 1 ) 𝑘 / 𝜆
+ 𝛼 3 ∑ 𝑚 1 = 0 ( − 1 ) 𝑚 1 Γ ( 𝛼 3 + 1 ) 𝑚 1 ! Γ ( 𝛼 3 − 𝑚 1 + 1 ) ( 𝑚 1 ) 𝑘 / 𝜆
2 𝐼 2 ( 𝑘 ) = 𝑘 𝜎 𝑘 Γ ( − 𝑘 / 𝜆 ) 𝜆 𝛼 1 ∑ 𝑚 1 = 0 𝛼 2 ∑ 𝑚 2 = 0 ( − 1 ) 𝑚 1 + 𝑚 2 Γ ( 𝛼 1 + 1 ) Γ ( 𝛼 2 + 1 ) 𝑚 1 ! 𝑚 2 ! Γ ( 𝛼 1 − 𝑚 1 + 1 ) Γ ( 𝛼 2 − 𝑚 2 + 1 ) ( 𝑚 1 + 𝑚 2 ) 𝑘 / 𝜆
+ 𝛼 1 ∑ 𝑚 1 = 0 𝛼 3 ∑ 𝑚 2 = 0 ( − 1 ) 𝑚 1 + 𝑚 2 Γ ( 𝛼 1 + 1 ) Γ ( 𝛼 3 + 1 ) 𝑚 1 ! 𝑚 2 ! Γ ( 𝛼 1 − 𝑚 1 + 1 ) Γ ( 𝛼 3 − 𝑚 2 + 1 ) ( 𝑚 1 + 𝑚 2 ) 𝑘 / 𝜆
+ 𝛼 2 ∑ 𝑚 1 = 0 𝛼 3 ∑ 𝑚 2 = 0 ( − 1 ) 𝑚 1 + 𝑚 2 Γ ( 𝛼 2 + 1 ) Γ ( 𝛼 3 + 1 ) 𝑚 1 ! 𝑚 2 ! Γ ( 𝛼 2 − 𝑚 1 + 1 ) Γ ( 𝛼 3 − 𝑚 2 + 1 )
× ( 𝑚 1 + 𝑚 2 ) 𝑘 / 𝜆
3 𝐼 3 ( 𝑘 ) = 𝑘 𝜎 𝑘 Γ ( − 𝑘 / 𝜆 ) 𝜆 𝛼 1 ∑ 𝑚 1 = 0 𝛼 2 ∑ 𝑚 2 = 0 𝛼 3 ∑ 𝑚 3 = 0 ( − 1 ) 𝑚 1 + 𝑚 2 + 𝑚 3 Γ ( 𝛼 1 + 1 ) Γ ( 𝛼 2 + 1 ) Γ ( 𝛼 3 + 1 ) 𝑚 1 ! 𝑚 2 ! 𝑚 3 ! Γ ( 𝛼 1 − 𝑚 1 + 1 ) Γ ( 𝛼 2 − 𝑚 2 + 1 ) Γ ( 𝛼 2 − 𝑚 2 + 1 )
× ( 𝑚 1 + 𝑚 2 + 𝑚 3 ) 𝑘 / 𝜆