Table 1 Distortion function q for a coherent system with 1–4 i.i.d. components
From: Extended cumulative entropy based on kth lower record values for the coherent systems lifetime
N | T | q(u) |
|---|---|---|
1 | \(X_{1}\) | u |
2 | \(X_{1:2}=\min (X_{1},X_{2})\) | \(2u-u^{2}\) |
3 | \(X_{2:2}=\max (X_{1},X_{2})\) | \(u^{2}\) |
4 | \(X_{1:3}=\min (X_{1},X_{2},X_{3})\) | \(3u-3u^{2}+u^{3}\) |
5 | \(\min (X_{1},\max (X_{2},X_{3}))\) | \(u+u^{2}-u^{3}\) |
6 | \(X_{2:3}\) | \(3u^{2}-2u^{3}\) |
7 | \(\max (X_{1},\min (X_{2},X_{3}))\) | \(2u^{2}-u^{3}\) |
8 | \(X_{3:3}=\max (X_{1},X_{2},X_{3})\) | \(u^{3}\) |
9 | \(X_{1:4}=\min (X_{1},X_{2},X_{3},X_{4})\) | \(4u-6u^{2}+4u^{3}-u^{4}\) |
10 | \(\max (\min (X_{1},X_{2}, X_{3}),\min (X_{2},X_{3},X_{4}))\) | \(2u-2u^{3}+u^{4}\) |
11 | \(\min (X_{2:3},X_{4})\) | \(u+3u^{2}-5u^{3}+2u^{4}\) |
12 | \(\min (X_{1},\max (X_{2},X_{3}),\max (X_{2},X_{4}))\) | \(u+2u^{2}-3u^{3}+u^{4}\) |
13 | \(\min (X_{1},\max (X_{2},X_{3},X_{4}))\) | \(u+u^{3}-u^{4}\) |
14 | \(X_{2:4}\) | \(6u^{2}-8u^{3}+3u^{4}\) |
15 | \(\max (\min (X_{1},X_{2}),\min (X_{1},X_{3},X_{4}),\min (X_{2},X_{3},X_{4}))\) | \(5u^{2}-6u^{3}+2u^{4}\) |
16 | \(\max (\min (X_{1},X_{2}),\min (X_{3},X_{4}))\) | \(4u^{2}-4u^{3}+u^{4}\) |
17 | \(\max (\min (X_{1},X_{2}),\min (X_{1},X_{3}),\min (X_{2},X_{3},X_{4}))\) | \(4u^{2}-4u^{3}+u^{4}\) |
18 | \(\max (\min (X_{1},X_{2}),\min (X_{2},X_{3}),\min (X_{3},X_{4}))\) | \(3u^{2}-2u^{3}\) |
19 | \(\min (\max (X_{1},X_{2}),\max (X_{2},X_{3}),\max (X_{3},X_{4}))\) | \(3u^{2}-2u^{3}\) |
20 | \(\min (\max (X_{1},X_{2}),\max (X_{1},X_{3}),\max (X_{2},X_{3},X_{4}))\) | \(2u^{2}-u^{4}\) |
21 | \(\min (\max (X_{1},X_{2}),\max (X_{3},X_{4}))\) | \(2u^{2}-u^{4}\) |
22 | \(\min (\max (X_{1},X_{2}),\max (X_{1},X_{3},X_{4}),\max (X_{2},X_{3},X_{4}))\) | \(u^{2}+2u^{3}-2u^{4}\) |
23 | \(X_{3:4}\) | \(4u^{3}-3u^{4}\) |
24 | \(\max (X_{1},\min (X_{2},X_{3},X_{4}))\) | \(3u^{2}-3u^{3}+u^{4}\) |
25 | \(\max (X_{1},\min (X_{2},X_{3}),\min (X_{2},X_{4}))\) | \(u^{2}+u^{3}-u^{4}\) |
26 | \(\max (X_{2:3},X_{4})\) | \(3u^{3}-2u^{4}\) |
27 | \(\max (\min (X_{1},X_{2},X_{3}),\min (X_{2},X_{3},X_{4}))\) | \(2u^{3}-u^{4}\) |
28 | \(X_{4:4}=\max (X_{1},X_{2},X_{3},X_{4})\) | \(u^{4}\) |