ABSTRACT
Example 8.1.1. To illustrate the hazard plotting technique and parameter estimation from progressively censored samples, we have selected a sample that was originally used as an illustration by Nelson (1969). Sample observations consist of times to failure and/or times to censoring for 70 generator fans. Sample data together with calculated values of the hazard function h(x) and the cumulative hazard function H(x) are entered in Table 8.1. In summary, N = 70, n = 12, X1:1o = 4500, ST = 3,444,400, ST!n = 287,033.33, "kJ: 1 X; = 365,700, and "kj= 1 c;Tj = 3,078,700. A rectangular coordinate plot of the cumulative hazard function for this example is given as Figure 8.3 with a straight line fitted by eye. The straight-line fit indicates that the exponential distribution is the appropriate model for these data. From the fitted line, we read H(268,000) = 100% and H(450) = 0. Accordingly 'Y* = 450, 'Y* + 13* = 268,000, and 13* = 275,550. We let k = 1/2 in equation (8.1.33) and use this equation to calculate
S* = -in 2 In H[ (132*) + 'Y*] = - 1.442695 ln(0.5) = 1.00. This result confirms our earlier conclusion that this sample is from an exponential distribution. For comparison with the hazard plot estimates. we employ equations (8. I. 14) to calculate modified maximum likelihood estimates as
"' = 12(4500) - 3,444,400/70 = 436 I ll '
Estimate~ of the mean and standard deviation are E(X) = "! + ~ = 284,927 and &x = 13 = 284,491. Differences between these estimates and the hazard
Table 8.1 A Progressively Censored Sample Consisting of Life-Span Observations of 70 Generator Fans
Reverse Time Reverse Time rank 1000 hr h(x) H(x) rank 1000 hr h(x) H(x)
Rank (m) (x) (%) (%) Rank (m) (x) (%) (%)
1 70 4.5 1.43 1.43 36 35 43.0+ 2 69 4.6+ 37 34 46.0 2.94 18.78 3 68 11.5 1.47 2.90 38 33 48.5+ 4 67 11.5 1.49 4.39 39 32 48.5+ 5 66 15.6+ 40 31 48.5+
6 65 16.0 1.54 5.93 41 30 48.5+ 7 64 16.6+ 42 29 50.0+ 8 63 18.5+ 43 28 50.0+ 9 62 18.5+ 44 27 50.0+
10 61 18.5+ 45 26 61.0+
11 60 18.5+ 46 25 61.0 4.00 22.78 12 59 18.5+ 47 24 61.0+ 13 58 20.3+ 48 23 61.0+ 14 57 20.3+ 49 22 63.0+ 15 56 20.3+ 50 21 64.5+
16 55 20.7 1.82 7.75 51 20 64.5+ 17 54 20.7 1.85 9.60 52 19 67.0+ 18 53 20.8 1.89 11.49 53 18 74.5+ 19 52 22.0+ 54 17 78.0+ 20 51 30.0+ 55 16 78.0+
21 50 30.0+ 56 15 81.0+ 22 50 30.0+ 57 14 81.0+ 23 48 30.0+ 58 13 82.0+ 24 47 31.0 2.13 13.62 59 12 85.0+ 25 46 32.0+ 60 11 85.0+
26 45 34.5 2.22 15.84 61 10 85.0+ 27 44 37.5+ 62 9 87.5+ 28 43 37.5+ 63 8 87.5 12.50 35.28 29 42 41. 5+ 64 7 87.5+ 30 41 41. 5+ 65 6 94.0+
31 40 41. 5+ 66 5 99.0+ 32 39 41.5+ 67 4 101.0+ 33 38 43.0+ 68 3 101.0+ 34 37 43.0+ 69 2 101.0+ 35 36 43.0+ 70 1 115.0+
"'
Figure 8.3 A cumulative hazard plot of generator fan data. Reprinted from Cohen and Whitten (1988), Fig. 7.3, p. 125, by courtesy of Marcel Dekker, Inc.
