General Examples Using the Lognormal Distribution

Lognormal Distribution Example 9

Determine the lognormal parameter estimates for the data given in Table 9.3.

Table 9.3 - Non-Grouped Data Times-to-Failure with Intervals (Interval and Left Censored)

Data point index	Last Inspected	State End Time
1	30	32
2	32	35
3	35	37
4	37	40
5	40	42
6	20	45
7	10	50
8	55	55

Solution to Lognormal Distribution Example 9

This is a sequence of interval times-to-failure where the intervals vary substantially in length. Using Weibull++, the computed parameters for maximum likelihood are calculated to be:

For rank regression on X:

For rank regression on Y:

Lognormal Distribution Example 10

Determine the lognormal parameter estimates for the data given in Table 9.4.

Table 9.4 - Non-Grouped Data Times-to-Failure with Suspensions (Right Censored)

Data point index	State F or S	State End Time
1	F	2
2	F	5
3	F	11
4	F	23
5	F	29
6	F	37
7	F	43
8	F	59

Solution to Lognormal Distribution Example 10

Using Weibull++, the computed parameters for maximum likelihood are:

For rank regression on X:

For rank regression on Y:

Lognormal Distribution Example 11

From Kececioglu [19, p. 406]. Nine identical units are tested continuously to failure and their times-to-failure were recorded at 30.4, 36.7, 53.3, 58.5, 74.0, 99.3, 114.3, 140.1, and 257.9 hours.

Solution to Lognormal Distribution Example 11

The results published were obtained by using the unbiased model.

Published Results (using MLE):

This same data set can be entered into Weibull++ by creating a data sheet capable of handling non-grouped time-to-failure data. Since the results shown above are unbiased, the Use Unbiased Std on Normal Data option in the User Setup must be selected in order to duplicate these results.

Weibull++ computed parameters for maximum likelihood are:

Lognormal Distribution Example 12

From Kececioglu [20, p. 347]. Fifteen identical units were tested to failure and following is a table of their times-to-failure:

Table 9.5 - Data of Example 11

Data Point Index	Time-to-Failure, hr
1	62.5
2	91.9
3	100.3
4	117.4
5	141.1
6	146.8
7	172.7
8	192.5
9	201.6
10	235.8
11	249.2
12	297.5
13	318.3
14	410.6
15	550.5

Solution to Lognormal Distribution Example 12

Published results (using probability plotting):

Weibull++ computed parameters for rank regression on X are:

The small differences are due to the precision errors when fitting a line manually, whereas in Weibull++ the line was fitted mathematically.

Lognormal Distribution Example 13

From Nelson [30, p. 324]. Ninety-six locomotive controls were tested, 37 failed and 59 were suspended after running for 135,000 miles. Table 9.6 shows their times-to-failure.

Table 9.6 - Nelson's Locomotive Data

	Number in State	F or S	Time
1	1	F	22.5
2	1	F	37.5
3	1	F	46
4	1	F	48.5
5	1	F	51.5
6	1	F	53
7	1	F	54.5
8	1	F	57.5
9	1	F	66.5
10	1	F	68
11	1	F	69.5
12	1	F	76.5
13	1	F	77
14	1	F	78.5
15	1	F	80
16	1	F	81.5
17	1	F	82
18	1	F	83
19	1	F	84
20	1	F	91.5
21	1	F	93.5
22	1	F	102.5
23	1	F	107
24	1	F	108.5
25	1	F	112.5
26	1	F	113.5
27	1	F	116
28	1	F	117
29	1	F	118.5
30	1	F	119
31	1	F	120
32	1	F	122.5
33	1	F	123
34	1	F	127.5
35	1	F	131
36	1	F	132.5
37	1	F	134
38	59	S	135

Solution to Lognormal Distribution Example 13

The distribution used in the publication was the base-10 lognormal.

Published results (using MLE):

Published 95% confidence limits on the parameters:

Published variance/covariance matrix:

To replicate the published results (since Weibull++ uses a lognormal to the base e), take the base-10 logarithm of the data and estimate the parameters using the Normal distribution and MLE.

Weibull++ computed parameters for maximum likelihood are:

Weibull++ computed 95% confidence limits on the parameters:

Weibull++ computed/variance covariance matrix: