(4)This section includes the following subsections:
Using Probability Plotting to Calculate the Parameters of the Lognormal Distribution
Using Rank Regression on Y to Calculate the Parameters of the Lognormal Distribution
Using Rank Regression on X to Calculate the Parameters of the Lognormal Distribution
Using Maximum Likelihood Estimation to Calculate the Parameters of the Lognormal Distribution
As described before, probability plotting involves plotting the failure times and associated unreliability estimates on specially constructed probability plotting paper. The form of this paper is based on a linearization of the cdf of the specific distribution. For the lognormal distribution, the cumulative density function can be written as:
(4)
or:
(5)
where:
Now, let:
(6)
(7)
and:
(8)
which results in the linear equation of:
The normal probability paper resulting from this linearized cdf function is shown next.
The process for reading the parameter estimate values from the lognormal probability plot is very similar to the method employed for the normal distribution (see the Normal Distribution chapter). However, since the lognormal distribution models the natural logarithms of the times-to-failure, the values of the parameter estimates must be read and calculated based on a logarithmic scale, as opposed to the linear time scale as it was done with the normal distribution. This parameter scale appears at the top of the lognormal probability plot.
The process of lognormal probability plotting is illustrated in the following example.
Eight units are put on a life test and tested to failure. The failures occurred at 45, 140, 260, 500, 850, 1400, 3000, and 9000 hours. Estimate the parameters for the lognormal distribution using probability plotting.
In order to plot the points for the probability plot, the appropriate unreliability estimate values must be obtained. These will be estimated through the use of median ranks, which can be obtained from statistical tables or the Quick Statistical Reference in Weibull++. The following table shows the times-to-failure and the appropriate median rank values for this example:
|
Time-to-Failure (hr.) |
Median Rank (%) |
|
45 |
8.30% |
|
140 |
20.11% |
|
260 |
32.05% |
|
500 |
44.02% |
|
850 |
55.98% |
|
1400 |
67.95% |
|
3000 |
79.89% |
|
9000 |
91.70% |
These points may now be plotted on normal probability plotting paper as shown in the next figure.
Draw the best possible line through the plot points. The time values where this line intersects the 15.85% and 50% unreliability values should be projected up to the logarithmic scale, as shown in the following plot.
The natural logarithm of the time where the fitted line intersects Q(t)
= 50% is equivalent
to 
. In this case, = 6.45. The value for
is equal to the difference between the natural logarithms
of the times where the fitted line crosses Q(t) =
50% and Q(t)
= 15.85%. At Q(t)
= 15.85%, ln(t) =
4.55. Therefore, = 6.45 - 4.55 = 1.9.
Performing a rank regression on Y requires that a straight line be fitted to a set of data points such that the sum of the squares of the vertical deviations from the points to the line is minimized.
The least squares parameter estimation method, or regression analysis, was discussed in the Statistical Background chapter and the following equations for regression on Y were derived, and are again applicable:
(9)
and:
(10)
In our case the equations for yi and xi are:
and:
where the 
is estimated from the median ranks. Once
and 
are obtained, then 
and 
can easily be obtained from Eqns. (7)
and (8).
The estimator of ρ is the sample correlation
coefficient, 
, given by:
(11)
Fourteen units were reliability tested and the following life test data were obtained:
Table 9.1 - The test data for Example 2
|
Data point index |
Time-to-failure |
|
1 |
5 |
|
2 |
10 |
|
3 |
15 |
|
4 |
20 |
|
5 |
25 |
|
6 |
30 |
|
7 |
35 |
|
8 |
40 |
|
9 |
50 |
|
10 |
60 |
|
11 |
70 |
|
12 |
80 |
|
13 |
90 |
|
14 |
100 |
Assuming the data follow a lognormal distribution, estimate the parameters and the correlation coefficient, ρ, using rank regression on Y.
Construct Table 9.2, as shown next.
Table 9.2 - Least Squares
Analysis
The median rank values (F(Ti)) can be found in rank tables or by using the Quick Statistical Reference in Weibull++.
The yi values were obtained from the standardized normal distribution's area tables by entering for F(z) and getting the corresponding z value (yi).
Given the values in the table above, calculate and using Eqns. (9) and (10):
or:
and:
or:
Therefore, from Eqn. (8):
and from Eqn. (7):
or:
The mean and the standard deviation of the lognormal distribution are obtained using Eqns. (2) and (3):
and:
The correlation coefficient can be estimated using Eqn. (11):
The above example can be repeated using Weibull++, using RRY.
The mean can be obtained from the QCP and both the mean and the standard deviation can be obtained from the Function Wizard.
Performing a rank regression on X requires that a straight line be fitted to a set of data points such that the sum of the squares of the horizontal deviations from the points to the line is minimized.
Again, the first task is to bring our cdf function into a linear form. This step is exactly the same as in regression on Y analysis and Eqns. (5), (6), (7) and (8) apply in this case too. The deviation from the previous analysis begins on the least squares fit part, where in this case we treat x as the dependent variable and y as the independent variable. The best-fitting straight line to the data, for regression on X (see the Statistical Background chapter), is the straight line:
(12)
The corresponding equations for and are:
and:
where:
and:
and the 
is estimated from the median ranks. Once and are obtained, solve Eqn. (12) for the
unknown y,
which corresponds to:
Solving for the parameters from Eqns. (8) and (7) we get:
(13)
and:
(14)
The correlation coefficient is evaluated as before using Eqn. (11).
Using the data of Example 2 and assuming a lognormal distribution, estimate the parameters and estimate the correlation coefficient, ρ, using rank regression on X.
Table 9.2 constructed in Example 2 applies to this example as well. Using the values in this table we get:
or:
and:
or:
Therefore, from Eqn. (14):
and from Eqn. (13):
Using Eqns. (2) and (3) we get:
and:
The correlation coefficient is found using Eqn. (11):
Note that the regression on Y analysis is not necessarily the same as the regression on X. The only time when the results of the two regression types are the same (i.e. will yield the same equation for a line) is when the data lie perfectly on a line.
Using Weibull++, with the Rank Regression on X option, the results are:
As it was outlined in the Statistical Background chapter, maximum likelihood estimation works by developing a likelihood function based on the available data and finding the values of the parameter estimates that maximize the likelihood function. This can be achieved by using iterative methods to determine the parameter estimate values that maximize the likelihood function. However, this can be rather difficult and time-consuming, particularly when dealing with the three-parameter distribution. Another method of finding the parameter estimates involves taking the partial derivatives of the likelihood equation with respect to the parameters, setting the resulting equations equal to zero, and solving simultaneously to determine the values of the parameter estimates. The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the lognormal distribution are covered in Appendix C.
See Also:
The Lognormal Distribution
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