The logistic distribution has been used for growth models, and is used in a certain type of regression known as the logistic regression. It has also applications in modeling life data. The shape of the logistic distribution and the normal distribution are very similar [27]. There are some who argue that the logistic distribution is inappropriate for modeling lifetime data because the left-hand limit of the distribution extends to negative infinity. This could conceivably result in modeling negative times-to-failure. However, provided that the distribution in question has a relatively high mean and a relatively small location parameter, the issue of negative failure times should not present itself as a problem.
This section includes the following subsections:
The logistic pdf is given by:
where:
μ = location
parameter (also denoted as 
)
σ
= scale parameter
The logistic mean or MTTF is actually one of the parameters of the distribution,
usually denoted as μ. Since the logistic
distribution is symmetrical, the median and the mode are always equal
to the mean, μ = 
= 
.
The standard deviation of the logistic distribution, σT, is given by:
The reliability for a mission of time T, starting at age 0, for the logistic distribution is determined by:
or:
The unreliability function is:
(14)
where:
(15)
The logistic conditional reliability function is given by:
The logistic reliable life is given by:
The logistic failure rate function is given by:
The logistic distribution has no shape parameter. This means that the logistic pdf has only one shape, the bell shape, and this shape does not change. The shape of the logistic distribution is very similar to that of the normal distribution.
The mean, μ,
or the mean life or the MTTF,
is also the location parameter of the logistic pdf,
as it locates the pdf along the
abscissa. It can assume values of -
< < .
As μ decreases, the pdf is shifted to the left.
As μ increases, the pdf is shifted to the right.
As σ decreases, the pdf gets pushed toward the mean, or it becomes narrower and taller.
As σ increases, the pdf spreads out away from the mean, or it becomes broader and shallower.
The scale parameter can assume values
of 0 <
σ < .
The logistic pdf
starts at T = - with an f(T)
= 0. As T
increases, f(T)
also increases, goes through its point of inflection and reaches its maximum
value at T = . Thereafter,
f(T)
decreases, goes through its point of inflection and assumes a value of
f(T)
= 0 at T = +.
For 
the
pdf equals 0. The maximum value
of the pdf occurs at T
= μ
and equals 
The point of inflection
of the pdf plot is the point
where the second derivative of the pdf
equals zero. The inflection point occurs at 
or 
.
If the location parameter μ decreases, the reliability plot is shifted to the left. If μ increases, the reliability plot is shifted to the right.
If T = μ then R = 0.5. T = μ is the inflection point. If T < μ then R(t) is concave (concave down); if T > μ then R(t) is convex (concave up). For T < μ, λ(t) is convex (concave up), for T > μ; λ(t) is concave (concave down).
The main difference between the normal distribution and logistic distribution lies in the tails and in the behavior of the failure rate function. The logistic distribution has slightly longer tails compared to the normal distribution. Also, in the upper tail of the logistic distribution, the failure rate function levels out for large t approaching 1/δ.
If location parameter μ decreases, the failure rate plot is shifted to the left. Vice versa if μ increases, the failure rate plot is shifted to the right.
λ
always increases. For 
for 
It is always 
If σ increases, then λ(t)
increases more slowly and smoothly. The segment of time where 
increases, too, whereas the region where λ(t)
is close to 0
or 
gets narrower. Conversely, if σ decreases, then λ(t)
increases more quickly and sharply. The segment of time where 0 <
decreases, too, whereas the region where λ(t)
is close to 0
or gets broader.
One of the disadvantages of using the logistic distribution for reliability calculations is the fact that the logistic distribution starts at negative infinity. This can result in negative values for some of the results. Negative values for time are not accepted in most of the components of Weibull++, nor are they implemented. Certain components of the application reserve negative values for suspensions, or will not return negative results. For example, the Quick Calculation Pad will return a null value (zero) if the result is negative. Only the Free-Form (Probit) data sheet can accept negative values for the random variable (x-axis values).
The form of the Logistic probability paper is based on linearizing the cdf.
From Eqn. (14), z can be calculated as a function of the cdf F as follows:
(16)
or using Eqn. (15)
Then:
(17)
Now let:
and:
(18)
(19)
which results in the following linear equation:
The logistic probability paper resulting from this linearized cdf function is shown next.
Since the logistic distribution is symmetrical, the area under the pdf curve from - to μ
is 0.5, as is the area from μ
to +. Consequently, the value of μ is
said to be the point where R(t)
= Q(t) = 50%.
This means that the estimate of μ
can be read from the point where the plotted line crosses the 50% unreliability
line.
For z = 1, σ = t - μ and 
Therefore,
σ
can be found by subtracting μ
from the time value where the plotted probability line crosses the 73.10%
unreliability (26.89% reliability) horizontal line.
In this section, we present the methods used in the application to estimate the different types of confidence bounds for logistically distributed data. The complete derivations were presented in detail (for a general function) in the Confidence Bounds chapter.
The lower and upper bounds on the location parameter 
are
estimated from:
The lower and upper bounds on the scale parameter 
are
estimated from:
where Kα is defined by:
If δ
is the confidence level, then 
for the two-sided bounds,
and α = 1 - δ
for the one-sided bounds.
The variances and covariances of and are
estimated from the Fisher matrix, as follows:
Λ is the log-likelihood function of the normal distribution, described in the Statistical Background chapter and Appendix C.
The reliability of the logistic distribution is:
where:
Here - < T < , - < μ < , 0 < σ < . Therefore, z also is
changing from - to +. Then the bounds
on z
are estimated from:
where:
or:
The upper and lower bounds on reliability are:
The bounds around time for a given logistic percentile (unreliability) are estimated by first solving the reliability equation with respect to time as follows:
where:
or:
The upper and lower bounds are then found by:
The lifetime of a mechanical valve is known to follow a logistic distribution. Ten units were tested for 28 months and the following months-to-failure data was collected.
Table 10.1- Times-to-Failure Data with Suspensions
|
Data Point Index |
State F or S |
State End Time |
|
1 |
F |
8 |
|
2 |
F |
10 |
|
3 |
F |
15 |
|
4 |
F |
17 |
|
5 |
F |
19 |
|
6 |
F |
26 |
|
7 |
F |
27 |
|
8 |
S |
28 |
|
9 |
S |
28 |
|
10 |
S |
28 |
Determine the valve's design life if specifications call for a reliability goal of 0.90.
The valve is to be used in a pumping device that requires 1 month of continuous operation. What is the probability of the pump failing due to the valve?
This data set can be entered into Weibull++ as follows:
The computed parameters for maximum likelihood are:
The valve's design life, along with 90% two sided confidence bounds, can be obtained using the QCP as follows:
The probability, along with 90% two sided confidence bounds, that the pump fails due to a valve failure during the first month is obtained as follows:
See Also:
Other Distributions
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