Engineering Statistics Handbook Episode 2 Part 4 docx

1.3.6.6.12. Double Exponential Distribution
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Cumulative
Distribution
Function
The formula for the cumulative distribution function of the power
normal distribution is
where is the cumulative distribution function of the standard normal
distribution.
The following is the plot of the power normal cumulative distribution
function with the same values of p as the pdf plots above.
1.3.6.6.13. Power Normal Distribution
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Percent
Point
Function
The formula for the percent point function of the power normal
distribution is
where is the percent point function of the standard normal
distribution.
The following is the plot of the power normal percent point function
with the same values of p as the pdf plots above.
1.3.6.6.13. Power Normal Distribution
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Hazard
Function
The formula for the hazard function of the power normal distribution is
The following is the plot of the power normal hazard function with the
same values of p as the pdf plots above.
1.3.6.6.13. Power Normal Distribution

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Cumulative
Hazard
Function
The formula for the cumulative hazard function of the power normal
distribution is
The following is the plot of the power normal cumulative hazard
function with the same values of p as the pdf plots above.
Survival
Function
The formula for the survival function of the power normal distribution is
The following is the plot of the power normal survival function with the
same values of p as the pdf plots above.
1.3.6.6.13. Power Normal Distribution
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Inverse
Survival
Function
The formula for the inverse survival function of the power normal
distribution is
The following is the plot of the power normal inverse survival function
with the same values of p as the pdf plots above.
1.3.6.6.13. Power Normal Distribution
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Common
Statistics
The statistics for the power normal distribution are complicated and
require tables. Nelson discusses the mean, median, mode, and standard
deviation of the power normal distribution and provides references to
the appropriate tables.

Software Most general purpose statistical software programs do not support the
probability functions for the power normal distribution. Dataplot does
support them.
1.3.6.6.13. Power Normal Distribution
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Cumulative
Distribution
Function
The formula for the cumulative distribution function of the power lognormal
distribution is
where is the cumulative distribution function of the standard normal distribution.
The following is the plot of the power lognormal cumulative distribution function
with the same values of p as the pdf plots above.
1.3.6.6.14. Power Lognormal Distribution
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Percent
Point
Function
The formula for the percent point function of the power lognormal distribution is
where is the percent point function of the standard normal distribution.
The following is the plot of the power lognormal percent point function with the
same values of p as the pdf plots above.
1.3.6.6.14. Power Lognormal Distribution
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Hazard
Function
The formula for the hazard function of the power lognormal distribution is
where is the cumulative distribution function of the standard normal distribution,
and
is the probability density function of the standard normal distribution.

Note that this is simply a multiple (p) of the lognormal hazard function.
The following is the plot of the power lognormal hazard function with the same
values of p as the pdf plots above.
Cumulative
Hazard
Function
The formula for the cumulative hazard function of the power lognormal
distribution is
The following is the plot of the power lognormal cumulative hazard function with
the same values of p as the pdf plots above.
1.3.6.6.14. Power Lognormal Distribution
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Survival
Function
The formula for the survival function of the power lognormal distribution is
The following is the plot of the power lognormal survival function with the same
values of p as the pdf plots above.
1.3.6.6.14. Power Lognormal Distribution
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Inverse
Survival
Function
The formula for the inverse survival function of the power lognormal distribution is
The following is the plot of the power lognormal inverse survival function with the
same values of p as the pdf plots above.
Common
Statistics
The statistics for the power lognormal distribution are complicated and require
tables. Nelson discusses the mean, median, mode, and standard deviation of the
power lognormal distribution and provides references to the appropriate tables.

Parameter
Estimation
Nelson discusses maximum likelihood estimation for the power lognormal
distribution. These estimates need to be performed with computer software.
Software for maximum likelihood estimation of the parameters of the power
lognormal distribution is not as readily available as for other reliability
distributions such as the exponential, Weibull, and lognormal.
Software Most general purpose statistical software programs do not support the probability
functions for the power lognormal distribution. Dataplot does support them.
1.3.6.6.14. Power Lognormal Distribution
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Cumulative
Distribution
Function
The Tukey-Lambda distribution does not have a simple, closed form. It
is computed numerically.
The following is the plot of the Tukey-Lambda cumulative distribution
function with the same values of
as the pdf plots above.
Percent
Point
Function
The formula for the percent point function of the standard form of the
Tukey-Lambda distribution is
The following is the plot of the Tukey-Lambda percent point function
with the same values of
as the pdf plots above.
1.3.6.6.15. Tukey-Lambda Distribution
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Other

Probability
Functions
The Tukey-Lambda distribution is typically used to identify an
appropriate distribution (see the comments below) and not used in
statistical models directly. For this reason, we omit the formulas, and
plots for the hazard, cumulative hazard, survival, and inverse survival
functions. We also omit the common statistics and parameter estimation
sections.
Comments The Tukey-Lambda distribution is actually a family of distributions that
can approximate a number of common distributions. For example,
= -1
approximately Cauchy
= 0
exactly logistic
= 0.14
approximately normal
= 0.5
U-shaped
= 1
exactly uniform (from -1 to +1)
The most common use of this distribution is to generate a
Tukey-Lambda PPCC plot of a data set. Based on the ppcc plot, an
appropriate model for the data is suggested. For example, if the
maximum correlation occurs for a value of
at or near 0.14, then the
data can be modeled with a normal distribution. Values of less than
this imply a heavy-tailed distribution (with -1 approximating a Cauchy).
That is, as the optimal value of
goes from 0.14 to -1, increasingly
heavy tails are implied. Similarly, as the optimal value of becomes

greater than 0.14, shorter tails are implied.
1.3.6.6.15. Tukey-Lambda Distribution
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As the Tukey-Lambda distribution is a symmetric distribution, the use
of the Tukey-Lambda PPCC plot to determine a reasonable distribution
to model the data only applies to symmetric distributuins. A histogram
of the data should provide evidence as to whether the data can be
reasonably modeled with a symmetric distribution.
Software Most general purpose statistical software programs do not support the
probability functions for the Tukey-Lambda distribution. Dataplot does
support them.
1.3.6.6.15. Tukey-Lambda Distribution
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The general formula for the probability density function of the Gumbel
(maximum) distribution is
where is the location parameter and is the scale parameter. The
case where
= 0 and = 1 is called the standard Gumbel
distribution. The equation for the standard Gumbel distribution
(maximum) reduces to
The following is the plot of the Gumbel probability density function for
the maximum case.
1.3.6.6.16. Extreme Value Type I Distribution
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Since the general form of probability functions can be expressed in
terms of the standard distribution, all subsequent formulas in this section
are given for the standard form of the function.
Cumulative
Distribution
Function

The formula for the cumulative distribution function of the Gumbel
distribution (minimum) is
The following is the plot of the Gumbel cumulative distribution function
for the minimum case.
1.3.6.6.16. Extreme Value Type I Distribution
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The formula for the cumulative distribution function of the Gumbel
distribution (maximum) is
The following is the plot of the Gumbel cumulative distribution function
for the maximum case.
1.3.6.6.16. Extreme Value Type I Distribution
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Percent
Point
Function
The formula for the percent point function of the Gumbel distribution
(minimum) is
The following is the plot of the Gumbel percent point function for the
minimum case.
The formula for the percent point function of the Gumbel distribution
(maximum) is
The following is the plot of the Gumbel percent point function for the
maximum case.
1.3.6.6.16. Extreme Value Type I Distribution
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Hazard
Function
The formula for the hazard function of the Gumbel distribution
(minimum) is
The following is the plot of the Gumbel hazard function for the

minimum case.
The formula for the hazard function of the Gumbel distribution
1.3.6.6.16. Extreme Value Type I Distribution
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