Statistics: BirnbaumSaundersDistribution

Class Definition: `BirnbaumSaundersDistribution`

statistics: BirnbaumSaundersDistribution

Birnbaum-Saunders probability distribution object.

A BirnbaumSaundersDistribution object consists of parameters, a model description, and sample data for a Birnbaum-Saunders probability distribution.

The Birnbaum-Saunders distribution is a continuous probability distribution that models the time to failure of materials subjected to cyclic loading. It is defined by scale parameter beta and shape parameter gamma.

There are several ways to create a BirnbaumSaundersDistribution object.

Fit a distribution to data using the fitdist function.
Create a distribution with fixed parameter values using the makedist function.
Use the constructor BirnbaumSaundersDistribution (beta, gamma) to create a Birnbaum-Saunders distribution with fixed parameter values beta and gamma.
Use the static method BirnbaumSaundersDistribution.fit (x, alpha, censor, freq, options) to fit a distribution to the data in x using the same input arguments as the bisafit function.

It is highly recommended to use fitdist and makedist functions to create probability distribution objects, instead of the class constructor or the aforementioned static method.

Further information about the Birnbaum-Saunders distribution can be found at https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders_distribution

See also: fitdist, makedist, bisacdf, bisainv, bisapdf, bisarnd, bisafit, bisalike, bisastat

Source Code: BirnbaumSaundersDistribution

The BirnbaumSaundersDistribution class contains the following properties:

A positive scalar value characterizing the scale of the Birnbaum-Saunders distribution. You can access the beta property using dot name assignment.

Example: 1

Create a Birnbaum-Saunders distribution with default parameters

 pd = makedist ("BirnbaumSaunders")

pd =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta = 1
    gamma = 1

Query parameter 'beta' (scale parameter)

 pd.beta

ans = 1

Set parameter 'beta'

 pd.beta = 2

pd =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta = 2
    gamma = 1

Use this to initialize or modify the scale parameter of a Birnbaum-Saunders distribution. The scale parameter must be a positive real scalar.

Example: 2

Create a Birnbaum-Saunders distribution object by calling its constructor

 pd = BirnbaumSaundersDistribution (1.5, 0.5)

pd =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta = 1.5
    gamma = 0.5

Query parameter 'beta'

 pd.beta

ans = 1.5000

This demonstrates direct construction with a specific scale parameter, useful for modeling time-to-failure data with a known scale.

A positive scalar value characterizing the shape of the Birnbaum-Saunders distribution. You can access the gamma property using dot name assignment.

Example: 1

Create a Birnbaum-Saunders distribution with default parameters

 pd = makedist ("BirnbaumSaunders")

pd =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta = 1
    gamma = 1

Query parameter 'gamma' (shape parameter)

 pd.gamma

ans = 1

Set parameter 'gamma'

 pd.gamma = 0.8

pd =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta =   1
    gamma = 0.8

Use this to initialize or modify the shape parameter in a Birnbaum-Saunders distribution. The shape parameter must be a positive real scalar.

Example: 2

Create a Birnbaum-Saunders distribution object by calling its constructor

 pd = BirnbaumSaundersDistribution (1.5, 0.5)

pd =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta = 1.5
    gamma = 0.5

Query parameter 'gamma'

 pd.gamma

ans = 0.5000

This shows how to set the shape parameter directly via the constructor, ideal for modeling specific variability in failure times.

A character vector specifying the name of the probability distribution object. This property is read-only.

A scalar integer value specifying the number of parameters characterizing the probability distribution. This property is read-only.

A $2×1$ cell array of character vectors with each element containing the name of a distribution parameter. This property is read-only.

A $2×1$ cell array of character vectors with each element containing a short description of a distribution parameter. This property is read-only.

A $2×1$ numeric vector containing the values of the distribution parameters. This property is read-only. You can change the distribution parameters by assigning new values to the beta and gamma properties.

A $2×2$ numeric matrix containing the variance-covariance of the parameter estimates. Diagonal elements contain the variance of each estimated parameter, and non-diagonal elements contain the covariance between the parameter estimates. The covariance matrix is only meaningful when the distribution was fitted to data. If the distribution object was created with fixed parameters, or a parameter of a fitted distribution is modified, then all elements of the variance-covariance are zero. This property is read-only.

A $1×2$ logical vector specifying which parameters are fixed and which are estimated. true values correspond to fixed parameters, false values correspond to parameter estimates. This property is read-only.

A $1×2$ numeric vector specifying the truncation interval for the probability distribution. First element contains the lower boundary, second element contains the upper boundary. This property is read-only. You can only truncate a probability distribution with the truncate method.

A logical scalar value specifying whether a probability distribution is truncated or not. This property is read-only.

A scalar structure containing the following fields:

data: a numeric vector containing the data used for distribution fitting.
cens: a numeric vector of logical values indicating censoring information corresponding to the elements of the data used for distribution fitting. If no censoring vector was used for distribution fitting, then this field defaults to an empty array.
freq: a numeric vector of non-negative integer values containing the frequency information corresponding to the elements of the data used for distribution fitting. If no frequency vector was used for distribution fitting, then this field defaults to an empty array.

The BirnbaumSaundersDistribution class offers the following public methods:

BirnbaumSaundersDistribution: p = cdf (pd, x)
BirnbaumSaundersDistribution: p = cdf (pd, x, 'upper')

p = cdf (pd, x) computes the CDF of the probability distribution object, pd, evaluated at the values in x.

p = cdf (…, 'upper') returns the complement of the CDF of the probability distribution object, pd, evaluated at the values in x.

Example: 1

Plot various CDFs from the Birnbaum-Saunders distribution

 x = 0:0.01:5;
 pd1 = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.2);
 pd2 = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.5);
 pd3 = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.8);
 p1 = cdf (pd1, x);
 p2 = cdf (pd2, x);
 p3 = cdf (pd3, x);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r")
 grid on
 legend ({"beta = 1, gamma = 0.2", "beta = 1, gamma = 0.5", "beta = 1, gamma = 0.8"}, ...
         "location", "southeast")
 title ("Birnbaum-Saunders CDF")
 xlabel ("Time to failure")
 ylabel ("Cumulative probability")

Use this to compute and visualize the cumulative distribution function for different Birnbaum-Saunders distributions, showing how probability accumulates over time-to-failure.

BirnbaumSaundersDistribution: x = icdf (pd, p)

x = icdf (pd, p) computes the quantile (the inverse of the CDF) of the probability distribution object, pd, evaluated at the values in p.

Example: 1

Plot various iCDFs from the Birnbaum-Saunders distribution

 p = 0.001:0.001:0.999;
 pd1 = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.2);
 pd2 = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.5);
 pd3 = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.8);
 x1 = icdf (pd1, p);
 x2 = icdf (pd2, p);
 x3 = icdf (pd3, p);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r")
 grid on
 legend ({"beta = 1, gamma = 0.2", "beta = 1, gamma = 0.5", "beta = 1, gamma = 0.8"}, ...
         "location", "northwest")
 title ("Birnbaum-Saunders iCDF")
 xlabel ("Probability")
 ylabel ("Time to failure")

This demonstrates the inverse CDF (quantiles) for Birnbaum-Saunders distributions, useful for finding the time-to-failure corresponding to given probabilities.

BirnbaumSaundersDistribution: r = iqr (pd)

r = iqr (pd) computes the interquartile range of the probability distribution object, pd.

Example: 1

Compute the interquartile range for a Birnbaum-Saunders distribution

 pd = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.5)

pd =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta =   1
    gamma = 0.5

 iqr_value = iqr (pd)

iqr_value = 0.6840

Use this to calculate the interquartile range, which measures the spread of the middle 50% of the distribution, useful for understanding variability in failure times.

BirnbaumSaundersDistribution: m = mean (pd)

m = mean (pd) computes the mean of the probability distribution object, pd.

Example: 1

Compute the mean for different Birnbaum-Saunders distributions

 pd1 = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.2);
 pd2 = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.5);
 mean1 = mean (pd1)

mean1 = 1.0200

 mean2 = mean (pd2)

mean2 = 1.1250

This shows how to compute the expected time to failure for Birnbaum-Saunders distributions with different shape parameters.

BirnbaumSaundersDistribution: m = median (pd)

m = median (pd) computes the median of the probability distribution object, pd.

Example: 1

Compute the median for different Birnbaum-Saunders distributions

 pd1 = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.2);
 pd2 = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.5);
 median1 = median (pd1)

median1 = 1

 median2 = median (pd2)

median2 = 1

Use this to find the median time to failure, which splits the distribution into two equal probability halves.

BirnbaumSaundersDistribution: nlogL = negloglik (pd)

nlogL = negloglik (pd) computes the negative loglikelihood of the probability distribution object, pd.

Example: 1

Compute the negative loglikelihood for a fitted Birnbaum-Saunders distribution

 pd = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.5)

pd =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta =   1
    gamma = 0.5

 randg ("seed", 21);
 data = random (pd, 100, 1);
 pd_fitted = fitdist (data, "BirnbaumSaunders")

pd_fitted =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta = 1.00751   [0.909051, 1.11663]
    gamma = 0.54495   [0.474425, 0.625959]

 nlogL = negloglik (pd_fitted)

nlogL = -78.996

This is useful for assessing the fit of a Birnbaum-Saunders distribution to data, lower values indicate a better fit.

BirnbaumSaundersDistribution: ci = paramci (pd)
BirnbaumSaundersDistribution: ci = paramci (pd, Name, Value)

ci = paramci (pd) computes the lower and upper boundaries of the 95% confidence interval for each parameter of the probability distribution object, pd.

ci = paramci (pd, Name, Value) computes the confidence intervals with additional options specified by Name-Value pair arguments listed below.

`Name`		`Value`
`'Alpha'`		A scalar value in the range $(0,1)$ specifying the significance level for the confidence interval. The default value 0.05 corresponds to a 95% confidence interval.
`'Parameter'`		A character vector or a cell array of character vectors specifying the parameter names for which to compute confidence intervals. By default, `paramci` computes confidence intervals for all distribution parameters.

paramci is meaningful only when pd is fitted to data, otherwise an empty array, [], is returned.

Example: 1

Compute confidence intervals for parameters of a fitted Birnbaum-Saunders distribution

 pd = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.5)

pd =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta =   1
    gamma = 0.5

 randg ("seed", 21);
 data = random (pd, 1000, 1);
 pd_fitted = fitdist (data, "BirnbaumSaunders")

pd_fitted =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta = 0.974238   [0.945964, 1.00336]
    gamma = 0.489944   [0.468936, 0.511894]

 ci = paramci (pd_fitted, "Alpha", 0.05)

ci =

   0.9460   0.4689
   1.0034   0.5119

Use this to obtain confidence intervals for the estimated parameters (beta and gamma), providing a range of plausible values given the data.

BirnbaumSaundersDistribution: y = pdf (pd, x)

y = pdf (pd, x) computes the PDF of the probability distribution object, pd, evaluated at the values in x.

Example: 1

Plot various PDFs from the Birnbaum-Saunders distribution

 x = 0:0.01:5;
 pd1 = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.2);
 pd2 = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.5);
 pd3 = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.8);
 y1 = pdf (pd1, x);
 y2 = pdf (pd2, x);
 y3 = pdf (pd3, x);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r")
 grid on
 legend ({"beta = 1, gamma = 0.2", "beta = 1, gamma = 0.5", "beta = 1, gamma = 0.8"}, ...
         "location", "northeast")
 title ("Birnbaum-Saunders PDF")
 xlabel ("Time to failure")
 ylabel ("Probability density")

This visualizes the probability density function for Birnbaum-Saunders distributions, showing the likelihood of different times to failure.

BirnbaumSaundersDistribution: plot (pd)
BirnbaumSaundersDistribution: plot (pd, Name, Value)
BirnbaumSaundersDistribution: h = plot (…)

plot (pd) plots a probability density function (PDF) of the probability distribution object pd. If pd contains data, which have been fitted by fitdist, the PDF is superimposed over a histogram of the data.

plot (pd, Name, Value) specifies additional options with the Name-Value pair arguments listed below.

`Name`		`Value`
`'PlotType'`		A character vector specifying the plot type. `'pdf'` plots the probability density function (PDF). When `pd` is fit to data, the PDF is superimposed on a histogram of the data. `'cdf'` plots the cumulative density function (CDF). When `pd` is fit to data, the CDF is superimposed over an empirical CDF. `'probability'` plots a probability plot using a CDF of the data and a CDF of the fitted probability distribution. This option is available only when `pd` is fitted to data.
`'Discrete'`		A logical scalar to specify whether to plot the PDF or CDF of a discrete distribution object as a line plot or a stem plot, by specifying `false` or `true`, respectively. By default, it is `true` for discrete distributions and `false` for continuous distributions. When `pd` is a continuous distribution object, option is ignored.
`'Parent'`		An axes graphics object for plot. If not specified, the `plot` function plots into the current axes or creates a new axes object if one does not exist.

h = plot (…) returns a graphics handle to the plotted objects.

Example: 1

Create a Birnbaum-Saunders distribution with fixed parameters β = 1 and γ = 0.5 and plot its PDF.

 pd = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.5)

pd =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta =   1
    gamma = 0.5

 plot (pd)
 title ("Fixed Birnbaum-Saunders distribution with beta = 1 and gamma = 0.5")

Example: 2

Generate a data set of 100 random samples from a Birnbaum-Saunders distribution with parameters β = 1 and γ = 0.5. Fit a Birnbaum-Saunders distribution to this data and plot its CDF superimposed over an empirical CDF.

 pd_fixed = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.5)

pd_fixed =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta =   1
    gamma = 0.5

 randg ("seed", 21);
 data = random (pd_fixed, 100, 1);
 pd_fitted = fitdist (data, "BirnbaumSaunders")

pd_fitted =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta =  1.07336   [0.971323, 1.18611]
    gamma = 0.528129   [0.459781, 0.606637]

 plot (pd_fitted, "PlotType", "cdf")
 txt = "Fitted Birnbaum-Saunders distribution with β = %0.2f and γ = %0.2f";
 title (sprintf (txt, pd_fitted.beta, pd_fitted.gamma))
 legend ({"empirical CDF", "fitted CDF"}, "location", "southeast")

Use this to visualize the fitted CDF compared to the empirical CDF of the data, useful for assessing model fit.

Example: 3

Generate a data set of 200 random samples from a Birnbaum-Saunders distribution with parameters β = 1 and γ = 0.5. Display a probability plot for the Birnbaum-Saunders distribution fit to the data.

 pd_fixed = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.5)

pd_fixed =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta =   1
    gamma = 0.5

 randg ("seed", 21);
 data = random (pd_fixed, 200, 1);
 pd_fitted = fitdist (data, "BirnbaumSaunders")

pd_fitted =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta =  1.07073   [1.00275, 1.14332]
    gamma = 0.487849   [0.442309, 0.538079]

 plot (pd_fitted, "PlotType", "probability")
 txt = strcat ("Probability plot of fitted Birnbaum-Saunders", ...
               " distribution with β = %0.2f and γ = %0.2f");
 title (sprintf (txt, pd_fitted.beta, pd_fitted.gamma))
 legend ({"empirical CDF", "fitted CDF"}, "location", "southeast")

This creates a probability plot to compare the fitted distribution to the data, useful for checking if the Birnbaum-Saunders model is appropriate.

BirnbaumSaundersDistribution: [nlogL, param] = proflik (pd, pnum)
BirnbaumSaundersDistribution: [nlogL, param] = proflik (pd, pnum, 'Display', display)
BirnbaumSaundersDistribution: [nlogL, param] = proflik (pd, pnum, setparam)
BirnbaumSaundersDistribution: [nlogL, param] = proflik (pd, pnum, setparam, 'Display', display)

[nlogL, param] = proflik (pd, pnum) returns a vector nlogL of negative loglikelihood values and a vector param of corresponding parameter values for the parameter in the position indicated by pnum. By default, proflik uses the lower and upper bounds of the 95% confidence interval and computes 100 equispaced values for the selected parameter. pd must be fitted to data.

[nlogL, param] = proflik (pd, pnum, 'Display', 'on') also plots the profile likelihood against the default range of the selected parameter.

[nlogL, param] = proflik (pd, pnum, setparam) defines a user-defined range of the selected parameter.

[nlogL, param] = proflik (pd, pnum, setparam, 'Display', 'on') also plots the profile likelihood against the user-defined range of the selected parameter.

For the Birnbaum-Saunders distribution, pnum = 1 selects the parameter beta and pnum = 2 selects the parameter gamma.

When opted to display the profile likelihood plot, proflik also plots the baseline loglikelihood computed at the lower bound of the 95% confidence interval and estimated maximum likelihood. The latter might not be observable if it is outside of the used-defined range of parameter values.

Example: 1

Compute and plot the profile likelihood for the shape parameter of a fitted Birnbaum-Saunders distribution

 pd = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.5)

pd =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta =   1
    gamma = 0.5

 randg ("seed", 21);
 data = random (pd, 1000, 1);
 pd_fitted = fitdist (data, "BirnbaumSaunders")

pd_fitted =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta =  1.01481   [0.98503, 1.04548]
    gamma = 0.495802   [0.474542, 0.518014]

 [nlogL, param] = proflik (pd_fitted, 2, "Display", "on");

Use this to analyze the profile likelihood of the shape parameter (gamma), helping to understand the uncertainty in parameter estimates.

BirnbaumSaundersDistribution: r = random (pd)
BirnbaumSaundersDistribution: r = random (pd, rows)
BirnbaumSaundersDistribution: r = random (pd, rows, cols, …)
BirnbaumSaundersDistribution: r = random (pd, [sz])

r = random (pd) returns a random number from the distribution object pd.

When called with a single size argument, bisarnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Example: 1

Generate random samples from a Birnbaum-Saunders distribution

 pd = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.5)

pd =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta =   1
    gamma = 0.5

 randg ("seed", 21);
 samples = random (pd, 500, 1);
 hist (samples, 50)
 title ("Histogram of 500 random samples from Birnbaum-Saunders(beta=1, gamma=0.5)")
 xlabel ("Time to failure")
 ylabel ("Frequency")

This generates random samples from a Birnbaum-Saunders distribution, useful for simulating time-to-failure data under cyclic loading.

BirnbaumSaundersDistribution: s = std (pd)

s = std (pd) computes the standard deviation of the probability distribution object, pd.

Example: 1

Compute the standard deviation for a Birnbaum-Saunders distribution

 pd = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.5)

pd =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta =   1
    gamma = 0.5

 std_value = std (pd)

std_value = 0.5728

Use this to calculate the standard deviation, which measures the variability in time to failure.

BirnbaumSaundersDistribution: t = truncate (pd, lower, upper)

t = truncate (pd, lower, upper) returns a probability distribution t, which is the probability distribution pd truncated to the specified interval with lower limit, lower, and upper limit, upper. If pd is fitted to data with fitdist, the returned probability distribution t is not fitted, does not contain any data or estimated values, and it is as it has been created with the makedist function, but it includes the truncation interval.

Example: 1

Plot the PDF of a Birnbaum-Saunders distribution, with parameters beta = 1 and gamma = 0.5, truncated at [0.5, 2] intervals. Generate 10000 random samples from this truncated distribution and superimpose a histogram scaled accordingly

 pd = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.5)

pd =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta =   1
    gamma = 0.5

 t = truncate (pd, 0.5, 2)

t =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta =   1
    gamma = 0.5
  Truncated to the interval [0.5, 2]

 randg ("seed", 21);
 data = random (t, 10000, 1);

Plot histogram and fitted PDF

 plot (t)
 hold on
 hist (data, 100, 50)
 hold off
 title ("Birnbaum-Saunders distribution (beta = 1, gamma = 0.5) truncated at [0.5, 2]")
 legend ("Truncated PDF", "Histogram")

This demonstrates truncating a Birnbaum-Saunders distribution to a specific range and visualizing the resulting distribution with random samples.

BirnbaumSaundersDistribution: v = var (pd)

v = var (pd) computes the variance of the probability distribution object, pd.

Example: 1

Compute the variance for a Birnbaum-Saunders distribution

 pd = makedist ("BirnbaumSaunders", "beta", 1, "gamma", 0.5)

pd =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta =   1
    gamma = 0.5

 var_value = var (pd)

var_value = 0.3281

Use this to calculate the variance, which quantifies the spread of the time to failure in the distribution.

Examples

 pd_fixed = makedist ('BirnbaumSaunders', 'beta', 1, 'gamma', 0.5)

pd_fixed =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta =   1
    gamma = 0.5

 randg ('seed', 21);
 data = random (pd_fixed, 5000, 1);
 pd_fitted = fitdist (data, 'BirnbaumSaunders')

pd_fitted =
  BirnbaumSaundersDistribution

  Birnbaum-Saunders distribution
     beta =  1.00789   [0.994443, 1.02153]
    gamma = 0.500467   [0.490753, 0.510373]

 plot (pd_fitted)
 msg = 'Fitted Birnbaum-Saunders distribution with beta = %0.2f and gamma = %0.2f';
 title (sprintf (msg, pd_fitted.beta, pd_fitted.gamma))

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Class Definition: BirnbaumSaundersDistribution

statistics: BirnbaumSaundersDistribution

beta

Example: 1

Example: 2

gamma

Example: 1

Example: 2

DistributionName

NumParameters

ParameterNames

ParameterDescription

ParameterValues

ParameterCovariance

ParameterIsFixed

Truncation

IsTruncated

InputData

cdf

BirnbaumSaundersDistribution: p = cdf (pd, x)

BirnbaumSaundersDistribution: p = cdf (pd, x, 'upper')

Example: 1

icdf

BirnbaumSaundersDistribution: x = icdf (pd, p)

Example: 1

iqr

BirnbaumSaundersDistribution: r = iqr (pd)

Example: 1

mean

BirnbaumSaundersDistribution: m = mean (pd)

Example: 1

median

BirnbaumSaundersDistribution: m = median (pd)

Example: 1

negloglik

BirnbaumSaundersDistribution: nlogL = negloglik (pd)

Example: 1

paramci

Class Definition: `BirnbaumSaundersDistribution`

`beta`

`gamma`

`DistributionName`

`NumParameters`

`ParameterNames`

`ParameterDescription`

`ParameterValues`

`ParameterCovariance`

`ParameterIsFixed`

`Truncation`

`IsTruncated`

`InputData`

`cdf`

`BirnbaumSaundersDistribution: p = cdf (pd, x)`

`BirnbaumSaundersDistribution: p = cdf (pd, x, 'upper')`

`icdf`

`BirnbaumSaundersDistribution: x = icdf (pd, p)`

`iqr`

`BirnbaumSaundersDistribution: r = iqr (pd)`

`mean`

`BirnbaumSaundersDistribution: m = mean (pd)`

`median`

`BirnbaumSaundersDistribution: m = median (pd)`

`negloglik`

`BirnbaumSaundersDistribution: nlogL = negloglik (pd)`

`paramci`

`BirnbaumSaundersDistribution: ci = paramci (pd)`

`BirnbaumSaundersDistribution: ci = paramci (pd, Name, Value)`

`pdf`

`BirnbaumSaundersDistribution: y = pdf (pd, x)`

`plot`

`BirnbaumSaundersDistribution: plot (pd)`

`BirnbaumSaundersDistribution: plot (pd, Name, Value)`

`BirnbaumSaundersDistribution: h = plot (…)`

`proflik`

`BirnbaumSaundersDistribution: [nlogL, param] = proflik (pd, pnum)`

`BirnbaumSaundersDistribution: [nlogL, param] = proflik (pd, pnum, 'Display', display)`

`BirnbaumSaundersDistribution: [nlogL, param] = proflik (pd, pnum, setparam)`

`BirnbaumSaundersDistribution: [nlogL, param] = proflik (pd, pnum, setparam, 'Display', display)`

`random`

`BirnbaumSaundersDistribution: r = random (pd)`

`BirnbaumSaundersDistribution: r = random (pd, rows)`

`BirnbaumSaundersDistribution: r = random (pd, rows, cols, …)`

`BirnbaumSaundersDistribution: r = random (pd, [sz])`

`std`

`BirnbaumSaundersDistribution: s = std (pd)`

`truncate`

`BirnbaumSaundersDistribution: t = truncate (pd, lower, upper)`

`var`

`BirnbaumSaundersDistribution: v = var (pd)`