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Class Definition: BurrDistribution

statistics: BurrDistribution

Burr probability distribution object.

A BurrDistribution object consists of parameters, a model description, and sample data for a Burr probability distribution.

The Burr distribution is a continuous probability distribution that models a non-negative random variable, commonly used to model household income. It is defined by a scale parameter alpha and two shape parameters c and k.

There are several ways to create a BurrDistribution object.

  • Fit a distribution to data using the fitdist function.
  • Create a distribution with fixed parameter values using the makedist function.
  • Use the constructor BurrDistribution (alpha, c, k) to create a Burr distribution with fixed parameter values alpha, c, and k.
  • Use the static method BurrDistribution.fit (x, alpha, censor, freq, options) to fit a distribution to the data in x using the same input arguments as the burrfit 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 Burr distribution can be found at https://en.wikipedia.org/wiki/Burr_distribution

See also: fitdist, makedist, burrcdf, burrinv, burrpdf, burrrnd, burrfit, burrlike, burrstat

Source Code: BurrDistribution

Properties

A positive scalar value characterizing the scale of the Burr distribution. You can access the alpha property using dot name assignment.

Example: 1

  

 ## Create a Burr distribution with default parameters
 pd = makedist ("Burr")

 ## Query parameter 'alpha' (scale parameter)
 pd.alpha

 ## Set parameter 'alpha'
 pd.alpha = 2

 ## Use this to initialize or modify the scale parameter of a Burr
 ## distribution. The scale parameter must be a positive real scalar.

pd =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 1
        k = 1

ans = 1
pd =
  BurrDistribution

  Burr distribution
    alpha = 2
        c = 1
        k = 1

Example: 2

  

 ## Create a Burr distribution object by calling its constructor
 pd = BurrDistribution (1.5, 2, 1)

 ## Query parameter 'alpha'
 pd.alpha

 ## This demonstrates direct construction with a specific scale parameter,
 ## ideal for modeling specific household income distributions.

pd =
  BurrDistribution

  Burr distribution
    alpha = 1.5
        c =   2
        k =   1

ans = 1.5000

A positive scalar value characterizing the first shape parameter of the Burr distribution. You can access the c property using dot name assignment.

Example: 1

  

 ## Create a Burr distribution with default parameters
 pd = makedist ("Burr")

 ## Query parameter 'c' (first shape parameter)
 pd.c

 ## Set parameter 'c'
 pd.c = 3

 ## Use this to initialize or modify the first shape parameter in a Burr
 ## distribution. The first shape parameter must be a positive real scalar.

pd =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 1
        k = 1

ans = 1
pd =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 3
        k = 1

Example: 2

  

 ## Create a Burr distribution object by calling its constructor
 pd = BurrDistribution (1, 3, 1)

 ## Query parameter 'c'
 pd.c

 ## This shows how to set the first shape parameter directly via the constructor,
 ## useful for modeling variability in non-negative data like household income.

pd =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 3
        k = 1

ans = 3

A positive scalar value characterizing the second shape parameter of the Burr distribution. You can access the k property using dot name assignment.

Example: 1

  

 ## Create a Burr distribution with default parameters
 pd = makedist ("Burr")

 ## Query parameter 'k' (second shape parameter)
 pd.k

 ## Set parameter 'k'
 pd.k = 2

 ## Use this to initialize or modify the second shape parameter in a Burr
 ## distribution. The second shape parameter must be a positive real scalar.

pd =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 1
        k = 1

ans = 1
pd =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 1
        k = 2

Example: 2

  

 ## Create a Burr distribution object by calling its constructor
 pd = BurrDistribution (1, 2, 2)

 ## Query parameter 'k'
 pd.k

 ## This shows how to set the second shape parameter directly via the constructor,
 ## ideal for tailoring the tail behavior in distributions for income data.

pd =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 2
        k = 2

ans = 2

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 3×1 cell array of character vectors with each element containing the name of a distribution parameter. This property is read-only.

A 3×1 cell array of character vectors with each element containing a short description of a distribution parameter. This property is read-only.

A 3×1 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 alpha, c, and k properties.

A 3×3 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 1×3 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: an empty array, since BurrDistribution does not allow censoring.
  • 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.

Methods

BurrDistribution: p = cdf (pd, x)
BurrDistribution: 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 Burr distribution
 x = 0:0.01:5;
 pd1 = makedist ("Burr", "alpha", 1, "c", 2, "k", 1);
 pd2 = makedist ("Burr", "alpha", 1, "c", 3, "k", 1);
 pd3 = makedist ("Burr", "alpha", 1, "c", 4, "k", 1);
 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 ({"alpha=1, c=2, k=1", "alpha=1, c=3, k=1", "alpha=1, c=4, k=1"}, ...
         "location", "southeast")
 title ("Burr CDF")
 xlabel ("values in x")
 ylabel ("Cumulative probability")

 ## Use this to compute and visualize the cumulative distribution function
 ## for different Burr distributions, showing how probability accumulates
 ## over non-negative values like income levels.
plotted figure

BurrDistribution: 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 Burr distribution
 p = 0.001:0.001:0.999;
 pd1 = makedist ("Burr", "alpha", 1, "c", 2, "k", 1);
 pd2 = makedist ("Burr", "alpha", 1, "c", 3, "k", 1);
 pd3 = makedist ("Burr", "alpha", 1, "c", 4, "k", 1);
 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 ({"alpha=1, c=2, k=1", "alpha=1, c=3, k=1", "alpha=1, c=4, k=1"}, ...
         "location", "northwest")
 title ("Burr iCDF")
 xlabel ("Probability")
 ylabel ("values in x")

 ## This demonstrates the inverse CDF (quantiles) for Burr distributions,
 ## useful for finding the value (e.g., income level) corresponding to given
 ## probabilities.
plotted figure

BurrDistribution: r = iqr (pd)

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

Example: 1

  

 ## Compute the interquartile range for a Burr distribution
 pd = makedist ("Burr", "alpha", 1, "c", 2, "k", 1)
 iqr_value = iqr (pd)

 ## Use this to calculate the interquartile range, which measures the spread
 ## of the middle 50% of the distribution, useful for understanding variability
 ## in data like household income.

pd =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 2
        k = 1

iqr_value = 1.1547
BurrDistribution: m = mean (pd)

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

Example: 1

  

 ## Compute the mean for different Burr distributions
 pd1 = makedist ("Burr", "alpha", 1, "c", 2, "k", 1);
 pd2 = makedist ("Burr", "alpha", 1, "c", 3, "k", 1);
 mean1 = mean (pd1)
 mean2 = mean (pd2)

 ## This shows how to compute the expected value for Burr distributions with
 ## different shape parameters, provided the parameters allow a finite mean
 ## (e.g., when c * k > 1).

mean1 = 1.5708
mean2 = 1.2092
BurrDistribution: m = median (pd)

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

Example: 1

  

 ## Compute the median for different Burr distributions
 pd1 = makedist ("Burr", "alpha", 1, "c", 2, "k", 1);
 pd2 = makedist ("Burr", "alpha", 1, "c", 3, "k", 1);
 median1 = median (pd1)
 median2 = median (pd2)

 ## Use this to find the median value, which splits the distribution into
 ## two equal probability halves, e.g., median household income.

median1 = 1
median2 = 1
BurrDistribution: 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 Burr distribution
 pd = makedist ("Burr", "alpha", 1, "c", 2, "k", 1)
 rand ("seed", 21);
 data = random (pd, 100, 1);
 pd_fitted = fitdist (data, "Burr")
 nlogL = negloglik (pd_fitted)

 ## This is useful for assessing the fit of a Burr distribution to data;
 ## lower values indicate a better fit, e.g., for income modeling.

pd =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 2
        k = 1

pd_fitted =
  BurrDistribution

  Burr distribution
    alpha = 1.18986   [1.01271, 1.39799]
        c = 1.91252   [1.62314, 2.25348]
        k = 1.30385   [0.625566, 2.71757]

nlogL = -123.35
BurrDistribution: ci = paramci (pd)
BurrDistribution: 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.

NameValue
"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 Burr distribution
 pd = makedist ("Burr", "alpha", 1, "c", 2, "k", 1)
 rand ("seed", 21);
 data = random (pd, 1000, 1);
 pd_fitted = fitdist (data, "Burr")
 ci = paramci (pd_fitted, "Alpha", 0.05)

 ## Use this to obtain confidence intervals for the estimated parameters (alpha,
 ## c, k), providing a range of plausible values given the data, e.g., for
 ## uncertainty in income distribution parameters.

pd =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 2
        k = 1

pd_fitted =
  BurrDistribution

  Burr distribution
    alpha =  1.00315   [0.951509, 1.0576]
        c =  2.04261   [1.93888, 2.1519]
        k = 0.974629   [0.777103, 1.22236]

ci =

   0.9515   1.9389   0.7771
   1.0576   2.1519   1.2224
BurrDistribution: 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 Burr distribution
 x = 0:0.01:5;
 pd1 = makedist ("Burr", "alpha", 1, "c", 2, "k", 1);
 pd2 = makedist ("Burr", "alpha", 1, "c", 3, "k", 1);
 pd3 = makedist ("Burr", "alpha", 1, "c", 4, "k", 1);
 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 ({"alpha=1, c=2, k=1", "alpha=1, c=3, k=1", "alpha=1, c=4, k=1"}, ...
         "location", "northeast")
 title ("Burr PDF")
 xlabel ("values in x")
 ylabel ("Probability density")

 ## This visualizes the probability density function for Burr distributions,
 ## showing the likelihood of different values, e.g., income levels.
plotted figure

BurrDistribution: plot (pd)
BurrDistribution: plot (pd, Name, Value)
BurrDistribution: 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.

NameValue
"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 Burr distribution with fixed parameters alpha = 1, c = 2, k = 1
 ## and plot its PDF.

 pd = makedist ("Burr", "alpha", 1, "c", 2, "k", 1)
 plot (pd)
 title ("Fixed Burr distribution with alpha = 1, c = 2, k = 1")

pd =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 2
        k = 1
plotted figure

Example: 2

  

 ## Generate a data set of 100 random samples from a Burr distribution with
 ## parameters alpha = 1, c = 2, k = 1. Fit a Burr distribution to this data
 ## and plot its CDF superimposed over an empirical CDF.

 pd_fixed = makedist ("Burr", "alpha", 1, "c", 2, "k", 1)
 rand ("seed", 21);
 data = random (pd_fixed, 100, 1);
 pd_fitted = fitdist (data, "Burr")
 plot (pd_fitted, "PlotType", "cdf")
 txt = "Fitted Burr distribution with alpha = %0.2f, c = %0.2f, k = %0.2f";
 title (sprintf (txt, pd_fitted.alpha, pd_fitted.c, pd_fitted.k))
 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 in income data.

pd_fixed =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 2
        k = 1

pd_fitted =
  BurrDistribution

  Burr distribution
    alpha = 1.18986   [1.01271, 1.39799]
        c = 1.91252   [1.62314, 2.25348]
        k = 1.30385   [0.625566, 2.71757]
plotted figure

Example: 3

  

 ## Generate a data set of 200 random samples from a Burr distribution with
 ## parameters alpha = 1, c = 2, k = 1. Display a probability plot for the
 ## Burr distribution fit to the data.

 pd_fixed = makedist ("Burr", "alpha", 1, "c", 2, "k", 1)
 rand ("seed", 21);
 data = random (pd_fixed, 200, 1);
 pd_fitted = fitdist (data, "Burr")
 plot (pd_fitted, "PlotType", "probability")
 txt = strcat ("Probability plot of fitted Burr distribution with ", ...
               "alpha = %0.2f, c = %0.2f, k = %0.2f");
 title (sprintf (txt, pd_fitted.alpha, pd_fitted.c, pd_fitted.k))
 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 Burr model is appropriate for the dataset.

pd_fixed =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 2
        k = 1

pd_fitted =
  BurrDistribution

  Burr distribution
    alpha = 1.23729   [1.10047, 1.39113]
        c = 1.88943   [1.68443, 2.11939]
        k = 1.27534   [0.720502, 2.25744]
plotted figure

BurrDistribution: [nlogL, param] = proflik (pd, pnum)
BurrDistribution: [nlogL, param] = proflik (pd, pnum, "Display", display)
BurrDistribution: [nlogL, param] = proflik (pd, pnum, setparam)
BurrDistribution: [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 Burr distribution, pnum = 1 selects the parameter alpha, pnum = 2 selects the parameter c, and pnum = 3 selects the parameter k.

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 first shape parameter of a
 ## fitted Burr distribution
 pd = makedist ("Burr", "alpha", 1, "c", 2, "k", 1)
 rand ("seed", 21);
 data = random (pd, 1000, 1);
 pd_fitted = fitdist (data, "Burr")
 [nlogL, param] = proflik (pd_fitted, 2, "Display", "on");

 ## Use this to analyze the profile likelihood of the first shape parameter (c),
 ## helping to understand the uncertainty in parameter estimates for models
 ## like household income.

pd =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 2
        k = 1

pd_fitted =
  BurrDistribution

  Burr distribution
    alpha =  1.00315   [0.951509, 1.0576]
        c =  2.04261   [1.93888, 2.1519]
        k = 0.974629   [0.777103, 1.22236]
plotted figure

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

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

When called with a single size argument, burrrnd 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 Burr distribution
 pd = makedist ("Burr", "alpha", 1, "c", 2, "k", 3)
 rand ("seed", 21);
 samples = random (pd, 50, 1);
 hist (samples, 90)
 title ("Histogram of 50 random samples from Burr(alpha=1, c=2, k=3)")
 xlabel ("values in x")
 ylabel ("Frequency")

 ## This generates random samples from a Burr distribution, useful for
 ## simulating non-negative data like household income under specific parameters.

pd =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 2
        k = 3
plotted figure

BurrDistribution: s = std (pd)

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

Example: 1

  

 ## Compute the standard deviation for a Burr distribution
 pd = makedist ("Burr", "alpha", 1, "c", 2, "k", 1)
 std_value = std (pd)

 ## Use this to calculate the standard deviation, which measures the variability
 ## in the distribution, e.g., spread of household income.

pd =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 2
        k = 1

std_value = Inf
BurrDistribution: 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 Burr distribution, with parameters alpha = 1, c = 2, k = 1,
 ## truncated at [0.5, 2.5] intervals. Generate 10000 random samples from this
 ## truncated distribution and superimpose a histogram scaled accordingly

 pd = makedist ("Burr", "alpha", 1, "c", 2, "k", 1)
 t = truncate (pd, 0.5, 2.5)
 rand ("seed", 21);
 data = random (t, 10000, 1);

 ## Plot histogram and fitted PDF
 plot (t)
 hold on
 hist (data, 100, 50)
 hold off
 title ("Burr distribution (alpha = 1, c = 2, k = 1) truncated at [0.5, 2.5]")
 legend ("Truncated PDF", "Histogram")

 ## This demonstrates truncating a Burr distribution to a specific range and
 ## visualizing the resulting distribution with random samples, useful for
 ## bounded data scenarios.

pd =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 2
        k = 1

t =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 2
        k = 1
  Truncated to the interval [0.5, 2.5]
plotted figure

BurrDistribution: v = var (pd)

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

Example: 1

  

 ## Compute the variance for a Burr distribution
 pd = makedist ("Burr", "alpha", 1, "c", 2, "k", 1)
 var_value = var (pd)

 ## Use this to calculate the variance, which quantifies the spread of the
 ## distribution, e.g., variability in household income.

pd =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 2
        k = 1

var_value = Inf

Examples

  
 pd = makedist ("Burr", "alpha", 1, "c", 2, "k", 1)
 rand ("seed", 21);
 data = random (pd, 5000, 1);
 pd = fitdist (data, "Burr")
 plot (pd)
 msg = strcat (["Fitted Burr type XII distribution with"], ...
                [" alpha = %0.2f, c =  %0.2f, and k = %0.2f"]);
 title (sprintf (msg, pd.alpha, pd.c, pd.k))
  
pd =
  BurrDistribution

  Burr distribution
    alpha = 1
        c = 2
        k = 1

pd =
  BurrDistribution

  Burr distribution
    alpha = 0.943213   [0.920634, 0.966345]
        c =  2.01899   [1.9726, 2.06648]
        k = 0.946371   [0.857098, 1.04494]
plotted figure