ShiftedGammaDistribution

R2026b

Shifted gamma probability distribution object

Since R2026b

Description

A ShiftedGammaDistribution object consists of parameters, a model description, and sample data for a shifted gamma probability distribution.

The shifted gamma distribution generalizes the gamma distribution by adding a location parameter θ. When θ = 0, the shifted gamma distribution reduces to the two-parameter gamma distribution. The gamma distribution is used to model the sum of exponentially distributed random variables.

The shifted gamma distribution uses the following parameters.

Parameter	Description	Support
a	Shape parameter	$$a>0$$
b	Scale parameter	$$b>0$$
theta	Location (threshold) parameter	$$-\infty <\theta <\infty$$

Creation

There are several ways to create a ShiftedGammaDistribution probability distribution object:

• Create a distribution with specified parameter values using makedist.

• Fit a distribution to data using fitdist.

• Interactively fit a distribution to data using the Distribution Fitter app.

Properties

Distribution Parameters

a — Shape parameter

positive scalar value

Shape parameter for the shifted gamma distribution, specified as a positive scalar value.

Data Types: single | double

b — Scale parameter

positive scalar value

Scale parameter for the shifted gamma distribution, specified as a positive scalar value.

Data Types: single | double

theta — Location (threshold) parameter

numeric scalar value

Location (threshold) parameter for the shifted gamma distribution, specified as a numeric scalar value.

Data Types: single | double

Distribution Characteristics

IsTruncated — Logical flag for truncated distribution

Read-only: 0 (default) | 1

This property is read-only.

Logical flag for a truncated distribution, represented as 0 or 1. If IsTruncated is 0, the distribution is not truncated. If IsTruncated is 1, the distribution is truncated.

Data Types: logical

NumParameters — Number of parameters

Read-only: positive integer value

This property is read-only.

Number of parameters for the probability distribution, represented as a positive integer value.

Data Types: double

ParameterCovariance — Covariance matrix of parameter estimates

Read-only: matrix of scalar values

This property is read-only.

Covariance matrix of the parameter estimates, represented as a p-by-p matrix, where p is the number of parameters in the distribution. The (i,j) element is the covariance between the estimates of the ith parameter and the jth parameter. The (i,i) element is the estimated variance of the ith parameter. If parameter i is fixed, and not estimated by fitting the distribution to data, then the (i,i) elements of the covariance matrix are 0.

Data Types: double

ParameterIsFixed — Logical flag for fixed parameters

Read-only: array of logical values

This property is read-only.

Logical flag for fixed parameters, represented as an array of logical values. If the value is 0, the corresponding parameter in the ParameterNames array is not fixed. If the value is 1, the corresponding parameter in the ParameterNames array is fixed.

Data Types: logical

ParameterValues — Distribution parameter values

Read-only: vector of scalar values

This property is read-only.

Distribution parameter values, represented as a vector of scalar values.

Data Types: single | double

Truncation — Truncation interval

Read-only: vector of numeric scalars

This property is read-only.

Truncation interval for the probability distribution, represented as a vector of numeric scalars containing the lower and upper truncation boundaries.

Data Types: single | double

Other Object Properties

DistributionName — Probability distribution name

Read-only: character vector

This property is read-only.

Probability distribution name, represented as a character vector.

Data Types: char

InputData — Data used for distribution fitting

Read-only: structure

This property is read-only.

Data used for distribution fitting, represented as a structure containing the following:

• data – Data vector used for distribution fitting

• cens – Censoring vector (empty if none)

• freq – Frequency vector (empty if none)

Data Types: struct

ParameterDescription — Distribution parameter descriptions

Read-only: cell array of character vectors

This property is read-only.

Distribution parameter descriptions, represented as a cell array of character vectors. Each cell contains a short description of one distribution parameter.

Data Types: cell

ParameterNames — Distribution parameter names

Read-only: cell array of character vectors

This property is read-only.

Distribution parameter names, represented as a cell array of character vectors.

Data Types: cell

Object Functions

cdf	Cumulative distribution function
icdf	Inverse cumulative distribution function
iqr	Interquartile range of probability distribution
mean	Mean of probability distribution
median	Median of probability distribution
negloglik	Negative loglikelihood of probability distribution
paramci	Confidence intervals for probability distribution parameters
pdf	Probability density function
plot	Plot probability distribution object
proflik	Profile likelihood function for probability distribution
random	Random numbers
std	Standard deviation of probability distribution
truncate	Truncate probability distribution object
var	Variance of probability distribution

Examples

Shifted Gamma Distribution with Default Parameters

Create a shifted gamma distribution object using the default parameter values.

pd = makedist("ShiftedGamma")

pd = 
  ShiftedGammaDistribution

  Shifted (3-parameter) Gamma distribution
        a = 1
        b = 1
    theta = 0

By default, the threshold parameter theta is 0, making the distribution equivalent to the two-parameter gamma distribution.

Shifted Gamma Distribution with Specified Parameters

Create a shifted gamma distribution object using a shape parameter value of 2, scale parameter value of 4, and location (threshold) parameter value of 1.

pd = makedist("ShiftedGamm",a=2,b=4,theta=1)

pd = 
  ShiftedGammaDistribution

  Shifted (3-parameter) Gamma distribution
        a = 2
        b = 4
    theta = 1

Generate 100 random numbers from the shifted gamma distribution.

r = random(pd,100,1)

r = 100×1

   22.3288
   13.1836
    2.9372
    9.5971
   38.8278
    7.3463
    6.6633
   18.6977
   17.9896
    3.1732
   20.1259
   14.5644
    6.2193
    4.3716
    3.3244
      ⋮

Plot a histogram of the random numbers.

histogram(r)

Version History

Introduced in R2026b