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**Package: **prob**Superclasses: **prob.ToolboxFittableParametricDistribution

Loglogistic probability distribution object

`prob.LoglogisticDistribution`

is an object
consisting of parameters, a model description, and sample data for
a loglogistic probability distribution.

Create a probability distribution object with specified parameter
values using `makedist`

. Alternatively,
fit a distribution to data using `fitdist`

or
the Distribution Fitting app.

creates
a loglogistic probability distribution object using the default parameter
values.`pd`

= makedist('Loglogistic')

creates
a loglogistic probability distribution object using the specified
parameter values .`pd`

= makedist('Loglogistic','mu',`mu`

,'sigma',`sigma`

)

cdf | Cumulative distribution function of probability distribution object |

icdf | Inverse cumulative distribution function of probability distribution object |

iqr | Interquartile range of probability distribution object |

median | Median of probability distribution object |

Probability density function of probability distribution object | |

random | Generate random numbers from probability distribution object |

truncate | Truncate probability distribution object |

mean | Mean of probability distribution object |

negloglik | Negative log likelihood of probability distribution object |

paramci | Confidence intervals for probability distribution parameters |

proflik | Profile likelihood function for probability distribution object |

std | Standard deviation of probability distribution object |

var | Variance of probability distribution object |

The loglogistic distribution is closely related to the logistic
distribution. If *x* is distributed loglogistically
with parameters *μ* and *σ*,
then log(*x*) is distributed logistically with mean
and standard deviation. This distribution is often used in survival
analysis to model events that experience an initial rate increase,
followed by a rate decrease.

The loglogistic distribution uses the following parameters.

Parameter | Description | Support |
---|---|---|

`mu` | Log mean | $$\mu >0$$ |

`sigma` | Log scale parameter | $$\sigma >0$$ |

The probability density function (pdf) is

$$f(x|\mu ,\sigma )=\frac{1}{\sigma}\frac{1}{x}\frac{{e}^{z}}{{\left(1+{e}^{z}\right)}^{2}}\text{\hspace{1em}};\text{\hspace{1em}}x\ge 0\text{\hspace{0.17em}},$$

where $$z=\frac{\mathrm{log}\left(x\right)-\mu}{\sigma}$$.

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