# Log-logistic distribution

**Fisk distributionlog logistic distributionlog-logisticLogLogisticloglogistic distributionshifted log-logistic distribution**

In probability and statistics, the log-logistic distribution (known as the Fisk distribution in economics) is a continuous probability distribution for a non-negative random variable.wikipedia

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### Heavy-tailed distribution

**heavy tailsheavy-tailedheavy tail**

It is similar in shape to the log-normal distribution but has heavier tails.

Many other heavy-tailed distributions such as the log-logistic and Pareto distribution are, however, also fat-tailed.

### Survival analysis

**hazard ratesurvivalevent history analysis**

It is used in survival analysis as a parametric model for events whose rate increases initially and decreases later, as, for example, mortality rate from cancer following diagnosis or treatment.

### Shape parameter

**shape**

The parameter \beta>0 is a shape parameter.

### Logistic distribution

**logisticbell-shaped curvelogistical**

The log-logistic distribution is the probability distribution of a random variable whose logarithm has a logistic distribution.

### Cumulative frequency analysis

**cumulative frequencyCumFreqfrequency distributions**

The rainfall data are represented by the plotting position r/(n+1) as part of the cumulative frequency analysis.

### Quantile function

**quantileinverse distribution functionnormal quantile function**

The quantile function (inverse cumulative distribution function) is :

The evaluation of quantile functions often involves numerical methods, as the example of the exponential distribution above is one of the few distributions where a closed-form expression can be found (others include the uniform, the Weibull, the Tukey lambda (which includes the logistic) and the log-logistic).

### Shifted log-logistic distribution

**shifted log-logistic**

*The addition of another parameter (a shift parameter) formally results in a shifted log-logistic distribution, but this is usually considered in a different parameterization so that the distribution can be bounded above or bounded below.

The shifted log-logistic distribution can be obtained from the log-logistic distribution by addition of a shift parameter \delta.

### Accelerated failure time model

**AFT model**

The log-logistic distribution can be used as the basis of an accelerated failure time model by allowing \alpha to differ between groups, or more generally by introducing covariates that affect \alpha but not \beta by modelling as a linear function of the covariates.

The log-logistic distribution provides the most commonly used AFT model.

### Burr distribution

**Burr type XII distributionSingh–Maddala distributionBurr type 12**

These include the Burr Type XII distribution (also known as the Singh–Maddala distribution) and the Dagum distribution, both of which include a second shape parameter.

It is also known as the Singh–Maddala distribution and is one of a number of different distributions sometimes called the "generalized log-logistic distribution".

### Beta prime distribution

**Beta primecompound gamma distributionGeneralized Beta Prime**

### Probability

**probabilisticprobabilitieschance**

In probability and statistics, the log-logistic distribution (known as the Fisk distribution in economics) is a continuous probability distribution for a non-negative random variable.

### Statistics

**statisticalstatistical analysisstatistician**

In probability and statistics, the log-logistic distribution (known as the Fisk distribution in economics) is a continuous probability distribution for a non-negative random variable.

### Economics

**economiceconomisteconomic theory**

In probability and statistics, the log-logistic distribution (known as the Fisk distribution in economics) is a continuous probability distribution for a non-negative random variable. It has also been used in hydrology to model stream flow and precipitation, in economics as a simple model of the distribution of wealth or income, and in networking to model the transmission times of data considering both the network and the software.

### Probability distribution

**distributioncontinuous probability distributiondiscrete probability distribution**

In probability and statistics, the log-logistic distribution (known as the Fisk distribution in economics) is a continuous probability distribution for a non-negative random variable.

### Random variable

**random variablesrandom variationrandom**

The log-logistic distribution is the probability distribution of a random variable whose logarithm has a logistic distribution. In probability and statistics, the log-logistic distribution (known as the Fisk distribution in economics) is a continuous probability distribution for a non-negative random variable.

### Parametric model

**parametricregular parametric modelparameters**

It is used in survival analysis as a parametric model for events whose rate increases initially and decreases later, as, for example, mortality rate from cancer following diagnosis or treatment.

### Mortality rate

**mortalitymortality ratesdeath rates**

It is used in survival analysis as a parametric model for events whose rate increases initially and decreases later, as, for example, mortality rate from cancer following diagnosis or treatment.

### Hydrology

**hydrologicalhydrologisthydrologic**

It has also been used in hydrology to model stream flow and precipitation, in economics as a simple model of the distribution of wealth or income, and in networking to model the transmission times of data considering both the network and the software.

### Precipitation

**rainfallhydrometeorannual precipitation**

It has also been used in hydrology to model stream flow and precipitation, in economics as a simple model of the distribution of wealth or income, and in networking to model the transmission times of data considering both the network and the software.

### Distribution of wealth

**wealth inequalitywealth distributionWealth condensation**

It has also been used in hydrology to model stream flow and precipitation, in economics as a simple model of the distribution of wealth or income, and in networking to model the transmission times of data considering both the network and the software.

### Income distribution

**distribution of incomeincomedistribution**

### Computer network

**networkcomputer networkingnetworking**

### Logarithm

**logarithmsloglogarithmic function**

The log-logistic distribution is the probability distribution of a random variable whose logarithm has a logistic distribution.

### Log-normal distribution

**lognormallog-normallognormal distribution**

It is similar in shape to the log-normal distribution but has heavier tails.

### Cumulative distribution function

**distribution functionCDFcumulative probability distribution function**

Unlike the log-normal, its cumulative distribution function can be written in closed form.