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- Virtual Laboratories
- 4. Special Distributions
- The Log-Logistic Distribution

As the name suggests, the log-logistic distribution is the distribution of a variable whose logarithm has the logistic distribution. The log-logistic is a continuous distribution on \( [0, \infty) \) that is used to model random lifetimes, and hence has applications in reliability.

Random variable \( Z \) has the basic log-logistic distribution with shape parameter \( k \in (0, \infty) \) if \( Z \) has distribution function \( G \) given by \[ G(z) = \frac{z^k}{1 + z^k}, \quad z \in [0, \infty) \] In the special case that \( k = 1 \), \( Z \) has the standard log-logistic distribution.

\( G \) really is the distribution function for a continuous distribution on \( [0, \infty) \).

Note that \( G \) is continuous on \( [0, \infty) \) with \( G(0) = 0 \) and \( G(z) \to 1 \) as \( z \to \infty \). Moreover, \[ g(z) = G^\prime(z) = \frac{k z^{k-1}}{(1 + z^k)^2} \gt 0, \quad z \in (0, \infty) \] so \( G \) is strictly increasing on \( [0, \infty) \).

\( Z \) has probability density function function \( g \) given by \[ g(z) = \frac{k z^{k-1}}{(1 + z^k)^2}, \quad z \in (0, \infty) \]

- If \( 0 \lt k \lt 1 \) then \( g \) is decreasing on \( (0, \infty) \) with \( g(z) \to \infty \) as \( z \to 0^+ \).
- If \( k = 1 \) then \( g \) is deceasing on \( [0, \infty) \) with \( g(0) = 1 \).
- If \( k \gt 1 \) then \( g \) is unimodal with mode \( \left(\frac{k - 1}{k + 1}\right)^{1/k}. \)

\( g = G^\prime \) was given in (1). The rest follows from \[ g^{\prime}(z) = \frac{k z^{k-2}[(k - 1) - (k + 1) z^k]}{1 + z^k}, \quad z \in (0, \infty) \]

Open the special distribution simulator and select the log-logistic distribution. Vary the shape parameter and note the shape of the probability density function. For selected values of the shape parameter, run the simulation 1000 times and note the agreement between the empirical density function and the probability density function.

\( Z \) has quantile function \( G^{-1} \) given by \[ G^{-1}(p) = \left(\frac{p}{1 - p}\right)^{1/k}, \quad p \in [0, 1) \]

- The first quartile is \( q_1 = (1/3)^{1/k} \).
- The median is \( q_2 = 1 \).
- The third quartile is \( q_3 = 3^{1/k} \).

The formula for \( G^{-1} \) follows from (2) by solving \( p = G(z) \) for \( z \) in terms of \( p \).

Recall that \( p / (1 - p) \) is the odds ratio associated with probability \( p \in (0, 1) \). Thus, the quantile function of the basic log-logistic distribution with shape parameter \( k \) is the \( k \)th root of the odds ratio function. In particular, the quantile function of the standard log-logistic distribution is the odds ratio function itself.

Open the special distribution calculator and select the log-logistic distribution. Vary the shape parameter and note the shape of the distribution function. For selected values of the shape parameter, computer a few values of the distribution function and the quantile function.

\( Z \) has reliability function \( \bar{G} \) given by \[ \bar{G}(z) = \frac{1}{1 + z^k}, \quad z \in [0, \infty) \]

This follows trivially from (1) since \( \bar{G} = 1 - G \).

The basic log-logistic distribution has either decreasing failure rate, or mixed decreasing-increasing failure rate, depending on the shape parameter.

\( Z \) has failure rate function \( r \) given by \[ r(z) = \frac{k z^{k-1}}{1 + z^k}, \quad z \in (0, \infty) \]

- If \( 0 \lt k \le 1 \) then \( r \) is decreasing on \( (0, \infty) \).
- If \( k \gt 1 \) then \( r \) is decreasing on \( (0, z_k) \) and increasing on \( (z_k, \infty) \) where \( z_k = (k - 1)^{1/k} \).

Recall that the is \( r(z) = g(z) / \bar{G}(z) \) so the formula follows from (2) and (6). Parts (a) and (b) follow from \[ r^\prime(z) = \frac{k z^{k-1}[(k - 1) - z^k]}{(1 + z^k)^2}, \quad z \in (0, \infty) \]

The moments (about 0) of the basic log-logistic distribution have an interesting expression in terms of the beta function \( B \) and in terms of the sine function.

If \( n \ge k \) then \( \E(Z^n) = \infty \). If \( 0 \le n \lt k \) then \[ \E(Z^n) = B\left(1 - \frac{n}{k}, 1 + \frac{n}{k}\right) = \frac{\pi n / k}{\sin(\pi n / k)} \]

From (2), \[ \E(Z^n) = \int_0^\infty z^n \frac{k z^{k-1}}{(1 + z^k)^2} dz \] The substitution \( u = 1 / (1 + z^k) \), \( du = -k z^{k-1}/(1 + z^k)^2 \) gives \[ \E(Z^n) = \int_0^1 (1/u - 1)^{n/k} du = \int_0^1 u^{-n/k} (1 - u)^{n/k} du \] The result now follows from the definition of the beta function.

The mean and variance of the basic log-logistic distribution follow easily from (8):

If \( 0 \lt k \le 1 \) then \( \E(Z) = \infty \). If \( k \gt 1 \) then \[ \E(Z) = \frac{\pi/k}{\sin(\pi/k)} \]

If \( 0 \lt k \le 1 \) then \( \var(Z) \) does not exist. If \( 1 \lt k \le 2 \) then \( \var(Z) = \infty \). If \(k \gt 2 \) then \[ \var(Z) = \frac{2 \pi / k}{\sin(2 \pi / k)} - \frac{\pi^2 / k^2}{\sin^2(\pi / k)} \]

Open the special distribution simulator and select the log-logistic distribution. Vary the shape parameter and note the size and location of the mean/standard deviation bar. For selected values of the shape parameter, run the simulation 1000 times and note the agreement between the true mean and standard deviation, and the empirical mean and standard deviation.

The basic log-logistic distribution is preserved under power transformations.

If \( Z \) has the basic log-logistic distribution with shape parameter \( k \) and if \( n \gt 0 \), then \( W = Z^n \) has the basic log-logistic distribution with shape parameter \( k / n \).

For \( w \in [0, \infty) \), \[ \P(W \le w) = \P(Z \le w^{1/n}) = G\left(w^{1/n}\right) = \frac{w^{k/n}}{1 + w^{k/n}} \] As a function of \( w \), this is the CDF of the basic log-logistic distribution with shape parameter \( k/n \).

In particular, it follows that if \( V \) has the standard log-logistic distribution and \( k \gt 0 \), then \( Z = V^{1/k} \) has the basic log-logistic distribution with shape parameter \( k \).

Since the quantile function of the basic log-logistic distribution has a simple closed form, the distribution can be simulated using the random quantile method.

Suppose that \( k \in (0, \infty) \).

- If \( U \) has the standard uniform distribution then \( Z = [U / (1 - U)]^{1/k} \) has the basic log-logistic distribution with shape parameter \( k \).
- If \( Z \) has the basic log-logistic distribution with shape parameter \( k \) then \( U = Z^k / (1 + Z^k) \) has the standard uniform distribution.

These results follow immediately from (1) and (4).

Open the random quantile experiment and select the log-logistic distribution. Vary the shape parameter and note the shape of the distribution and probability density functions. For selected values of the parameter, run the simulation 1000 times and note the agreement between the empirical density function and the probability density function.

Of course, as mentioned in the introduction, the log-logistic distribution is related to the logistic distribution. The simplest result relates the standard forms of the two distributions.

Suppose that \( k, \, b \gt 0 \).

- If \( Z \) has the standard log-logistic distribution then \( Y = \ln(Z) \) has the standard logistic distribution.
- If \( Y \) has the standard logistic distribution then \( Z = e^Y \) has the standard log-logistic distribution.
- If \( Z \) has the basic log-logistic distribution with shape parameter \( k \) then \( Y = \ln(Z) \) has the logistic distribution with location parameter 0 and scale parameter \( 1/k \).
- If \( Y \) has the logistic distribution with location parameter \( 0 \) and scale parameter \( b \) then \( Z = e^Y \) has the basic log-logistic distribution with shape parameter \( 1 / b \).

For part (a), \[ \P(Y \le y) = \P\left(Z \le e^y\right) = \frac{e^y}{1 + e^y}, \quad y \in \R \] As a function of \( y \), this is the CDF of the standard logistic distribution.

Similarly, for part (b), \[ \P(Z \le z) = \P[Y \le \ln(z)] = \frac{e^{\ln(z)}}{1 + e^{\ln(z)}} = \frac{z}{1 + z}, \quad z \in (0, \infty) \] and as a function of \( z \), this is the CDF of the standard log-logistic distribution.

For part (c), by (12) we can take \( Z = W^{1/k} \) where \( W \) has the standard log-logistic distribution. Then \( Y = \ln(Z) = \frac{1}{k} \ln(W) \). But by (a), \( \ln(W) \) has the standard logistic distribution, and hence \( \frac{1}{k} \ln(W) \) has the logistic distribution with location parameter \( 0 \) and scale parameter \( 1/k \).

For part (d), we can take \( Y = b V \) where \( V \) has the standard logistic distribution. Hence \( Z = e^Y = e^{b V} = \left(e^V\right)^b \). But by (b), \( e^V \) has the standard log-logistic distribution, and so by (12) \( \left(e^V\right)^b \) has the log-logistic distribution with shape parameter \( 1 / b \).

The standard log-logistic distribution is the same as the standard beta prime distribution.

The PDF of the standard log-logistic distribution is \( g(z) = 1 / (1 + z)^2 \) for \( z \in [0, \infty) \), which is the same as the PDF of the standard beta prime distribution.

The basic log-logistic distribution is generalized, like so many distributions on \( [0, \infty) \), by adding a scale parameter. Thus, if \( Z \) has the basic log-logistic distribution with shape parameter \( k \in (0, \infty) \) and if \( b \in (0, \infty) \) then \( X = b Z \) has the log-logistic distribution with shape parameter \( k \) and scale parameter \( b \). In particular, if \( V \) has the standard log-logistic distribution, then \( Z = b V^{1/k} \) has the log-logistic distribution with shape parameter \( k \) and scale parameter \( b \).

\( X \) has distribution function \( F \) given by \[ F(x) = \frac{x^k}{b^k + x^k}, \quad x \in [0, \infty) \]

Recall that \( F(x) = G(x / b) \) where \( G \) is the distribution function of the basic log-logistic distribution with shape parameter \( k \).

\( X \) has probability density function \( f \) given by \[ f(x) = \frac{b^k k x^{k-1}}{(b^k + x^k)^2}, \quad x \in (0, \infty) \]

- If \( 0 \lt k \lt 1 \) then \( f \) is decreasing on \( (0, \infty) \) with \( f(x) \to \infty \) as \( x \to 0^+ \).
- f If \( k = 1 \) then \( f \) is deceasing on \( [0, \infty) \) with \( f(0) = 1/b \).
- If \( k \gt 1 \) then \( g \) is unimodal with mode \( b \left(\frac{k - 1}{k + 1}\right)^{1/k}. \)

Recall that \( f(x) = \frac{1}{b} g\left(\frac{x}{b}\right) \) where \( g \) is the PDF of the basic log-logistic distribution given in (2). Also of course, \( f = F^\prime \).

Open the special distribution simulator and select the log-logistic distribution. Vary the shape and scale parameters and note the shape of the probability density function. For selected values of the parameters, run the simulation 1000 times and note the agreement between the empirical density function and the probability density function.

\( X \) has quantile function \( F^{-1} \) given by \[ F^{-1}(p) = b \left(\frac{p}{1 - p}\right)^{1/k}, \quad p \in [0, 1) \]

- The first quartile is \( q_1 = b (1/3)^{1/k} \).
- The median is \( q_2 = b \).
- The third quartile is \( q_3 = b 3^{1/k} \).

Recall that \( F^{-1}(p) = b G^{-1}(p) \) for \( p \in [0, 1) \) where \( G^{-1} \) is the quantile function of the basic log-logistic distribution given in (4).

Open the special distribution calculator and select the log-logistic distribution. Vary the shape and sclae parameters and note the shape of the distribution function. For selected values of the parameters, computer a few values of the distribution function and the quantile function.

\( X \) has reliability function \( \bar{F} \) given by \[ \bar{F}(x) = \frac{b^k}{b^k + x^k}, \quad x \in [0, \infty) \]

This follows trivially from (19) since \( \bar{F} = 1 - F \).

The log-logistic distribution has either decreasing failure rate, or mixed decreasing-increasing failure rate, depending on the shape parameter.

\( X \) has failure rate function \( R \) given by \[ R(x) = \frac{k x^{k-1}}{b^k + x^k}, \quad x \in (0, \infty) \]

- If \( 0 \lt k \le 1 \) then \( r \) is decreasing on \( (0, \infty) \).
- If \( k \gt 1 \) then \( r \) is decreasing on \( (0, x_k) \) and increasing on \( (x_k, \infty) \) where \( x_k = b (k - 1)^{1/k} \).

Recall that \( R(x) = \frac{1}{b} r\left(\frac{x}{b}\right) \) where \( r \) is the failure rate function of the basic log-logistic distribution. Hence the results follow from (7).

The moments of the general log-logistic distribution can be computed easily from the representation \( X = b Z \) where \( Z \) has the basic log-logistic distribution.

If \( n \ge k \) then \( \E(X^n) = \infty \). If \( 0 \le n \lt k \) then \[ \E(X^n) = b^n B\left(1 - \frac{n}{k}, 1 + \frac{n}{k}\right) = b^n \frac{\pi n / k}{\sin(\pi n / k)} \]

If \( 0 \lt k \le 1 \) then \( \E(X) = \infty \). If \( k \gt 1 \) then \[ \E(X) = b \frac{\pi/k}{\sin(\pi/k)} \]

If \( 0 \lt k \le 1 \) then \( \var(X) \) does not exist. If \( 1 \lt k \le 2 \) then \( \var(X) = \infty \). If \(k \gt 2 \) then \[ \var(X) = b^2 \left[\frac{2 \pi / k}{\sin(2 \pi / k)} - \frac{\pi^2 / k^2}{\sin^2(\pi / k)} \right] \]

Open the special distribution simulator and select the log-logistic distribution. Vary the shape and scale parameters and note the size and location of the mean/standard deviation bar. For selected values of the parameters, run the simulation 1000 times and note the agreement between the true mean and standard deviation, and the empirical mean and standard deviation.

Since the log-logistic distribution is a scale family for each value of the shape parameter, it is trivially closed under scale transformations.

If \( X \) has the log-logistic distribution with shape parameter \( k \) and scale parameter \( b \), and if \( c \in (0, \infty) \), then \( Y = c X \) has the log-logistic distribution with shape parameter \( k \) and scale parameter \( b c \).

The log-logistic distribution is preserved under power transformations.

If \( X \) has the log-logistic distribution with shape parameter \( k \) and scale parameter \( b \), and if \( n \in (0, \infty) \), then \( Y = X^n \) has the log-logistic distribution with shape parameter \( k / n \) and scale parameter \( b^n \).

We can take \( X = b Z \) where \( Z \) has the basic log-logistic distribution with shape parameter \( k \). Then \( X^n = b^n Z^n \). But by (12), \( Z^n \) has the basic log-logistic distribution with shape parameter \( k / n \) and hence \( X \) has the log-logistic distribution with shape parameter \( k / n \) and scale parameter \( b^n \).

Again, since the quantile function of the log-logistic distribution has a simple closed form, the distribution can be simulated using the random quantile method.

Suppose that \( k, \, b \in (0, \infty) \).

- If \( U \) has the standard uniform distribution then \( X = b [U / (1 - U)]^{1/k} \) has the log-logistic distribution with shape parameter \( k \) and scale parameter \( b \).
- If \( X \) has the log-logistic distribution with shape parameter \( k \) and scale parameter \( b \), then \( U = X^k/(b^k + X^k) \) has the standard uniform distribution.

Open the random quantile experiment and select the log-logistic distribution. Vary the shape and scale parameters and note the shape and location of the distribution and probability density functions. For selected values of the parameters, run the simulation 1000 times and note the agreement between the empirical density function and the probability density function.

Again, the logarithm of a log-logistic variable has the logistic distribution.

Suppose that \( k, \, b, \, c \in (0, \infty) \) and \( a \in \R \).

- If \( X \) has the log-logistic distribution with shape parameter \( k \) and scale parameter \( b \) then \( Y = \ln(X) \) has the logistic distribution with location parameter \( \ln(b) \) and scale parameter \( 1 / k \).
- If \( Y \) has the logistic distribution with location parameter \( a \) and scale parameter \( c \) then \( X = e^Y \) has the log-logistic distribution with shape parameter \( 1/c \) and scale parameter \( e^a \).

For part (a), we can take \( X = b V^{1/k} \) where \( V \) has the standard log-logistic distribution. Then \( Y = \ln(X) = \ln(b) + \frac{1}{k} \ln(V) \). But \( \ln(V) \) has the standard logistic distribution, so \( Y \) has the logistic distribution with location parameter \( \ln(b) \) and scale parameter \( 1/k \). For part (b), we can take \( Y = a + c U \) where \( U \) has the standard logistic distribution. Hence \( X = e^Y = e^a e^{c U} = e^a \left(e^U\right)^c \). But \( e^U \) has the standard log-logistic distribution so \( X \) has the log-logistic distribution with shape parameter \( 1/c \) and scale parameter \( e^a \).