mgf_lognormal: Moment Generating Function for a Lognormal Distribution

In flexCountReg: Estimation of a Variety of Count Regression Models

Moment Generating Function for a Lognormal Distribution

Description

Computes the value of the moment generating function (MGF) for a lognormal distribution at a given point through numerical integration. This function is particularly useful for distributions where the MGF does not have a closed-form solution. The lognormal distribution is specified by its log-mean (\mu) and log-standard deviation (\sigma).

Usage

mgf_lognormal(mu, sigma, n)

Arguments

mu	The mean of the log-transformed variable, corresponding to \mu in the lognormal distribution's parameters.
sigma	The standard deviation of the log-transformed variable, corresponding to \sigma in the lognormal distribution's parameters.
n	The point at which to evaluate the MGF, often denoted as t in the definition of the MGF. This parameter essentially specifies the order of the moment generating function.

Details

The moment generating function (MGF) for the lognormal distribution does not have a closed form solution. The MGF is defined as:

M_x(n) = \int_0^\infty e^{nx}\frac{1}{x\sigma\sqrt{2\pi}} e^{-\frac{(\ln(x)-\mu)^2}{2\sigma^2}}\,dx

The MGF for the lognormal distribution is useful for adjusting the predictions of generalized linear mixed models (GLMMs) that have parameters that follow a lognormal distribution and use a log link function. The adjustment for the mean value is the MGF with n=1 or E[e^x]=M_x(n=1). The variance for the lognormal random parameter is:

Var(e^x)=E[e^{2x}]-E[e^x]^2=M_x(n=2)-M_x(n=1)^2

Value

The estimated value of the moment generating function (MGF) for the specified lognormal distribution at the given point.

Examples

mu <- 0
sigma <- 1
n <- 1
mgf_value <- mgf_lognormal(mu, sigma, n)
print(mgf_value)

flexCountReg documentation built on Jan. 20, 2026, 1:06 a.m.