Moment Generating Function for a Lognormal Distribution
mgf_lognormal.RdComputes 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 (\(\sigma\)).
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.
Value
The estimated value of the moment generating function (MGF) for the specified lognormal distribution at the given point.
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: $$Var(e^x)=E[e^{2x}]-E[e^x]^2=M_x(n=2)-M_x(n=1)^2$$
Examples
mu <- 0
sigma <- 1
n <- 1
mgf_value <- mgf_lognormal(mu, sigma, n)
print(mgf_value)
#> [1] 2.821428e+36