Nakagami Distribution

Generalization of the chi distribution. Shape parameter is $\nu >0.$ The support is $x\ge 0.$

where $\gamma$ is the lower incomplete gamma function, $\gamma \left(\nu ,x\right)={\int }_0^x{t}^{\nu -1}{e}^{-t}dt$.

Implementation: scipy.stats.nakagami

MLE of the Nakagami Distribution in SciPy (nakagami.fit)

The probability density function of the nakagami distribution in SciPy is

for $x$ such that $\frac{x-\mu }{\sigma }\ge 0$, where $\nu \ge \frac{1}{2}$ is the shape parameter, $\mu$ is the location, and $\sigma$ is the scale.

The log-likelihood function is therefore

which can be expanded as

Leaving supports constraints out, the first-order condition for optimality on the likelihood derivatives gives estimates of parameters:

where ${\psi }^{(0)}$ is the polygamma function of order $0$; i.e. ${\psi }^{(0)}(\nu )=\frac{d}{d\nu }\log \Gamma (\nu )$.

However, the support of the distribution is the values of $x$ for which $\frac{x-\mu }{\sigma }\ge 0$, and this provides an additional constraint that

For $\nu =\frac{1}{2}$, the partial derivative of the log-likelihood with respect to $\mu$ reduces to:

which is positive when the support constraint is satisfied. Because the partial derivative with respect to $\mu$ is positive, increasing $\mu$ increases the log-likelihood, and therefore the constraint is active at the maximum likelihood estimate for $\mu$

For $\nu$ sufficiently greater than $\frac{1}{2}$, the likelihood equation $\frac{\partial l}{\partial \mu }(\nu ,\mu ,\sigma )=0$ has a solution, and this solution provides the maximum likelihood estimate for $\mu$. In either case, however, the condition $\mu ={\min }_i{x}_i$ provides a reasonable initial guess for numerical optimization.

Furthermore, the likelihood equation for $\sigma$ can be solved explicitly, and it provides the maximum likelihood estimate

Hence, the _fitstart method for nakagami uses

as initial guesses for numerical optimization accordingly.