Find the mode of a probability distribution function

Question

I am trying the find mode of a probability distribution function given by

\begin{equation} g(x/\alpha,\beta,\sigma)=\frac{1}{\Gamma \left( \alpha \right)\beta^{\alpha}}exp\left\{{-\frac{x^2}{2\sigma^{2}}\frac{1}{\beta}}\right\}\frac{x^{2\alpha-1}}{2^{\alpha-1}\sigma^{2\alpha}}I_{{\rm I\!R}^{+}}(x) \end{equation} where $\alpha, \beta, \sigma >0$.

The limit of this density function as $x \to \infty$ is $0$ and the limit of its hazard function as $x \to \infty$ is $\infty$. Also the limit of this density function and its hazard function as $x \to 0^{+}$ is given by

\begin{equation} \lim_{x \to 0^{+}} g(x)=\lim_{x \to 0^{+}} h(x) = \begin{cases} 0 & \alpha > \dfrac{1}{2} \\ \dfrac{\sqrt{2}}{\sigma\sqrt{\pi \beta}} & \alpha= \dfrac{1}{2} \\ \infty & \alpha < \dfrac{1}{2} \end{cases} \end{equation}

The mode can be obtained by taking the derivative of $g(x)$ and setting it to zero. The obtained mode is an increasing function of $\alpha$, $\beta$ and $\sigma$. Thus, The mode given as \begin{equation} k(x)=\sigma \sqrt{ \beta(2\alpha -1)}. \end{equation}

However, the definition of square root, the mode is only defined when $\alpha \geq \dfrac{1}{2}$. However, my $\alpha$ parameter is defined on $(0, \infty)$. There's something I am missing out!!

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Answer 1

Consider that there are shapes of pdf that have a mode, but at which the derivative of the pdf is not zero (the Laplace being an obvious example).

There are also cases where there's no mode in the domain of the variable (examples below).

That is, we can't say as a general statement "the mode can be obtained by taking the derivative of $g(x)$ and setting it to zero".

You must give up trying to look at the value where the derivative is zero when it's clear that the reasoning underlying its use (that the derivative is zero at the mode) fails. Here's a simple example where it works for part of the domain of a parameter, but not everywhere:

(Here I've include $x=0$ in the domain for the exponential case; if it was excluded, strictly there's no mode in the domain for that case (the supremum would be $1$, which is the value of $\frac{1}{\mu} \exp(-\frac{x}{\mu})$ at $0$, but the density would not be defined there). For the case where the shape parameters $<1$, you couldn't include $x=0$, and there would be no mode.)

So there's no use looking for where the derivative is zero when it isn't zero at the mode - indeed it may not be zero anywhere.

Further, for some densities, even when the derivative is 0, it doesn't imply there's a mode there. Consider this density (a beta density):

There's not a local mode where the derivative is zero; it's an antimode. You can also have a density with a horizontal point of inflexion which will be neither a mode nor an antimode:

As a result, it's not sufficient to simply calculate a formula at which the derivative is zero; even if you can calculate such values, that may not tell you where the modes are.

You must work out when that calculation corresponds to modes of the density; where that fails, quite often the location of the mode is obvious (if it exists at all).

A first exercise would be to draw the density at a few values of $\alpha$ and see how it behaves. Then you should be able to bring some reasoning to bear on the problem, which will tell you where the mode is when $\alpha <\frac{1}{2}$.

It's possible for the function to be everywhere decreasing in the domain; if it's on an open interval there may be no mode in that interval. For example, in the case of the gamma with $\alpha<1$, it's not uncommon to say "there's a mode at 0" even though the limit isn't in the interval - it's strictly incorrect to say there's a mode in that case (but usually one can understand the actual intent if someone says there's a mode at 0 even though the function is unbounded in the limit).

Here's an example of what your density looks like at $\beta=\sigma=1$ for three different values of $\alpha$ near $0.5$:

It does clearly suggest the behavior for $\alpha<\frac{1}{2}$ may be monotonically decreasing. The obvious thing to do is to try to check whether it's the case that $g(x+\epsilon)<g(x)$ with $\alpha<\frac{1}{2}$ when $\epsilon>0$ (e.g. is the derivative always negative when $\alpha<\frac{1}{2}$?)

If it's the case that it is monotonic decreasing (you should carry out such a check for yourself), it doesn't strictly have a mode; best to simply describe the behavior near 0 for $\alpha<\frac{1}{2}$.