Is the harmonic mean the maximum likelihood estimator for some common continous distribution's parameter?

Question

If $y$ is a vector of continuous data the arithmetic mean is the maximum likelihood estimator for $\mu$ when assuming $y \sim \text{Normal}(\mu,\sigma)$ (not uniquely, of course). The geometric mean is the maximum likelihood estimator of $\mu$ when assuming $y \sim \text{Log-Normal}(\mu,\sigma)$. Similarly, is the harmonic mean the maximum likelihood estimator for a parameter of some common distribution?

Answer 1

This will happen for the recirpocal of a random variable following an Exponential distribution.

Let $X$ follow an Exponential distribution with probability density function

$$f_X(x) = \alpha e^{-\alpha x}$$

Consider $$ Y = \frac 1{X} \Rightarrow X = 1/Y \Rightarrow \frac{\partial X}{\partial Y} = -Y^{-2}$$

Then

$$f_Y(y) = \left|\frac{\partial X}{\partial Y}\right|\cdot f_X(1/y)= \alpha y^{-2}e^{-\alpha/y},\;\; y\in (0,\infty)$$

The log-likelihood of a sample of $n$ observations will be

$$\ln L = n\ln\alpha -2\sum_{i=1}^n\ln y_i -\alpha \sum_{i=1}^n(1/y_i)$$

and

$$\frac{\partial \ln L}{\partial \alpha} = \frac n{\alpha}-\sum_{i=1}^n(1/y_i) = 0 \Rightarrow \hat \alpha_{ML} = \frac n{\sum_{i=1}^n(1/y_i)}$$

which is the harmonic mean of the $y$'s.