I seek a univariate distribution with analytically expressible density function that approximates (a vertically scaled version of) the standard normal distribution around the origin, but with a heavy tail extending in the positive direction (only).
The distribution should be smooth and unimodal, with at least one parameter for specifying the heaviness of the one tail.
My first thought is to mix a standard normal distribution with an exponential distribution (which decays slower than a normal distribution), but this causes a step change at the origin, making the distribution unsmooth.
Trying to replace the exponential distribution with other positive distributions (such as Chi-squared, Beta prime, Chi, Dagum, F-distribution, Gamma, Erlang), it is possible to preserve smoothness, but this potentially introduces bimodality, depending on the parameters chosen.
Can you suggest any idea for a distribution that:
- has analytically expressible PDF
- approximates (a vertically scaled version of) the standard normal distribution for $x<0$
- is unimodal
- is smooth
- has a heavy tail in the $+x$ direction (with at least one parameter for specifying heaviness of the tail)
For a very-heavy tailed distribution on the right try:
$$f(x)= \frac{1}{\sqrt{2\pi}}e^{-x^2/2} \text{ when } x \le 0$$ $$f(x)= \frac{1}{\sqrt{2\pi}}\left(\frac{1}{1+\pi x^2/2} \right)\text{ when } x \ge 0$$
where the left hand side is part of a standard normal distribution with half the probability and the right hand side is part of a scaled Cauchy distribution with the other half. This is so heavy that the mean is infinite and the density overall looks like this