Is there a common name to refer to distributions that are bounded on one side, and unbounded on the other side? For example, log-normal distribution, where the minimum value is zero, the maximum is +infinity.
Solved – Common name for distributions that are bounded on one side
distributionslognormal distribution
Related Solutions
There's no one universal list, and there can't be an exhaustive list. However, you can find a list of some continuous densities with bounds in wiki: https://en.wikipedia.org/wiki/List_of_probability_distributions#Supported_on_a_bounded_interval
Also, remember that you can take any distribution and bound it in an interval. These are called truncated distributions, e.g. see truncated normal.
I am guessing that you are looking for a positive, continuous probability distribution with infinite mean and with a maximum density away from zero.
I thought that by analogy with a Gamma distribution ($p(x) \propto x^a \exp(-x) \, dx$), we could try something with a rational (polynomial) rather than an exponential tail. After a little bit of lazy R and Python (sympy) experimentation, I came up with
$$ p(x) = \frac{1}{2\sqrt{3}\cdot \pi/9} \cdot \frac{x}{1+x^3} \, dx $$
(I initially tried $p(x) \propto x/(1+x^2)$, but its integral diverges.) $\int_0^\infty p(x) \, dx$ is 1, as required, and $\int_0^\infty x p(x) \, dx$ diverges.
I don't know if this distribution has a name/literature associated with it.
The CDF is available in closed form but is pretty horrible ... (see Python code below ...)
$$ 3 \sqrt{3} \left(- \frac{\log{\left(X + 1 \right)}}{6 \pi} + \frac{\log{\left(X^{2} - X + 1 \right)}}{12 \pi} + \frac{\sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} X}{3} - \frac{\sqrt{3}}{3} \right)}}{6 \pi}\right) + \frac{1}{4} $$
Without actually trying anything, I would guess that distributions of the form $x^a/(1+x^{a+2})$ will generally have these properties (but the computations will get progressively nastier). Someone with more analysis skills could probably prove a bunch of things.
An extremely knowledgeable colleague identified this as almost the same as a "Beta-Type 2 (m=2/3,n=1/3)" distribution (a Beta-Type 2 distribution has a term of the form $(1+x)^n$ in the denominator rather than the $1+x^n$ given above). You might want to use the Beta-Type 2 instead of my version; since you know what it's called you can search for useful code, or literature on its properties (e.g. here or here or McDonald et al 2013), or cite it in a paper: "Beta-Type 2" sounds so much better than "a distribution that some guy on CrossValidated made up".
... the Beta-Type 2 family with its density as
$$ f(x) = \frac{1}{\textrm{Beta}(m,n)} \frac{x^{m-1}}{(1+x)^{m+n}} $$
over the support $(0,\infty)$
It is evident that if $m$ is chosen to be > 1, the mode will be away from 0. Also, if $n$ is chosen to be $\leq 1$, then the mean will be infinite. This family will produce [an] uncountable number of models with the property you are looking for. ... If you set $Y=X^3$ in your model, then it becomes a Beta-Type 2 $(m=2/3,n=1/3)$ and you can see that this fits the description I have given above.
They also identified the name of @ThomasLumley's contribution:
... it is called power gamma model or exponential-gamma model.
McDonald, James B., Jeff Sorensen, and Patrick A. Turley. “Skewness and Kurtosis Properties of Income Distribution Models.” Review of Income and Wealth 59, no. 2 (2013): 360–74. https://doi.org/10.1111/j.1475-4991.2011.00478.x.
R code:
f <- function(x) 1/(2*sqrt(3)*pi/9)*x/(1+x^3)
integrate(f, 0, Inf) ## 1 with absolute error < 4e-07
curve(f, from=0, to=10)
Python code (because I'm too lazy to integrate):
from sympy import *
x, n, N = symbols('x,n,N')
n=integrate(x/(1+x**3), (x, 0, oo)) ## 2*sqrt(3)*pi/9
integrate(x**2/(1+x**3), (x, 0, oo)) ## infinite mean
cdf = integrate(1/n*x/(1+x**3), (x, 0, X))
print(latex(cdf))
Best Answer
I would say no, as there are quite a few distributions in addition to log-normal that have support on $[ 0, \infty)$, such as the $\chi^2$ or gamma. (Wikipedia even has a list devoted to this criterion.)
In practice there are different circumstances in which such distributions are useful to approximate actually observed and measured phenomena. For one example, in some situations you may use the term censored at the boundary to describe how observations can go above and below the boundary, but they can only be measured within a certain support (and when they go outside this support they are recorded as being at the end point and/or beyond). This is typically referred to when you have an instrument that can not measure the numerical value outside of the bounds, it only knows it is at or below the boundary. For log-normal an example is the measurement of $\log(\text{wages})$. I believe for the US census they censor the distribution at $0$, although people who own their own businesses can be in the negative. (Some macro economic variables are well approximated by a log-normal distribution, but the support of the actual micro level units is partly in the negative.)
Another example (as user41315 mentioned) are truncated distributions. Truncated means "chopped off". Sometimes we only observe/record the measurement if it exceeds the boundary. For another economic example of wages, lets say you only had to file taxes if your wages were above $0$. So it is not like the census that just records $0$ even if you have less than $0$, you just simply don't observe the individuals with less than $0$ wages. As whuber stated in the comment, you can take take any distribution and re-express it as a truncated one.
The descriptions of censored or truncated refer to how the data are measured, and not to particular distributions. Not all measurements that are bounded on $[ 0, \infty)$ are necessarily truncated or censored though. For example distances or squares of values we know can not go below $0$.