What is the statistics term for exact value that occurs in otherwise continous distribution?

Question

For some continuous quantities (e.g. daily rainfall at a certain location), there is one exact value that occurs often (in the case of daily rainfall that's the value of zero: there are days on which it does not rain). However, for continuous distributions, the chance of finding some other exact number (say 3.2000... mm) is zero. I remember from statistics class that an exact number that does occur more often has a name. But I can't remember it! Can anyone help?

Answer 1

The common term here is a "point mass" in an otherwise continuous distribution. In the specific case where there is a point mass at zero we refer to the distribution as being zero inflated.

Answer 2

I can't find a good technical reference for this, but I would call this a point mass (this has also been pointed out in comments). For example, this introduction to Bayesian statistics gives the example:

This time, the Bayesian believes that the probability p of an RU-486 baby is uniformly distributed between 0 and one-half, but has a point mass of 0.5 at one-half. That is, she believes there is a 50% chance that no difference exists between standard therapy and RU-486. But if a difference exists, she thinks that RU-486 is better, but she is completely unsure about how much better it would be.

Or, Trong et al. say:

Hence, the distribution of fitness effects of the mutations is a mixture of the point-mass and the continuous distribution.

In physics (since the question was originally asked on Physics SE) a point mass is sometimes referred to as (a multiple of) a Dirac delta function (there is an example/section on "probability theory" in the linked Wikipedia article ...)

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Trong, Dang Duc, Nguyen Hoang Thanh, Nguyen Dang Minh, and Nguyen Nhu Lan. “Density Estimation of a Mixture Distribution with Unknown Point-Mass and Normal Error.” Journal of Statistical Planning and Inference 215 (December 1, 2021): 268–88. https://doi.org/10.1016/j.jspi.2021.04.002.

Answer 3

As a comment noted, these situations are not well described by a continuous distribution, but by a mixed one (part discrete, part continuous)

$$F_R(r) = \Pr(R\leq r) = \begin{cases} {\rm p}_0 & r=0 \\ F_{Rc}(r) & r>0, \end{cases}$$

where ${\rm p}_0$ is the strictly positive probability that the r.v. $R$ will take the value $0$, and $F_{Rc}(r)$ has the properties of a distribution function, except that its range is $({\rm p}_0, 1]$. The whole has a continuous graph, since $\lim_{r\downarrow 0} F_{Rc}(r) = {\rm p}_0$.

Sometimes for convenience we take a proper distribution function of a continuous random variable $G$, ranging in $[0,1]$, and we define something compact like $$F_{R}(r) = {\rm p}_0 + (1-{\rm p}_0) G(r).$$

In this formulation, the conceptual lines between a "mixed" distribution and a "mixture" distribution blur, since the formula above can also be seen as a "mixture" / convex combination of the degenerate distribution function (that takes the value $1$) and of $G$.

But you may also attempt to use a purely discrete distribution with perhaps a dense grid for support, if the possible inaccuracy (in case rainfall for some days is measured in between the designated values of the support), does not harm you.

Answer 4

A hurdle model has this characteristic. Here's the summary from the link:

Hurdle models were introduced by John G. Cragg in 1971, where the non-zero values of x were modelled using a normal model, and a probit model was used to model the zeros. The probit part of the model was said to model the presence of "hurdles" that must be overcome for the values of x to attain non-zero values, hence the designation hurdle model. Hurdle models were later developed for count data, with Poisson, geometric, and negative binomial models for the non-zero counts.

Answer 5

In a discrete distribution the value that has the highest probability of occurring is called the "mode" of the distribution. Similarly, the "mode" of a sample is the data value that occurs most often in that sample. The term "mode" can be applied to continuous distributions as well, but then it usually means a local maximum of the probability density function. The mode of a distribution is not necessarily unique - a distribution may have several modes, in which case it is a multimodal distribution.