# What does 'Parent Distribution' mean in statistics?

I am studying some articles related to statistics and some of them mention the term 'Parent Distribution'. What does that mean? Is it a distribution model that the authors decide to use as a basis of comparison with other distributions? I'm quite confused as I couldn't find the definition of this term anywhere.

• Welcome to Cross Validated! In what context are you reading this?
– Dave
Aug 10, 2022 at 19:46
• It might refer to the population distribution from which the sample was drawn, by contrast with some derived quantity -- the distribution of some statistic. But I agree with Dave; we should have context to be sure what you're looking at. Aug 11, 2022 at 6:32
• The easiest way to get a helpful answer is to be more specific about the "some articles related to statistics" you are reading. Add proper citations. Aug 11, 2022 at 8:52
• In my reading it is just a term of art. Just as parents produce children (they wouldn't be parents otherwise), so a parent distribution is whatever is presumed to produce your data. A fanciful picture is to imagine that it is as if Nature or Society has a bag or box with all possible values and is pulling the values you see out of that bag or box. The words as if mean a lot here. There was an entire book on that: en.wikipedia.org/wiki/The_Philosophy_of_%27As_if%27 Aug 11, 2022 at 9:42
• I haven't used the term myself that I can remember. Nor can I recall seeing a formal definition, but I think that's not a problem. Why would anyone ever use that term in statistics unless a loose analogy with parents in the biological sense was intended or in mind? I can't see that you're offering a better or different explanation. It would be correct but not interesting or helpful to say that it's a just a term some people use in statistics. Aug 11, 2022 at 10:26

Statistics sometimes gives words a special meaning that's not close to their "common" meaning recognizable to most people in the world. This can make for amusing conversations; here is an example where the term in question is "power".

So what does "parent distribution" mean? Actually it means different things to different statisticians.

In extreme value theory, the parent distribution generates a sample of iid observations and we are interested in the distribution of an extreme sample statistic such as the maximum:

For a set of observations $$(x_1, x_2, \ldots, x_k)$$ from an identically distributed and independent set of random variables $$(X_1, X_2, \ldots, X_k)$$, the distribution of $$X_i$$ is called the parent (or initial) distribution. The maximum extreme value of the observed values is a random variable $$M_k = \operatorname{Maximum}(X_1, X_2, \ldots, X_k)$$.

 B.M. Ayyub and R.H. McCuen. Probability, Statistics, and Reliability for Engineers and Scientists (p. 162). CRC Press.

In measurement theory, the "parent distribution" is the limiting distribution of a relative frequency distribution:

It is important to distinguish between the sample and parent distributions. In the theory of statistics, the parent distribution refers to the number of possible measured values, $$\xi_i$$; the parent population might consist of an infinite number of values. Two independent parameters, the mean $$\mu_{\text{parent}}$$, and a standard deviation, $$\sigma_{\text{parent}}$$, characterise the parent distribution. In practice when we take a series of measurements in an experiment, we take a selection, or sample, from this parent distribution which results in a distribution called the sample distribution. This distribution is centred on the mean of the data set, $$\bar{x}$$, and has a standard deviation: $$\sigma_{\text{sample}}$$.

 I. Hughes and T. Hase. Measurements and their Uncertainties (p. 13). OUP Oxford.

In a mixture $$h(y|\mu) = \int f(y|\mu,\nu)g(\nu)d\nu$$, the "parent distribution" is the distribution of the mixture components.

[In reference to a mixture of Poissons as a way to deal with overdispersion] Mixing causes the proportion of zero counts to increase. It exceeds the corresponsing proportion of zeros in the parent distribution. The overdispersion and excess of zeros, relative to the Poisson, are related consequences of unobserved heterogeneity. That is, irrespective of the form of the $$g(\nu_i)$$ for the mixture, and provided it is nondegenarete, for the parent and mixture distributions with the same mean $$\mu_i$$, it is true that $$h(y_i=0|\mu_i) \geq f(y_i=0|\mu_i,\nu_i)$$.

 A.C. Cameron and K. Trivedi. Regression Analysis of Count Data (p. 116). Cambridge University Press.

And sometimes the "parent distribution" refers to the true underlying distribution from which the data is sampled.

When we train a model, we use numbers, or sets of numbers, that we measure; we can think of those numbers as coming from a probability distribution. We’ll refer to that distribution as the parent distribution. Think of it as the thing that generates the data we’ll feed our model; another, more Platonic, way to think about it is as the ideal set of data that our data is approximating.

 R.T. Kneusel. Practical Deep Learning: A Python-Based Introduction (p. 7). No Starch Press.

So you see, the term "parent distribution" is confusing because it doesn't have a unified statistical definition accepted by all. It would help to include proper citation to the articles you are reading.