# Definition of Parameters [duplicate]

I imagine this either extremely simple or extremely complex. I am trying to understand the interpretation of the term 'parameter'.

A couple of quick online searches deliver an intuitive understanding of parameter, i.e. where a statistical model is a joint probability distribution over the variables of interest, parametric models denote a subset of such models (in the form of a set of probability distributions), and parameters are therefore unique finite-dimensional points that index a particular distribution.

All fine. I can see why means, variances, covariances, correlations, and any other such moments are parameters.

What I can't see is why numerous other features, such as regression coefficients, etc. (i.e. the Betas we try to estimate with a linear model) are called parameters.

Is is simply because things like regression coefficients are functions of parameters in the first sense? And when we estimate beta coefficients, we are really trying to estimate the ratio between parameters, or are the beta coefficients parameters in themselves?

(I assume it must be the latter, the OLS estimator, for example, is just a function of the sample that estimates a number called Beta; coverage of the OLS seems to imply that the OLS ratio of cov to var just happens to be a great estimator of that number, rather than of the cov to var ratio in the population...)

• Don't parameters just take you from a general case or a family of cases to a particular one? So a Gaussian distribution with $\mu=10$ and $\sigma^2=4$ is a particular case of Gaussian distributions, in the same way that a straight line with gradient $10$ and intercept $4$ is a particular case of straight lines. – Henry Oct 10 '14 at 23:26
• certainly, but that doesn't really couch with the definition of models as expressed above – Sue Doh Nimh Oct 10 '14 at 23:38
• If you think about a regression model as specifying the conditional distribution of $(Y|X=x)$, then the parameters of the line are parameters that specify the conditional mean $E(Y|X=x)$. – Glen_b Oct 11 '14 at 0:34
• Just wanted to add (for what it's worth) that the question which this was marked a duplicate of, while providing a great set of answers, does not address the specific query raised here (given that regression parameters index a given distribution in a family of distributions, are they parameters themselves or functions of parameters). Glen's answer resolves this, thank you! :) – Sue Doh Nimh Oct 24 '14 at 21:28

## 1 Answer

It's pretty simple, and not limited to statistics. A parameter is a basic feature of the universe. Like 100 is the parameter for the temperature that water boils at sea level. Or $c$ for the speed of light in $E=Mc^2$.

In the particular case of statistics, you have parameters of estimators, which you hope correspond to some parameter of the universe which is of interest. So, the fact that 2 is the mean of 1 and 3 is a basic feature of the universe, which contains 1 and 3 and the procedure for taking averages. Similarly, the universe often contains vectors of x and y, it contains OLS, and therefore any given set of x and y have associated OLS parameters. They are unchanging features of existence.

Again, whether these OLS parameters correspond the the parameters of the universe is another question.