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In some probability distributions, like normal or (non-standard) t distributions etc, there are location parameters such that a change to this parameter leads to the distribution moving rigidly to the left or right, i.e. with no change to its shape etc. Are location parameters defined strictly in these terms, i.e. if the location parameter changes the distribution must move rigidly to the left or right (assuming we are dealing with a univariate distribution)?

If this is the case, I presume it would be technically incorrect to call, e.g., the $\lambda$ parameter of a Poisson distribution its location parameter, despite it defining the distribution's mean, and thus roughly where it is centered?

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    $\begingroup$ The first question is answered in the affirmative at stats.stackexchange.com/questions/265939. This supports the conclusion that the Poisson parameter "technically" is not a location parameter. However, terminology varies: I wouldn't be surprised by an author loosely referring to the Poisson parameter as a "location." $\endgroup$
    – whuber ♦
    Commented Mar 5, 2019 at 14:39
  • $\begingroup$ Yes. In fact, you can invent "new" families of distributions by just adding a shift parameter, like the $\exp(\lambda, \theta)$ which is just an exponential $\lambda$ distribution that starts at $\theta$ instead of 0. Be careful, in robust statistics, they often talk of "robust measures of location" by speaking of a general class of estimators like the trimmed mean or the minimum absolute deviation parameter. Here "location" is a non-parametric measure rather than a strict parameter. $\endgroup$
    – AdamO
    Commented Mar 5, 2019 at 14:55

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You are right that the Poisson mean parameter $\lambda$ is not a location parameter, as it does not strictly shift the distribution right or left. The formal definition of a location parameter is as follows:

Let $f_0$ be a given probability density function (pdf) on the real line. Let $\mu\in\mathbb{R}$. Then the family of densities given by $f(x;\mu)=f_0(x-\mu)$ is a location (or translation) family, and $\mu$ is a location parameter.

This is the most important example of a group family, see Wikipedia. From a more practical point of view, you get a location family if you take a histogram (or pdf), cut it out "of the paper" with a pair of scissors, and then just slide it along the real line.

You can convince yourself you cannot do that with the Poisson distribution -- even when the above description is generalized to pmf's (probability mass functions) -- since the minimum of the support for the Poisson is always at zero, while the described process will slide the minimum along with $\mu$.

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    $\begingroup$ Binomial distributions raise similar issues, as changing the mean doesn't shift the support. So, generalising your definition to include discrete distributions seems indicated. One way to do that is just to be generous that a density can also be a density with respect to counting measure (excuse me if I get the terminology slightly wrong...). $\endgroup$
    – Nick Cox
    Commented Mar 5, 2019 at 14:50
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    $\begingroup$ @Nick Right--but it's simpler and fully general just to describe translation in terms of the CDF. $\endgroup$
    – whuber ♦
    Commented Mar 5, 2019 at 14:59

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