Finding location and scale parameters from PDF

Question

If I have given distribution family, say normal, is there a way how to derive what are the location and scale parameters based on the probability density function (PDF)?

I know in case of normal distribution, it is $\mu$ and $\sigma$ respectively. I know how to show, that these two are in fact location and scale parameters.

But what if I had no idea, what are the scale and location parameters and wanted to use PDF to find them?

Alternatively, if I can't tell from PDF as such, is there some other approach of deriving the location and scale?

It doesn't have to be restricted to Normal distribution. Alternatively, I would like to know, how to derive, that for distribution Uniform[a, b] i have location a and scale b - a.

Answer 1

Let's talk about what a location parameter is. (The discussion of scale parameters will exactly parallel it and offers little new.)

The setting concerns a set $\mathcal{F}$ of probability distributions $F_\theta$ indexed by a parameter $\theta \in \Theta \subset \mathbb{R}^p$. "Indexed by" not only means each $\theta\in\Theta$ denotes a distribution $F_\theta$: it also means that no distribution is identified by more than one such $\theta$.

Consider any location-covariant property of $F_\theta$, such as a specific quantile (which will always exist) or its first moment (which might exist for some distribution families). By "property" I mean quite generally some "nice" real-valued function $t$ ("nice" will be explained momentarily) defined on this family of distributions and "location-covariant" means that for any $F$ in the family, $$t(F^{(\mu)}) = t(F) + \mu$$

where the translate $F^{(\mu)}$ is the distribution function given by

$$F^{(\mu)}(x) = F(x-\mu).$$

Consider any $\theta_0$ in the interior of $\Theta$. Parameterizations are assumed to have the property that such a $\theta_0$ will have a $p$-dimensional neighborhood $\mathcal{B}$ in which $t$ is differentiable with nonzero derivative throughout (that's what "nice" means). The Implicit Function Theorem then implies there is a coordinate system in this neighborhood in which the first coordinate is $t$ and the remaining $p-1$ coordinates are differentiable functions. Locally, at least, the distributions can be parameterized by $t$ and by the codimension-1 subset of $\mathcal{B}$ whose first coordinate is $t(\theta_0)$. That gives a new set of coordinates $\gamma=(\gamma_1,\ldots, \gamma_p)$ with $\gamma_1=t$.

We say that $t$ is a location parameter for the family. It has the property that if we fix the last $p-1$ coordinates in $\mathcal{B}$, then the distribution function can be written in the form

$$F_{(t, \gamma_2, \ldots,\gamma_p)}(x) = H_{(\gamma_2,\ldots, \gamma_p)}(x-t)$$

where $H$ (and therefore anything equivalent to it, such as its pdf if it has one) depends only on the last $p-1$ parameters. This is how you recognize a location parameter: the argument of the distribution function shows up in the formula only as the combination $x-t$.

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Let's work an example. Suppose each $F_\theta$ in the interior of $\Theta = \{(\theta_1,\theta_2)\mid \theta_1 \ge 0\}$ is continuous and is given in terms of its pdf $f_\theta$ as

$$f_\theta(x) \propto \exp(-\theta_1 x^2 + \theta_2 x) \propto \exp\left(-\theta_1\left(x - \frac{\theta_2}{2\theta_1}\right)^2\right).\tag{1}$$

Is this a location family? If so, what is its location parameter?

In the first formula of $(1)$, there is no obvious location parameter: $f_\theta$ is not explicitly a function of $x-\theta_1$ or $x-\theta_2$.

The second formula of $(1)$ shows each pdf is symmetric around the value $$t(\theta) = \frac{\theta_2}{2\theta_1},$$ which therefore must be the median. Since the median is a location-covariant property, we can exploit this observation to construct a location parameter.

Consider $\theta_0 = (1,0)$ for instance. I chose this to make $t(\theta_0)=0$ have a simple value. The level set of $t$ passing through $\theta_0$ is given by

$$0 = t(1,0) = t(\theta_1, \theta_2) = \frac{\theta_2}{2\theta_1},$$

showing that (locally) it's the set where $\theta_2=0$ and we may parameterize it by $\theta_1$. Let us therefore change the parameterization from $\theta$ to

$$\gamma = (\gamma_1, \gamma_2) = (t(\theta_1, \theta_2), \theta_1) = \left( \frac{\theta_2}{2\theta_1}, \theta_1\right).$$

The base point $\theta_0$ corresponds to $\gamma_0 = (0, 1)$.

The inverse of this transformation from $\theta$ to $\gamma$ is

$$\theta = (\theta_1, \theta_2) = \left(\gamma_2, 2\gamma_1\gamma_2\right).$$

The new parameterization is therefore

$$G_\gamma(x) = F_{\left(\gamma_2, 2\gamma_1\gamma_2\right)}(x) \propto \exp\left(-\gamma_2(x - \gamma_1)^2\right).$$

Now it is perfectly obvious that $\gamma_1$ is a location parameter, because its sole effect is to shift $x$ in the formula.

By finding a candidate for a location parameter and demonstrating it acts as one, we have verified that this is a location family and we have found a location parameter for it. Incidentally, we have also identified this parameter with a location-covariant property: the median.

Answer 2

Consider a random variable $Z$ with any density function $f_Z(z)$. You can define a location–scale family with the transform $X=\phi Z +\theta$, & the new density function is given by

$$f_X(x;\theta,\phi) = \frac{1}{\phi}\cdot f_Z\left(\frac{x-\theta}{\phi}\right)$$

So any family of distributions whose density function can be written in this form is a location–scale family.

For example the family of uniform distributions on $(a,b)$ is defined by the density function

$$f_X(x;a,b) = \frac{1}{b-a}\cdot \boldsymbol{1}_{[a,b]}(x)$$

which can be re-written as

$$f_X(x;a,b) = \frac{1}{b-a}\cdot \boldsymbol{1}_{[0,1]}\left(\frac{x-a}{b-a}\right)$$

$$f_X(x;a,b) = \frac{1}{b-a}\cdot f_X\left(\frac{x-a}{b-a};0,1\right)$$

showing that $a$ is a location parameter & $b-a$ is a scale parameter.