Location parameter

Source: Let f(x) be any probability density function and let \mu and \sigma > 0 be any given constants. Then the function g(x| \mu, \sigma)= \frac{1}{\sigma} f{\left(\frac{x-\mu}{\sigma}\right)} is a probability density function. The location family is then defined as follows: Let f(x) be any probability density function. Then the family of probability density functions \mathcal{F} = \{f(x-\mu) : \mu \in \mathbb{R}\} is called the location family with standard probability density function f(x) , where \mu is called the location parameter for the family. ==Additive noise==

An alternative way of thinking of location families is through the concept of additive noise. If x_0 is a constant and W is random noise with probability density f_W(w), then X = x_0 + W has probability density f_{x_0}(x) = f_W(x-x_0) and its distribution is therefore part of a location family. ==Proofs==

For the continuous univariate case, consider a probability density function f(x | \theta), x \in [a, b] \subset \mathbb{R}, where \theta is a vector of parameters. A location parameter x_0 can be added by defining: g(x | \theta, x_0) = f(x - x_0 | \theta), \; x \in [a + x_0, b + x_0] it can be proved that g is a p.d.f. by verifying if it respects the two conditions g(x | \theta, x_0) \ge 0 and \int_{-\infty}^{\infty} g(x | \theta, x_0) dx = 1. g integrates to 1 because: \int_{-\infty}^{\infty} g(x | \theta, x_0) dx = \int_{a + x_0}^{b + x_0} g(x | \theta, x_0) dx = \int_{a + x_0}^{b + x_0} f(x - x_0 | \theta) dx now making the variable change u = x - x_0 and updating the integration interval accordingly yields: \int_{a}^{b} f(u | \theta) du = 1 because f(x | \theta) is a p.d.f. by hypothesis. g(x | \theta, x_0) \ge 0 follows from g sharing the same image of f, which is a p.d.f. so its range is contained in [0, 1]. ==See also==