If a family of
probability distributions is such that there is a parameter
s (and other parameters
θ) for which the
cumulative distribution function satisfies F(x;s,\theta) = F(x/s;1,\theta), then
s is called a
scale parameter, since its value determines the "
scale" or
statistical dispersion of the probability distribution. If
s is large, then the distribution will be more spread out; if
s is small then it will be more concentrated. If the
probability density exists for all values of the complete parameter set, then the density (as a function of the scale parameter only) satisfies f_s(x) = f(x/s)/s, where
f is the density of a standardized version of the density, i.e. f(x) \equiv f_{s=1}(x). An
estimator of a scale parameter is called an
estimator of scale. Families with Location Parameters In the case where a parametrized family has a
location parameter, a slightly different definition is often used as follows. If we denote the location parameter by m, and the scale parameter by s, then we require that F(x;s,m,\theta)=F((x-m)/s;1,0,\theta) where F(x,s,m,\theta) is the CDF for the parametrized family. This modification is necessary in order for the standard deviation of a non-central Gaussian to be a scale parameter, since otherwise the mean would change when we rescale x. However, this alternative definition is not consistently used.
Simple manipulations We can write f_s in terms of g(x) = x/s, as follows: f_s(x) = f\left(\frac{x}{s}\right) \cdot \frac{1}{s} = f(g(x))g'(x). Because
f is a probability density function, it integrates to unity: 1 = \int_{-\infty}^{\infty} f(x)\,dx = \int_{g(-\infty)}^{g(\infty)} f(x)\,dx. By the
substitution rule of integral calculus, we then have 1 = \int_{-\infty}^{\infty} f(g(x)) g'(x)\,dx = \int_{-\infty}^{\infty} f_s(x)\,dx. So f_s is also properly normalized. ==Rate parameter==