Scale parameter

In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. 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 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 where f is the density of a standardized version of the density, i.e. f ( x ) ≡ f s = 1 ( x ) {displaystyle f(x)equiv f_{s=1}(x)} . An estimator of a scale parameter is called an estimator of scale. 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 {displaystyle m} , and the scale parameter by s {displaystyle s} , then we require that F ( x ; s , m , θ ) = F ( ( x − m ) / s ; 1 , 0 , θ ) {displaystyle F(x;s,m, heta )=F((x-m)/s;1,0, heta )} where F ( x , s , m , θ ) {displaystyle F(x,s,m, heta )} is the cmd 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 {displaystyle x} . However, this alternative definition is not consistently used. We can write f s {displaystyle f_{s}} in terms of g ( x ) = x / s {displaystyle g(x)=x/s} , as follows: Because f is a probability density function, it integrates to unity: