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.
An estimator of a scale parameter is called an estimator of scale.
Rate parameter
Some families of distributions use a rate parameter which is simply the reciprocal of the scale parameter(尺度参数的倒数).
Examples
- The normal distribution has two parameters: a location parameter μ and a scale parameter σ. In practice the normal distribution is often parameterized in terms of the squared scale σ2, which corresponds to the variance of the distribution.
- The gamma distribution is usually parameterized in terms of a scale parameter θ or its inverse.
- Special cases of distributions where the scale parameter equals unity may be called "standard" under certain conditions. For example, if the location parameter equals zero and the scale parameter equals one, the normal distribution is known as the standard normal distribution, and the Cauchy distribution as the standard Cauchy distribution.