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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. .
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 , and the scale parameter by , then we require that where 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 . However, this alternative definition is not consistently used.
We can write in terms of , as follows:
Because f is a probability density function, it integrates to unity:
By the substitution rule of integral calculus, we then have
So is also properly normalized.
Some families of distributions use a rate parameter (or "inverse scale parameter"), which is simply the reciprocal of the scale parameter. So for example the exponential distribution with scale parameter β and probability density
could equivalently be written with rate parameter λ as
A statistic can be used to estimate a scale parameter so long as it:
Various measures of statistical dispersion satisfy these. In order to make the statistic a consistent estimator for the scale parameter, one must in general multiply the statistic by a constant scale factor. This scale factor is defined as the theoretical value of the value obtained by dividing the required scale parameter by the asymptotic value of the statistic. Note that the scale factor depends on the distribution in question.
For instance, in order to use the median absolute deviation (MAD) to estimate the standard deviation of the normal distribution, one must multiply it by the factor
where Φ−1 is the quantile function (inverse of the cumulative distribution function) for the standard normal distribution. (See MAD for details.) That is, the MAD is not a consistent estimator for the standard deviation of a normal distribution, but 1.4826... MAD is a consistent estimator. Similarly, the average absolute deviation needs to be multiplied by approximately 1.2533 to be a consistent estimator for standard deviation. Different factors would be required to estimate the standard deviation if the population did not follow a normal distribution.
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