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.

Definition

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.

An estimator of a scale parameter is called an estimator of scale.

Simple manipulations

We can write $f_{s}$ in terms of $g(x)=x/s$, as follows:

$f_{s}(x)=f(x/s)\times 1/s=f(g(x))\times g'(x).\!$

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

$1=\int _{-\infty }^{\infty }f(g(x))\times g'(x)\,dx=\int _{-\infty }^{\infty }f_{s}(x)\,dx.\!$

So $f_{s}$ is also properly normalized.

Rate parameter

Some families of distributions use a rate parameter which is simply the reciprocal of the scale parameter. So for example the exponential distributions with scale parameter β and probability density

$f(x;\beta )={\frac {1}{\beta }}e^{-x/\beta },\;x\geq 0$

could equally be written with rate parameter λ as

$f(x;\lambda )=\lambda e^{-\lambda x},\;x\geq 0.$

Examples

• The normal distribution has two parameters: a location parameter $\mu$ and a scale parameter $\sigma$. In practice the normal distribution is often parameterized in terms of the squared scale $\sigma ^{2}$, which corresponds to the variance of the distribution.

• The gamma distribution is usually parameterized in terms of a scale parameter $\theta$ 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.

Estimation

A statistic can be used to estimate a scale parameter so long as it is:

• location-invariant, and
• scale linearly with the scale parameter.

Various measures of statistical dispersion satisfy these.

In order to make the statistic a consistent estimator for the scale factor, one must in general multiply by a constant scale factor, namely the value of the scale factor, divided by the asymptotic value of the estimator. 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 σ factor in the normal distribution, one must multiply it by $1/\Phi ^{-1}(3/4)\approx 1.4826$, 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 σ.

See also

• Central tendency
• Invariant estimator
• Location-scale family
• Statistical dispersion