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 distribution with scale parameter β and probability density

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

could equally be written with rate parameter λ as

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

Examples

Estimation

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

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 standard deviation. Different factors would be required to estimate the standard deviation if the population did not follow a normal distribution.

See also

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