- Scale parameter
In

probability theory andstatistics , a**scale parameter**is a special kind ofnumerical parameter of aparametric family ofprobability distribution s. The larger the scale parameter, the more spread out the distribution.**Definition**If a family of

probability distribution s is such that there is a parameter "s" (and other parameters "θ") for which thecumulative distribution function satisfies:$F(x;s,\; heta)\; =\; F(x/s;1,\; heta),\; !$

then "s" is called a

**scale parameter**, since its value determines the "scale" orstatistical 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.****imple manipulations**We can write $f\_s$ in terms of $g(x)\; =\; x/s$, as follows:

:$f\_s(x)\; =\; f(x/s)\; imes\; 1/s\; =\; f(g(x))\; imes\; g\text{'}(x).\; !$

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:$1\; =\; int\_\{-infty\}^\{infty\}\; f(g(x))\; imes\; g\text{'}(x),dx\; =\; int\_\{-infty\}^\{infty\}\; f\_s(x),dx.$

!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 theexponential distribution s with scale parameter β and probability density :$f(x;eta\; )\; =\; frac\{1\}\{eta\}\; e^\{-x/eta\}\; ,;\; x\; ge\; 0$could equally be written with rate parameter λ as:$f(x;lambda)\; =\; lambda\; e^\{-lambda\; x\}\; ,;\; x\; ge\; 0.$**Examples*** The

normal distribution has two parameters: alocation 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 thevariance of the distribution.* The

gamma distribution is usually parameterized in terms of a scale parameter $heta$ 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 theCauchy 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 constantscale 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 thenormal distribution , one must multiply it by$1/Phi^\{-1\}(3/4)\; approx\; 1.4826$, where Φ^{-1}is thequantile function (inverse of thecumulative 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.**ee also***

Central tendency

*Invariant estimator

*Location-scale family

*Statistical dispersion

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