Scale parameter
Encyclopedia
In probability theory
and statistics
, a scale parameter is a special kind of numerical parameter of a parametric family
of probability distribution
s. The larger the scale parameter, the more spread out the distribution.
s 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.
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.
with scale parameter β and probability density
could equally be written with rate parameter λ as
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.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, a scale parameter is a special kind of numerical parameter of a parametric family
Parametric family
In mathematics and its applications, a parametric family or a parameterized family is a family of objects whose definitions depend on a set of parameters....
of probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
s. The larger the scale parameter, the more spread out the distribution.
Definition
If a family of probability distributionProbability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
s is such that there is a parameter s (and other parameters θ) for which the cumulative distribution function
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
satisfies
then s is called a scale parameter, since its value determines the "scale
Scale (ratio)
The scale ratio of some sort of model which represents an original proportionally is the ratio of a linear dimension of the model to the same dimension of the original. Examples include a 3-dimensional scale model of a building or the scale drawings of the elevations or plans of a building. In such...
" or statistical dispersion
Statistical dispersion
In statistics, statistical dispersion is variability or spread in a variable or a probability distribution...
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
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
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
Estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule and its result are distinguished....
of a scale parameter is called an estimator of scale.
Simple manipulations
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.Rate parameter
Some families of distributions use a rate parameter which is simply the reciprocal of the scale parameter. So for example the exponential distributionExponential distribution
In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...
with scale parameter β and probability density
could equally be written with rate parameter λ as
Examples
- The normal distribution has two parameters: a location parameterLocation parameterIn statistics, a location family is a class of probability distributions that is parametrized by a scalar- or vector-valued parameter μ, which determines the "location" or shift of the distribution...
and a scale parameter 
. In practice the normal distribution is often parameterized in terms of the squared scale 
, which corresponds to the varianceVarianceIn probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
of the distribution.
- The gamma distribution is usually parameterized in terms of a scale parameter
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 distributionCauchy distributionThe Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous probability distribution. As a probability distribution, it is known as the Cauchy distribution, while among physicists, it is known as the Lorentz distribution, Lorentz function, or Breit–Wigner...
as the standard Cauchy distribution.
Estimation
A statistic can be used to estimate a scale parameter so long as it:- Is location-invariant,
- Scales linearly with the scale parameter, and
- Converges as the sample size grows.
Various measures of statistical dispersion satisfy these.
In order to make the statistic a consistent estimator
Consistent estimator
In statistics, a sequence of estimators for parameter θ0 is said to be consistent if this sequence converges in probability to θ0...
for the scale parameter, one must in general multiply the statistic by a constant scale factor
Scale factor
A scale factor is a number which scales, or multiplies, some quantity. In the equation y=Cx, C is the scale factor for x. C is also the coefficient of x, and may be called the constant of proportionality of y to x...
. 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
Median absolute deviation
In statistics, the median absolute deviation is a robust measure of the variability of a univariate sample of quantitative data. It can also refer to the population parameter that is estimated by the MAD calculated from a sample....
(MAD) to estimate the standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
of the normal distribution, one must multiply it by the factor
where Φ−1 is the quantile function
Quantile function
In probability and statistics, the quantile function of the probability distribution of a random variable specifies, for a given probability, the value which the random variable will be at, or below, with that probability...
(inverse of the cumulative distribution function
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
) 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
- Central tendencyCentral tendencyIn statistics, the term central tendency relates to the way in which quantitative data is clustered around some value. A measure of central tendency is a way of specifying - central value...
- Invariant estimatorInvariant estimatorIn statistics, the concept of being an invariant estimator is a criterion that can be used to compare the properties of different estimators for the same quantity. It is a way of formalising the idea that an estimator should have certain intuitively appealing qualities...
- Location parameterLocation parameterIn statistics, a location family is a class of probability distributions that is parametrized by a scalar- or vector-valued parameter μ, which determines the "location" or shift of the distribution...
- Location-scale familyLocation-scale familyIn probability theory, especially as that field is used in statistics, a location-scale family is a family of univariate probability distributions parametrized by a location parameter and a non-negative scale parameter; if X is any random variable whose probability distribution belongs to such a...
- Statistical dispersionStatistical dispersionIn statistics, statistical dispersion is variability or spread in a variable or a probability distribution...