Scale parameter

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.


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, heta) = F(x/s;1, heta), !

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.

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'(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'(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;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.


* 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 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 the Cauchy distribution as the "standard" Cauchy distribution.


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 by1/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.

ee also

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

Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • Scale space — theory is a framework for multi scale signal representation developed by the computer vision, image processing and signal processing communities with complementary motivations from physics and biological vision. It is a formal theory for handling …   Wikipedia

  • Scale space implementation — Scale space Scale space axioms Scale space implementation Feature detection Edge detection Blob detection Corner detection …   Wikipedia

  • Scale-space segmentation — or multi scale segmentation is a general framework for signal and image segmentation, based on the computation of image descriptors at multiple scales of smoothing. One dimensional hierarchical signal segmentationWitkin s seminal work in scale… …   Wikipedia

  • Scale-inverse-chi-square distribution — Probability distribution name =Scale inverse chi square type =density pdf cdf parameters = u > 0, sigma^2 > 0, support =x in (0, infty) pdf =frac{(sigma^2 u/2)^{ u/2{Gamma( u/2)} frac{expleft [ frac{ u sigma^2}{2 x} ight] }{x^{1+ u/2 cdf… …   Wikipedia

  • Scale — NOTOC Scale can refer to:ystems of representation* Duraton scale, an ordering of time intervals from shortest to longest * Measurement, referring to the size of buildings or other structures * Scalability, a system s capacity to adapt to changes… …   Wikipedia

  • Scale invariance — In physics and mathematics, scale invariance is a feature of objects or laws that do not change if length scales (or energy scales) are multiplied by a common factor. The technical term for this transformation is a dilatation (also known as… …   Wikipedia

  • Scale factor (cosmology) — The scale factor or cosmic scale factor parameter of the Friedmann equations is a function of time which represents the relative expansion of the universe. It is sometimes called the Robertson Walker scale factor.[1] It is the (time dependent)… …   Wikipedia

  • Scale-invariant feature transform — Feature detection Output of a typical corner detection algorithm …   Wikipedia

  • Scale factor (Universe) — The scale factor, parameter of Friedmann Lemaître Robertson Walker model, is a function of time which represents the relative expansion of the universe. It relates physical coordinates (also called proper coordinates) to comoving coordinates.: L …   Wikipedia

  • scale — Synonyms and related words: Danish balance, Indian file, Lambert conformal projection, Mercator projection, Miller projection, Roman balance, Weightometer, accommodation ladder, adjust, aeronautical chart, alloy balance, amount, amplitude,… …   Moby Thesaurus

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”