# 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

${\displaystyle F(x;s,\theta )=F(x/s;1,\theta ),\!}$

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

${\displaystyle f_{s}(x)=f(x/s)/s,\!}$

where f is the density of a standardized version of the density, i.e. ${\displaystyle f(x)\equiv f_{s=1}(x)}$.

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

### Families with Location Parameters

In the case where a parametrized family has a location parameter, a slightly different definition is often used as follows. If we denote the location parameter by ${\displaystyle m}$, and the scale parameter by ${\displaystyle s}$, then we require that ${\displaystyle F(x;s,m,\theta )=F((x-m)/s;1,0,\theta )}$ where ${\displaystyle F(x,s,m,\theta )}$ is the cmd for the parametrized family[1]. This modification is necessary in order for the standard deviation of a non-central Gaussian to be a scale parameter, since otherwise the mean would change when we rescale ${\displaystyle x}$. However, this alternative definition is not consistently used[2].

### Simple manipulations

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

${\displaystyle f_{s}(x)=f\left({\frac {x}{s}}\right)\cdot {\frac {1}{s}}=f(g(x))g'(x).}$

Because f is a probability density function, it integrates to unity:

${\displaystyle 1=\int _{-\infty }^{\infty }f(x)\,dx=\int _{g(-\infty )}^{g(\infty )}f(x)\,dx.}$

By the substitution rule of integral calculus, we then have

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

So ${\displaystyle f_{s}}$ 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

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

could equivalently be written with rate parameter λ as

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

## Examples

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

${\displaystyle 1/\Phi ^{-1}(3/4)\approx 1.4826,}$

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.

## References

1. ^ Prokhorov, A.V. (7 February 2011). "Scale parameter". Encyclopedia of Mathematics. Springer. Retrieved 7 February 2019.
2. ^ Koski, Timo. "Scale parameter". KTH Royal Institute of Technology. Retrieved 7 February 2019.

• Mood, A. M.; Graybill, F. A.; Boes, D. C. (1974). "VII.6.2 Scale invariance". Introduction to the theory of statistics (3rd ed.). New York: McGraw-Hill.
Asymmetric Laplace distribution

In probability theory and statistics, the asymmetric Laplace distribution (ALD) is a continuous probability distribution which is a generalization of the Laplace distribution. Just as the Laplace distribution consists of two exponential distributions of equal scale back-to-back about x = m, the asymmetric Laplace consists of two exponential distributions of unequal scale back to back about x = m, adjusted to assure continuity and normalization. The difference of two variates exponentially distributed with different means and rate parameters will be distributed according to the ALD. When the two rate parameters are equal, the difference will be distributed according to the Laplace distribution.

Fréchet distribution

The Fréchet distribution, also known as inverse Weibull distribution, is a special case of the generalized extreme value distribution. It has the cumulative distribution function

${\displaystyle \Pr(X\leq x)=e^{-x^{-\alpha }}{\text{ if }}x>0.}$

where α > 0 is a shape parameter. It can be generalised to include a location parameter m (the minimum) and a scale parameter s > 0 with the cumulative distribution function

${\displaystyle \Pr(X\leq x)=e^{-\left({\frac {x-m}{s}}\right)^{-\alpha }}{\text{ if }}x>m.}$

Named for Maurice Fréchet who wrote a related paper in 1927, further work was done by Fisher and Tippett in 1928 and by Gumbel in 1958.

Gamma/Gompertz distribution

In probability and statistics, the Gamma/Gompertz distribution is a continuous probability distribution. It has been used as an aggregate-level model of customer lifetime and a model of mortality risks.

Gamma distribution

In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are three different parametrizations in common use:

With a shape parameter k and a scale parameter θ.

With a shape parameter α = k and an inverse scale parameter β = 1/θ, called a rate parameter.

With a shape parameter k and a mean parameter μ = kθ = α/β.In each of these three forms, both parameters are positive real numbers.

The gamma distribution is the maximum entropy probability distribution (both with respect to a uniform base measure and with respect to a 1/x base measure) for a random variable X for which E[X] = kθ = α/β is fixed and greater than zero, and E[ln(X)] = ψ(k) + ln(θ) = ψ(α) − ln(β) is fixed (ψ is the digamma function).

Generalised hyperbolic distribution

The generalised hyperbolic distribution (GH) is a continuous probability distribution defined as the normal variance-mean mixture where the mixing distribution is the generalized inverse Gaussian distribution (GIG). Its probability density function (see the box) is given in terms of modified Bessel function of the second kind, denoted by ${\displaystyle K_{\lambda }}$. It was introduced by Ole Barndorff-Nielsen, who studied it in the context of physics of wind-blown sand.

Gompertz distribution

In probability and statistics, the Gompertz distribution is a continuous probability distribution, named after Benjamin Gompertz. The Gompertz distribution is often applied to describe the distribution of adult lifespans by demographers and actuaries. Related fields of science such as biology and gerontology also considered the Gompertz distribution for the analysis of survival. More recently, computer scientists have also started to model the failure rates of computer codes by the Gompertz distribution. In Marketing Science, it has been used as an individual-level simulation for customer lifetime value modeling. In network theory, particularly the Erdős–Rényi model, the walk length of a random self-avoiding walk (SAW) is distributed according to the Gompertz distribution.

Holtsmark distribution

The (one-dimensional) Holtsmark distribution is a continuous probability distribution. The Holtsmark distribution is a special case of a stable distribution with the index of stability or shape parameter ${\displaystyle \alpha }$ equal to 3/2 and skewness parameter ${\displaystyle \beta }$ of zero. Since ${\displaystyle \beta }$ equals zero, the distribution is symmetric, and thus an example of a symmetric alpha-stable distribution. The Holtsmark distribution is one of the few examples of a stable distribution for which a closed form expression of the probability density function is known. However, its probability density function is not expressible in terms of elementary functions; rather, the probability density function is expressed in terms of hypergeometric functions.

The Holtsmark distribution has applications in plasma physics and astrophysics. In 1919, Norwegian physicist J. Holtsmark proposed the distribution as a model for the fluctuating fields in plasma due to chaotic motion of charged particles. It is also applicable to other types of Coulomb forces, in particular to modeling of gravitating bodies, and thus is important in astrophysics.

Location–scale family

In probability theory, especially in mathematical statistics, a location–scale family is a family of probability distributions parametrized by a location parameter and a non-negative scale parameter. For any random variable ${\displaystyle X}$ whose probability distribution function belongs to such a family, the distribution function of ${\displaystyle Y{\stackrel {d}{=}}a+bX}$ also belongs to the family (where ${\displaystyle {\stackrel {d}{=}}}$ means "equal in distribution"—that is, "has the same distribution as"). Moreover, if ${\displaystyle X}$ and ${\displaystyle Y}$ are two random variables whose distribution functions are members of the family, and assuming 1) existence of the first two moments and 2) ${\displaystyle X}$ has zero mean and unit variance, then ${\displaystyle Y}$ can be written as ${\displaystyle Y{\stackrel {d}{=}}\mu _{Y}+\sigma _{Y}X}$ , where ${\displaystyle \mu _{Y}}$ and ${\displaystyle \sigma _{Y}}$ are the mean and standard deviation of ${\displaystyle Y}$.

In other words, a class ${\displaystyle \Omega }$ of probability distributions is a location–scale family if for all cumulative distribution functions ${\displaystyle F\in \Omega }$ and any real numbers ${\displaystyle a\in \mathbb {R} }$ and ${\displaystyle b>0}$, the distribution function ${\displaystyle G(x)=F(a+bx)}$ is also a member of ${\displaystyle \Omega }$.

In decision theory, if all alternative distributions available to a decision-maker are in the same location–scale family, and the first two moments are finite, then a two-moment decision model can apply, and decision-making can be framed in terms of the means and the variances of the distributions.

Log-logistic distribution

In probability and statistics, the log-logistic distribution (known as the Fisk distribution in economics) is a continuous probability distribution for a non-negative random variable. It is used in survival analysis as a parametric model for events whose rate increases initially and decreases later, for example mortality rate from cancer following diagnosis or treatment. It has also been used in hydrology to model stream flow and precipitation, in economics as a simple model of the distribution of wealth or income, and in networking to model the transmission times of data considering both the network and the software.

The log-logistic distribution is the probability distribution of a random variable whose logarithm has a logistic distribution.

It is similar in shape to the log-normal distribution but has heavier tails. Unlike the log-normal, its cumulative distribution function can be written in closed form.

Lomax distribution

The Lomax distribution, conditionally also called the Pareto Type II distribution, is a heavy-tail probability distribution used in business, economics, actuarial science, queueing theory and Internet traffic modeling. It is named after K. S. Lomax. It is essentially a Pareto distribution that has been shifted so that its support begins at zero.

Normal-exponential-gamma distribution

In probability theory and statistics, the normal-exponential-gamma distribution (sometimes called the NEG distribution) is a three-parameter family of continuous probability distributions. It has a location parameter ${\displaystyle \mu }$, scale parameter ${\displaystyle \theta }$ and a shape parameter ${\displaystyle k}$ .

Probable error

In statistics, probable error defines the half-range of an interval about a central point for the distribution, such that half of the values from the distribution will lie within the interval and half outside. Thus for a symmetric distribution it is equivalent to half the interquartile range, or the median absolute deviation. One such use of the term probable error in this sense is as the name for the scale parameter of the Cauchy distribution, which does not have a standard deviation.

The probable error can also be expressed as a multiple of the standard deviation σ,, which requires that at least the second statistical moment of the distribution should exist, whereas the other definition does not. For a normal distribution this is ${\displaystyle \gamma =0.6745\times \sigma }$ (see details)

Rouse number

The Rouse number (P or Z) is a non-dimensional number in fluid dynamics which is used to define a concentration profile of suspended sediment and which also determines how sediment will be transported in a flowing fluid. It is a ratio between the sediment fall velocity ${\displaystyle w_{s}}$ and the upwards velocity on the grain as a product of the von Kármán constant ${\displaystyle \kappa }$ and the shear velocity ${\displaystyle u_{*}}$.

${\displaystyle \mathrm {P} ={\frac {w_{s}}{\kappa u_{*}}}}$

Occasionally the factor β is included before the von Kármán constant in the equation, which is a constant which correlates eddy viscosity to eddy diffusivity. This is generally taken to be equal to 1, and therefore is ignored in actual calculation. However, it should not be ignored when considering the full equation.

${\displaystyle \mathrm {P} ={\frac {w_{s}}{\beta \kappa u_{*}}}}$

It is named after the American fluid dynamicist Hunter Rouse. It is a characteristic scale parameter in the Rouse Profile of suspended sediment concentration with depth in a flowing fluid. The concentration of suspended sediment with depth goes as the power of the negative Rouse number. It also is used to determine how the particles will move in the fluid. The required Rouse numbers for transport as bed load, suspended load, and wash load, are given below.

Shape parameter

In probability theory and statistics, a shape parameter is a kind of numerical parameter of a parametric family of probability distributions.Specifically, a shape parameter is any parameter of a probability distribution that is neither a location parameter nor a scale parameter (nor a function of either or both of these only, such as a rate parameter). Such a parameter must affect the shape of a distribution rather than simply shifting it (as a location parameter does) or stretching/shrinking it (as a scale parameter does).

Shifted log-logistic distribution

The shifted log-logistic distribution is a probability distribution also known as the generalized log-logistic or the three-parameter log-logistic distribution. It has also been called the generalized logistic distribution, but this conflicts with other uses of the term: see generalized logistic distribution.

Tukey lambda distribution

Formalized by John Tukey, the Tukey lambda distribution is a continuous, symmetric probability distribution defined in terms of its quantile function. It is typically used to identify an appropriate distribution (see the comments below) and not used in statistical models directly.

The Tukey lambda distribution has a single shape parameter, λ, and as with other probability distributions, it can be transformed with a location parameter, μ, and a scale parameter, σ. Since the general form of probability distribution can be expressed in terms of the standard distribution, the subsequent formulas are given for the standard form of the function.

Variance-gamma distribution

The variance-gamma distribution, generalized Laplace distribution or Bessel function distribution is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the gamma distribution. The tails of the distribution decrease more slowly than the normal distribution. It is therefore suitable to model phenomena where numerically large values are more probable than is the case for the normal distribution. Examples are returns from financial assets and turbulent wind speeds. The distribution was introduced in the financial literature by Madan and Seneta. The variance-gamma distributions form a subclass of the generalised hyperbolic distributions.

The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available. The class of variance-gamma distributions is closed under convolution in the following sense. If ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ are independent random variables that are variance-gamma distributed with the same values of the parameters ${\displaystyle \alpha }$ and ${\displaystyle \beta }$, but possibly different values of the other parameters, ${\displaystyle \lambda _{1}}$, ${\displaystyle \mu _{1}}$ and ${\displaystyle \lambda _{2},}$ ${\displaystyle \mu _{2}}$, respectively, then ${\displaystyle X_{1}+X_{2}}$ is variance-gamma distributed with parameters ${\displaystyle \alpha }$, ${\displaystyle \beta }$, ${\displaystyle \lambda _{1}+\lambda _{2}}$ and ${\displaystyle \mu _{1}+\mu _{2}}$.

The variance-gamma distribution can also be expressed in terms of three inputs parameters (C,G,M) denoted after the initials of its founders. If the "C", ${\displaystyle \lambda }$ here, parameter is integer then the distribution has a closed form 2-EPT distribution. See 2-EPT Probability Density Function. Under this restriction closed form option prices can be derived.

If ${\displaystyle \alpha =1}$, ${\displaystyle \lambda =1}$ and ${\displaystyle \beta =0}$, the distribution becomes a Laplace distribution with scale parameter ${\displaystyle b=1}$. As long as ${\displaystyle \lambda =1}$, alternative choices of ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ will produce distributions related to the Laplace distribution, with skewness, scale and location depending on the other parameters.