Scale parameter

Statistical measure

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.
Find sources: "Scale parameter" – news · newspapers · books · scholar · JSTOR (December 2009) (Learn how and when to remove this message)

|

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

[edit]

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 , θ ) = F ( x / s ; 1 , θ ) , {\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.

Animation showing the effects of a scale parameter on a probability distribution supported on the positive real line. Effect of a scale parameter over a mixture of two normal probability distributions

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

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

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

### Families with Location Parameters

[edit]

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  m {\displaystyle m} , and the scale parameter by  s {\displaystyle s} , then we require that  F ( x ; s , m , θ ) = F ( ( x − m ) / s ; 1 , 0 , θ ) {\displaystyle F(x;s,m,\theta )=F((x-m)/s;1,0,\theta )} where  F ( x , s , m , θ ) {\displaystyle F(x,s,m,\theta )} is the CDF 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  x {\displaystyle x} . However, this alternative definition is not consistently used.[2]

### Simple manipulations

[edit]

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

f s ( x ) = f ( x s ) ⋅ 1 s = f ( g ( x ) ) g ′ ( x ) . {\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:

1 = ∫ − ∞ ∞ f ( x ) d x = ∫ g ( − ∞ ) g ( ∞ ) f ( x ) d x . {\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

1 = ∫ − ∞ ∞ f ( g ( x ) ) g ′ ( x ) d x = ∫ − ∞ ∞ f s ( x ) d x . {\displaystyle 1=\int _{-\infty }^{\infty }f(g(x))g'(x)\,dx=\int _{-\infty }^{\infty }f_{s}(x)\,dx.}

So  f s {\displaystyle f_{s}} is also properly normalized.

## Rate parameter

[edit]

Some families of distributions use a rate parameter (or inverse scale 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 ; β ) = 1 β e − x / β , x ≥ 0 {\displaystyle f(x;\beta )={\frac {1}{\beta }}e^{-x/\beta },\;x\geq 0}

could equivalently be written with rate parameter λ as

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

## Examples

[edit]

* The uniform distribution can be parameterized with a location parameter of  ( a + b ) / 2 {\displaystyle (a+b)/2} and a scale parameter  | b − a | {\displaystyle |b-a|} .
* The normal distribution has two parameters: a location parameter μ {\displaystyle \mu } and a scale parameter  σ {\displaystyle \sigma } . In practice the normal distribution is often parameterized in terms of the squared scale  σ 2 {\displaystyle \sigma ^{2}} , which corresponds to the variance of the distribution.
* The gamma distribution is usually parameterized in terms of a scale parameter  θ {\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

[edit]

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

1 / Φ − 1 ( 3 / 4 ) ≈ 1.4826 , {\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.

## See also

[edit]

* Central tendency
* Invariant estimator
* Location parameter
* Location-scale family
* Mean-preserving spread
* Scale mixture
* Shape parameter
* Statistical dispersion

## References

[edit]

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.

## General references

[edit]

* "1.3.6.4. Location and Scale Parameters". National Institute of Standards and Technology. Retrieved 2025-03-17.

## Further reading

[edit]

* 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.

* v
* t
* e

Statistics

* Outline
* Index

Descriptive statistics

| Continuous data|

| Center|

* Mean
* Arithmetic
* Arithmetic-Geometric
* Contraharmonic
* Cubic
* Generalized/power
* Geometric
* Harmonic
* Heronian
* Heinz
* Lehmer
* Median
* Mode

|

Dispersion|

* Average absolute deviation
* Coefficient of variation
* Interquartile range
* Percentile
* Range
* Standard deviation
* Variance

Shape|

* Central limit theorem
* Moments
* Kurtosis
* L-moments
* Skewness

Count data|

* Index of dispersion

Summary tables|

* Contingency table
* Frequency distribution
* Grouped data

Dependence|

* Partial correlation
* Pearson product-moment correlation
* Rank correlation
* Kendall's τ
* Spearman's ρ
* Scatter plot

Graphics|

* Bar chart
* Biplot
* Box plot
* Control chart
* Correlogram
* Fan chart
* Forest plot
* Histogram
* Pie chart
* Q–Q plot
* Radar chart
* Run chart
* Scatter plot
* Stem-and-leaf display
* Violin plot

Data collection

| Study design|

* Effect size
* Missing data
* Optimal design
* Population
* Replication
* Sample size determination
* Statistic
* Statistical power

|

Survey methodology|

* Sampling
* Cluster
* Stratified
* Opinion poll
* Questionnaire
* Standard error

Controlled experiments|

* Blocking
* Factorial experiment
* Interaction
* Random assignment
* Randomized controlled trial
* Randomized experiment
* Scientific control

Adaptive designs|

* Adaptive clinical trial
* Stochastic approximation
* Up-and-down designs

Observational studies|

* Cohort study
* Cross-sectional study
* Natural experiment
* Quasi-experiment

Statistical inference

| Statistical theory|

* Population
* Statistic
* Probability distribution
* Sampling distribution
* Order statistic
* Empirical distribution
* Density estimation
* Statistical model
* Model specification
* Lp space
* Parameter
* location
* scale
* shape
* Parametric family
* Likelihood (monotone)
* Location–scale family
* Exponential family
* Completeness
* Sufficiency
* Statistical functional
* Bootstrap
* U
* V
* Optimal decision
* loss function
* Efficiency
* Statistical distance
* divergence
* Asymptotics
* Robustness

|

Frequentist inference|

| Point estimation|

* Estimating equations
* Maximum likelihood
* Method of moments
* M-estimator
* Minimum distance
* Unbiased estimators
* Mean-unbiased minimum-variance
* Rao–Blackwellization
* Lehmann–Scheffé theorem
* Median unbiased
* Plug-in

|

Interval estimation|

* Confidence interval
* Pivot
* Likelihood interval
* Prediction interval
* Tolerance interval
* Resampling
* Bootstrap
* Jackknife

Testing hypotheses|

* 1- & 2-tails
* Power
* Uniformly most powerful test
* Permutation test
* Randomization test
* Multiple comparisons

Parametric tests|

* Likelihood-ratio
* Score/Lagrange multiplier
* Wald

Specific tests|

* Z-test (normal)
* Student's t-test
* F-test

Goodness of fit|

* Chi-squared
* G-test
* Kolmogorov–Smirnov
* Anderson–Darling
* Lilliefors
* Jarque–Bera
* Normality (Shapiro–Wilk)
* Likelihood-ratio test
* Model selection
* Cross validation
* AIC
* BIC

Rank statistics|

* Sign
* Sample median
* Signed rank (Wilcoxon)
* Hodges–Lehmann estimator
* Rank sum (Mann–Whitney)
* Nonparametric anova
* 1-way (Kruskal–Wallis)
* 2-way (Friedman)
* Ordered alternative (Jonckheere–Terpstra)
* Van der Waerden test

Bayesian inference|

* Bayesian probability
* prior
* posterior
* Credible interval
* Bayes factor
* Bayesian estimator
* Maximum posterior estimator

* Correlation
* Regression analysis

| Correlation|

* Pearson product-moment
* Partial correlation
* Confounding variable
* Coefficient of determination

|

Regression analysis|

* Errors and residuals
* Regression validation
* Mixed effects models
* Simultaneous equations models
* Multivariate adaptive regression splines (MARS)

Linear regression|

* Simple linear regression
* Ordinary least squares
* General linear model
* Bayesian regression

Non-standard predictors|

* Nonlinear regression
* Nonparametric
* Semiparametric
* Isotonic
* Robust
* Homoscedasticity and Heteroscedasticity

Generalized linear model|

* Exponential families
* Logistic (Bernoulli) / Binomial / Poisson regressions

Partition of variance|

* Analysis of variance (ANOVA, anova)
* Analysis of covariance
* Multivariate ANOVA
* Degrees of freedom

Categorical / multivariate / time-series / survival analysis

| Categorical|

* Cohen's kappa
* Contingency table
* Graphical model
* Log-linear model
* McNemar's test
* Cochran–Mantel–Haenszel statistics

|

Multivariate|

* Regression
* Manova
* Principal components
* Canonical correlation
* Discriminant analysis
* Cluster analysis
* Classification
* Structural equation model
* Factor analysis
* Multivariate distributions
* Elliptical distributions
* Normal

Time-series|

| General|

* Decomposition
* Trend
* Stationarity
* Seasonal adjustment
* Exponential smoothing
* Cointegration
* Structural break
* Granger causality

|

Specific tests|

* Dickey–Fuller
* Johansen
* Q-statistic (Ljung–Box)
* Durbin–Watson
* Breusch–Godfrey

Time domain|

* Autocorrelation (ACF)
* partial (PACF)
* Cross-correlation (XCF)
* ARMA model
* ARIMA model (Box–Jenkins)
* Autoregressive conditional heteroskedasticity (ARCH)
* Vector autoregression (VAR)

Frequency domain|

* Spectral density estimation
* Fourier analysis
* Least-squares spectral analysis
* Wavelet
* Whittle likelihood

Survival|

| Survival function|

* Kaplan–Meier estimator (product limit)
* Proportional hazards models
* Accelerated failure time (AFT) model
* First hitting time

|

Hazard function|

* Nelson–Aalen estimator

Test|

* Log-rank test

Applications

| Biostatistics|

* Bioinformatics
* Clinical trials / studies
* Epidemiology
* Medical statistics

|

Engineering statistics|

* Chemometrics
* Methods engineering
* Probabilistic design
* Process / quality control
* Reliability
* System identification

Social statistics|

* Actuarial science
* Census
* Crime statistics
* Demography
* Econometrics
* Jurimetrics
* National accounts
* Official statistics
* Population statistics
* Psychometrics

Spatial statistics|

* Cartography
* Environmental statistics
* Geographic information system
* Geostatistics
* Kriging

* Category
* Mathematics portal
* Commons
* WikiProject

*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template