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Moment (mathematics)

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In mathematics, a quantitative measure of the shape of a set of points

For the physical concept, seeMoment (physics).

Moments of afunction inmathematics are certain quantitative measures related to the shape of the function'sgraph. For example, if the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is thecenter of mass, and the second moment is themoment of inertia. If the function is aprobability distribution, then the first moment is theexpected value, the secondcentral moment is thevariance, the thirdstandardized moment is theskewness, and the fourth standardized moment is thekurtosis.

For a distribution of mass or probability on abounded interval, the collection of all the moments (of all orders, from0 to∞) uniquely determines the distribution (Hausdorff moment problem). The same is not true on unbounded intervals (Hamburger moment problem).

In the mid-nineteenth century,Pafnuty Chebyshev became the first person to think systematically in terms of the moments ofrandom variables.^[1]

Significance of the moments

Thenth raw moment (i.e., moment about zero) of a random variable $X {\displaystyle X}$ with density function $f(x)$ is defined by^[2] $\mu '_{n}=\langle X^{n}\rangle ~{\overset {\mathrm {def} }{=}}~{\begin{cases}\sum _{i}x_{i}^{n}f(x_{i}),&{\text{discrete distribution}}\\[1.2ex]\int x^{n}f(x)\,dx,&{\text{continuous distribution}}\end{cases}}$ Thenth moment of areal-valued continuous random variable with density function $f(x)$ about a value $c {\displaystyle c}$ is theintegral $\mu _{n}=\int _{-\infty }^{\infty }(x-c)^{n}\,f(x)\,\mathrm {d} x.$

It is possible to define moments forrandom variables in a more general fashion than moments for real-valued functions – seemoments in metric spaces. The moment of a function, without further explanation, usually refers to the above expression with $c=0$ .For the second and higher moments, thecentral moment (moments about the mean, withc being the mean) are usually used rather than the moments about zero, because they provide clearer information about the distribution's shape.

Other moments may also be defined. For example, thenth inverse moment about zero is $\operatorname {E} \left[X^{-n}\right]$ and thenth logarithmic moment about zero is $\operatorname {E} \left[\ln ^{n}(X)\right].$

Thenth moment about zero of a probability density function $f(x)$ is theexpected value of $X^{n}$ and is called araw moment orcrude moment.^[3] The moments about its mean $\mu$ are calledcentral moments; these describe the shape of the function, independently oftranslation.

If $f {\displaystyle f}$ is aprobability density function, then the value of the integral above is called thenth moment of theprobability distribution. More generally, ifF is acumulative probability distribution function of any probability distribution, which may not have a density function, then thenth moment of the probability distribution is given by theRiemann–Stieltjes integral $\mu '_{n}=\operatorname {E} \left[X^{n}\right]=\int _{-\infty }^{\infty }x^{n}\,\mathrm {d} F(x)$ whereX is arandom variable that has this cumulative distributionF, andE is theexpectation operator or mean.When $\operatorname {E} \left[\left|X^{n}\right|\right]=\int _{-\infty }^{\infty }\left|x^{n}\right|\,\mathrm {d} F(x)=\infty$ the moment is said not to exist. If thenth moment about any point exists, so does the(n − 1)th moment (and thus, all lower-order moments) about every point. The zeroth moment of anyprobability density function is1, since the area under anyprobability density function must be equal to one.

Significance of moments (raw, central, standardised) and cumulants (raw, normalised), in connection with named properties of distributions
Moment ordinal	Moment			Cumulant
Moment ordinal	Raw	Central	Standardized	Raw	Normalized
1	Mean	0	0	Mean	—
2	—	Variance	1	Variance	1
3	—	—	Skewness	—	Skewness
4	—	—	(Non-excess or historical)kurtosis	—	Excess kurtosis
5	—	—	Hyperskewness	—	—
6	—	—	Hypertailedness	—	—
7+	—	—	—	—	—

Standardized moments

Main article:Standardized moment

Thenormalisednth central moment or standardised moment is thenth central moment divided byσⁿ; the normalisednth central moment of the random variableX is ${\frac {\mu _{n}}{\sigma ^{n}}}={\frac {\operatorname {E} \left[(X-\mu )^{n}\right]}{\sigma ^{n}}}={\frac {\operatorname {E} \left[(X-\mu )^{n}\right]}{\operatorname {E} \left[(X-\mu )^{2}\right]^{\frac {n}{2}}}}.$

These normalised central moments aredimensionless quantities, which represent the distribution independently of any linear change of scale.

Notable moments

Mean

Main article:Mean

The first raw moment is themean, usually denoted $\mu \equiv \operatorname {E} [X].$

Variance

Main article:Variance

The secondcentral moment is thevariance. The positivesquare root of the variance is thestandard deviation $\sigma \equiv \left(\operatorname {E} \left[(x-\mu )^{2}\right]\right)^{\frac {1}{2}}.$

Skewness

Main article:Skewness

The third central moment is the measure of the lopsidedness of the distribution; any symmetric distribution will have a third central moment, if defined, of zero. The normalised third central moment is called theskewness, oftenγ. A distribution that is skewed to the left (the tail of the distribution is longer on the left) will have a negative skewness. A distribution that is skewed to the right (the tail of the distribution is longer on the right), will have a positive skewness.

For distributions that are not too different from thenormal distribution, themedian will be somewhere nearμ −γσ/6; themode aboutμ −γσ/2.

Kurtosis

Main article:Kurtosis

The fourth central moment is a measure of the heaviness of the tail of the distribution. Since it is the expectation of a fourth power, the fourth central moment, where defined, is always nonnegative; and except for apoint distribution, it is always strictly positive. The fourth central moment of a normal distribution is3σ⁴.

Thekurtosisκ is defined to be the standardized fourth central moment. (Equivalently, as in the next section, excess kurtosis is the fourthcumulant divided by the square of the secondcumulant.)^[4]^[5] If a distribution has heavy tails, the kurtosis will be high (sometimes called leptokurtic); conversely, light-tailed distributions (for example, bounded distributions such as the uniform) have low kurtosis (sometimes called platykurtic).

The kurtosis can be positive without limit, butκ must be greater than or equal toγ² + 1; equality only holds forbinary distributions. For unbounded skew distributions not too far from normal,κ tends to be somewhere in the area ofγ² and2γ².

The inequality can be proven by considering $\operatorname {E} \left[\left(T^{2}-aT-1\right)^{2}\right]$ whereT = (X −μ)/σ. This is the expectation of a square, so it is non-negative for alla; however it is also a quadraticpolynomial ina. Itsdiscriminant must be non-positive, which gives the required relationship.

Higher moments

High-order moments are moments beyond 4th-order moments.

As with variance, skewness, and kurtosis, these arehigher-order statistics, involving non-linear combinations of the data, and can be used for description or estimation of furthershape parameters. The higher the moment, the harder it is to estimate, in the sense that larger samples are required in order to obtain estimates of similar quality. This is due to the excessdegrees of freedom consumed by the higher orders. Further, they can be subtle to interpret, often being most easily understood in terms of lower order moments – compare the higher-order derivatives ofjerk andjounce inphysics. For example, just as the 4th-order moment (kurtosis) can be interpreted as "relative importance of tails as compared to shoulders in contribution to dispersion" (for a given amount of dispersion, higher kurtosis corresponds to thicker tails, while lower kurtosis corresponds to broader shoulders), the 5th-order moment can be interpreted as measuring "relative importance of tails as compared to center (mode and shoulders) in contribution to skewness" (for a given amount of skewness, higher 5th moment corresponds to higher skewness in the tail portions and little skewness of mode, while lower 5th moment corresponds to more skewness in shoulders).

Mixed moments

Mixed moments are moments involving multiple variables.

The value $E[X^{k}]$ is called the moment of order $k {\displaystyle k}$ (moments are also defined for non-integral $k {\displaystyle k}$ ). The moments of the joint distribution of random variables $X_{1}...X_{n}$ are defined similarly. For any integers $k_{i}\geq 0$ , the mathematical expectation $E[{X_{1}}^{k_{1}}\cdots {X_{n}}^{k_{n}}]$ is called a mixed moment of order $k {\displaystyle k}$ (where $k=k_{1}+...+k_{n}$ ), and $E[(X_{1}-E[X_{1}])^{k_{1}}\cdots (X_{n}-E[X_{n}])^{k_{n}}]$ is called a central mixed moment of order $k {\displaystyle k}$ . The mixed moment $E[(X_{1}-E[X_{1}])(X_{2}-E[X_{2}])]$ is called the covariance and is one of the basic characteristics of dependency between random variables.

Some examples arecovariance,coskewness andcokurtosis. While there is a unique covariance, there are multiple co-skewnesses and co-kurtoses.

Properties of moments

Transformation of center

Since $(x-b)^{n}=(x-a+a-b)^{n}=\sum _{i=0}^{n}{n \choose i}(x-a)^{i}(a-b)^{n-i}$ where ${\textstyle {\binom {n}{i}}}$ is thebinomial coefficient, it follows that the moments aboutb can be calculated from the moments abouta by: $E\left[(x-b)^{n}\right]=\sum _{i=0}^{n}{n \choose i}E\left[(x-a)^{i}\right](a-b)^{n-i}.$

Moment of a convolution of function

Main article:Convolution

The raw moment of a convolution ${\textstyle h(t)=(f*g)(t)=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau }$ reads $\mu _{n}[h]=\sum _{i=0}^{n}{n \choose i}\mu _{i}[f]\mu _{n-i}[g]$ where $\mu _{n}[\,\cdot \,]$ denotes the $n {\displaystyle n}$ th moment of the function given in the brackets. This identity follows by the convolution theorem for moment generating function and applying the chain rule fordifferentiating a product.

Cumulants

Main article:Cumulant

The first raw moment and the second and thirdunnormalized central moments are additive in the sense that ifX andY areindependent random variables then ${\begin{aligned}m_{1}(X+Y)&=m_{1}(X)+m_{1}(Y)\\\operatorname {Var} (X+Y)&=\operatorname {Var} (X)+\operatorname {Var} (Y)\\\mu _{3}(X+Y)&=\mu _{3}(X)+\mu _{3}(Y)\end{aligned}}$

(These can also hold for variables that satisfy weaker conditions than independence. The first always holds; if the second holds, the variables are calleduncorrelated).

These are the first three cumulants and all cumulants share this additivity property.

Sample moments

For allk, thekth raw moment of a population can be estimated using thekth raw sample moment ${\frac {1}{n}}\sum _{i=1}^{n}X_{i}^{k}$ applied to a sampleX₁, ...,X_n drawn from the population.

It can be shown that the expected value of the raw sample moment is equal to thekth raw moment of the population, if that moment exists, for any sample sizen. It is thus an unbiased estimator. This contrasts with the situation for central moments, whose computation uses up a degree of freedom by using the sample mean. So for example an unbiased estimate of the population variance (the second central moment) is given by ${\frac {1}{n-1}}\sum _{i=1}^{n}\left(X_{i}-{\bar {X}}\right)^{2}$ in which the previous denominatorn has been replaced by the degrees of freedomn − 1, and in which ${\bar {X}}$ refers to the sample mean. This estimate of the population moment is greater than the unadjusted observed sample moment by a factor of ${\tfrac {n}{n-1}},$ and it is referred to as the "adjusted sample variance" or sometimes simply the "sample variance".

Problem of moments

Main article:Moment problem

Problems of determining a probability distribution from its sequence of moments are calledproblem of moments. Such problems were first discussed by P.L. Chebyshev (1874)^[6] in connection with research on limit theorems. In order that the probability distribution of a random variable $X {\displaystyle X}$ be uniquely defined by its moments $\alpha _{k}=E\left[X^{k}\right]$ it is sufficient, for example, that Carleman's condition be satisfied: $\sum _{k=1}^{\infty }{\frac {1}{\alpha _{2k}^{1/2k}}}=\infty$ A similar result even holds for moments of random vectors. Theproblem of moments seeks characterizations of sequences ${{\mu _{n}}':n=1,2,3,\dots }$ that are sequences of moments of some functionf, all moments $\alpha _{k}(n)$ of which are finite, and for each integer $k\geq 1$ let $\alpha _{k}(n)\rightarrow \alpha _{k},n\rightarrow \infty ,$ where $\alpha _{k}$ is finite. Then there is a sequence ${\mu _{n}}'$ that weakly converges to a distribution function $\mu$ having $\alpha _{k}$ as its moments. If the moments determine $\mu$ uniquely, then the sequence ${\mu _{n}}'$ weakly converges to $\mu$ .

Partial moments

Partial moments are sometimes referred to as "one-sided moments". Thenth order lower and upper partial moments with respect to a reference pointr may be expressed as $\mu _{n}^{-}(r)=\int _{-\infty }^{r}(r-x)^{n}\,f(x)\,\mathrm {d} x,$ $\mu _{n}^{+}(r)=\int _{r}^{\infty }(x-r)^{n}\,f(x)\,\mathrm {d} x.$

If the integral function does not converge, the partial moment does not exist.

Partial moments are normalized by being raised to the power1/n. Theupside potential ratio may be expressed as a ratio of a first-order upper partial moment to a normalized second-order lower partial moment.

Central moments in metric spaces

Let(M,d) be ametric space, and letB(M) be theBorelσ-algebra onM, theσ-algebra generated by thed-open subsets ofM. (For technical reasons, it is also convenient to assume thatM is aseparable space with respect to themetricd.) Let1 ≤p ≤ ∞.

Thepth central moment of a measureμ on themeasurable space(M, B(M)) about a given pointx₀ ∈M is defined to be $\int _{M}d\left(x,x_{0}\right)^{p}\,\mathrm {d} \mu (x).$

μ is said to havefinitepth central moment if thepth central moment ofμ aboutx₀ is finite for somex₀ ∈M.

This terminology for measures carries over to random variables in the usual way: if(Ω, Σ,P) is aprobability space andX : Ω →M is a random variable, then thepth central moment ofX aboutx₀ ∈M is defined to be $\int _{M}d\left(x,x_{0}\right)^{p}\,\mathrm {d} \left(X_{*}\left(\mathbf {P} \right)\right)(x)=\int _{\Omega }d\left(X(\omega ),x_{0}\right)^{p}\,\mathrm {d} \mathbf {P} (\omega )=\operatorname {\mathbf {E} } [d(X,x_{0})^{p}],$ andX hasfinitepth central moment if thepth central moment ofX aboutx₀ is finite for somex₀ ∈M.

See also

References

Text was copied fromMoment at the Encyclopedia of Mathematics, which is released under aCreative Commons Attribution-Share Alike 3.0 (Unported) (CC-BY-SA 3.0) license and theGNU Free Documentation License.

^George Mackey (July 1980). "HARMONIC ANALYSIS AS THE EXPLOITATION OF SYMMETRY - A HISTORICAL SURVEY".Bulletin of the American Mathematical Society. New Series.3 (1): 549.
^Papoulis, A. (1984).Probability, Random Variables, and Stochastic Processes, 2nd ed. New York:McGraw Hill. pp. 145–149.
^"Raw Moment -- from Wolfram MathWorld".Archived from the original on 2009-05-28. Retrieved2009-06-24. Raw Moments at Math-world
^Casella, George;Berger, Roger L. (2002).Statistical Inference (2 ed.). Pacific Grove: Duxbury.ISBN 0-534-24312-6.
^Ballanda, Kevin P.;MacGillivray, H. L. (1988). "Kurtosis: A Critical Review".The American Statistician.42 (2). American Statistical Association:111–119.doi:10.2307/2684482.JSTOR 2684482.
^Feller, W. (1957-1971).An introduction to probability theory and its applications. New York: John Wiley & Sons. 419 p.

Further reading

Spanos, Aris (1999).Probability Theory and Statistical Inference. New York: Cambridge University Press. pp. 109–130.ISBN 0-521-42408-9.
Walker, Helen M. (1929).Studies in the history of statistical method, with special reference to certain educational problems. Baltimore, Williams & Wilkins Co. p. 71.

External links

"Moment",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
Moments at Mathworld

v t e Theory ofprobability distributions
probability mass function (pmf) probability density function (pdf) cumulative distribution function (cdf) quantile function
raw moment central moment mean variance standard deviation skewness kurtosis L-moment
moment-generating function (mgf) characteristic function probability-generating function (pgf) cumulant combinant

v
t
e

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

Index of dispersion

Summary tables

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

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

Z-test(normal) Student'st-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

Correlation	Pearson product-moment Partial correlation Confounding variable Coefficient of determination
Regression analysis (see alsoTemplate:Least squares and 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

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) (Autoregressive model (AR))
Frequency domain	Spectral density estimation Fourier analysis Least-squares spectral analysis Wavelet Whittle likelihood

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

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

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