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Multivariate normal

Probability density function Many sample points from a multivariate normal distribution with μ=[00]{displaystyle {boldsymbol {mu }}=left[{begin{smallmatrix}0\0end{smallmatrix}}right]} and Σ=[13/53/52]{displaystyle {boldsymbol {Sigma }}=left[{begin{smallmatrix}1&3/5\3/5&2end{smallmatrix}}right]}, shown along with the 3-sigma ellipse, the two marginal distributions, and the two 1-d histograms.
Notation	N(μ,Σ){displaystyle {mathcal {N}}({boldsymbol {mu }},,{boldsymbol {Sigma }})}
Parameters	μ ∈ Rk — location Σ ∈ Rk×k — covariance (positive semi-definite matrix)
Support	x ∈ μ + span(Σ) ⊆ Rk
PDF	(2π)−k2det⁡(Σ)−12e−12(x−μ)′Σ−1(x−μ),{displaystyle (2pi )^{-{frac {k}{2}}}operatorname {det} ({boldsymbol {Sigma }})^{-{frac {1}{2}}},e^{-{frac {1}{2}}(mathbf {x} -{boldsymbol {mu }})'{boldsymbol {Sigma }}^{-1}(mathbf {x} -{boldsymbol {mu }})},} exists only when Σ is positive-definite
Mean	μ
Mode	μ
Variance	Σ
Entropy	12ln⁡det⁡(2πeΣ){displaystyle {frac {1}{2}}ln operatorname {det} left(2pi mathrm {e} {boldsymbol {Sigma }}right)}
MGF	exp(μ′t+12t′Σt){displaystyle exp !{Big (}{boldsymbol {mu }}'mathbf {t} +{tfrac {1}{2}}mathbf {t} '{boldsymbol {Sigma }}mathbf {t} {Big )}}
CF	exp(iμ′t−12t′Σt){displaystyle exp !{Big (}i{boldsymbol {mu }}'mathbf {t} -{tfrac {1}{2}}mathbf {t} '{boldsymbol {Sigma }}mathbf {t} {Big )}}

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value.

Notation and parametrization

The multivariate normal distribution of a k-dimensional random vector X=(X1,…,Xk)T{displaystyle mathbf {X} =(X_{1},ldots ,X_{k})^{T}} can be written in the following notation:

X ∼ N(μ,Σ),{displaystyle mathbf {X} sim {mathcal {N}}({boldsymbol {mu }},,{boldsymbol {Sigma }}),}

or to make it explicitly known that X is k-dimensional,

X ∼ Nk(μ,Σ),{displaystyle mathbf {X} sim {mathcal {N}}_{k}({boldsymbol {mu }},,{boldsymbol {Sigma }}),}

with k-dimensional mean vector

μ=E⁡[X]=[E⁡[X1],E⁡[X2],…,E⁡[Xk]]T,{displaystyle {boldsymbol {mu }}=operatorname {E} [mathbf {X} ]=[operatorname {E} [X_{1}],operatorname {E} [X_{2}],ldots ,operatorname {E} [X_{k}]]^{rm {T}},}

and k×k{displaystyle ktimes k} covariance matrix

Σi,j=:E⁡[(Xi−μi)(Xj−μj)]=Cov⁡[Xi,Xj]{displaystyle Sigma _{i,j}=:operatorname {E} [(X_{i}-mu _{i})(X_{j}-mu _{j})]=operatorname {Cov} [X_{i},X_{j}]}

Σ=:E⁡[(X−μ)(X−μ)T]=[Cov⁡[Xi,Xj];1≤i,j≤k].{displaystyle {boldsymbol {Sigma }}=:operatorname {E} [(mathbf {X} -{boldsymbol {mu }})(mathbf {X} -{boldsymbol {mu }})^{rm {T}}]=[operatorname {Cov} [X_{i},X_{j}];1leq i,jleq k].}

The inverse of the covariance matrix is called the precision matrix, denoted by Q=Σ−1{displaystyle {boldsymbol {Q}}={boldsymbol {Sigma }}^{-1}}.

Definitions

Standard normal random vector

A real random vector X=(X1,…,Xk)T{displaystyle mathbf {X} =(X_{1},ldots ,X_{k})^{mathrm {T} }} is called a standard normal random vector if all of its components Xn{displaystyle X_{n}} are independent and each is a zero-mean unit-variance normally distributed random variable, i.e. if Xn∼ N(0,1){displaystyle X_{n}sim {mathcal {N}}(0,1)} for all n{displaystyle n}.^{[1]:p. 454}

Centered normal random vector

A real random vector X=(X1,…,Xk)T{displaystyle mathbf {X} =(X_{1},ldots ,X_{k})^{mathrm {T} }} is called a centered normal random vector if there exists a deterministic k×ℓ{displaystyle ktimes ell } matrix A{displaystyle {boldsymbol {A}}} such that AZ{displaystyle {boldsymbol {A}}mathbf {Z} } has the same distribution as X{displaystyle mathbf {X} } where Z{displaystyle mathbf {Z} } is a standard normal random vector with ℓ{displaystyle ell } components.^{[1]:p. 454}

Normal random vector

A real random vector X=(X1,…,Xk)T{displaystyle mathbf {X} =(X_{1},ldots ,X_{k})^{mathrm {T} }} is called a normal random vector if there exists a random ℓ{displaystyle ell }-vector Z{displaystyle mathbf {Z} }, which is a standard normal random vector, a k{displaystyle k}-vector μ{displaystyle mathbf {mu } }, and a k×ℓ{displaystyle ktimes ell } matrix A{displaystyle {boldsymbol {A}}}, such that X=AZ+μ{displaystyle mathbf {X} ={boldsymbol {A}}mathbf {Z} +mathbf {mu } }.^{[2]:p. 454[1]:p. 455}

Formally:

Here the covariance matrix is Σ=AAT{displaystyle {boldsymbol {Sigma }}={boldsymbol {A}}{boldsymbol {A}}^{mathrm {T} }}.

In the degenerate case where the covariance matrix is singular, the corresponding distribution has no density; see the section below for details. This case arises frequently in statistics; for example, in the distribution of the vector of residuals in the ordinary least squares regression. Note also that the Xi{displaystyle X_{i}} are in general not independent; they can be seen as the result of applying the matrix A{displaystyle {boldsymbol {A}}} to a collection of independent Gaussian variables Z{displaystyle mathbf {Z} }.

Equivalent definitions

The following definitions are equivalent to the definition given above. A random vector X=(X1,…,Xk)T{displaystyle mathbf {X} =(X_{1},ldots ,X_{k})^{T}} has a multivariate normal distribution if it satisfies one of the following equivalent conditions.

• Every linear combination Y=a1X1+⋯+akXk{displaystyle Y=a_{1}X_{1}+cdots +a_{k}X_{k}} of its components is normally distributed. That is, for any constant vector a∈Rk{displaystyle mathbf {a} in mathbb {R} ^{k}}, the random variable Y=aTX{displaystyle Y=mathbf {a} ^{mathrm {T} }mathbf {X} } has a univariate normal distribution, where a univariate normal distribution with zero variance is a point mass on its mean.


• There is a k-vector μ{displaystyle mathbf {mu } } and a symmetric, positive semidefinite k×k{displaystyle ktimes k} matrix Σ{displaystyle {boldsymbol {Sigma }}}, such that the characteristic function of X{displaystyle mathbf {X} } is

φX(u)=exp⁡(iuTμ−12uTΣu).{displaystyle varphi _{mathbf {X} }(mathbf {u} )=exp {Big (}imathbf {u} ^{T}{boldsymbol {mu }}-{tfrac {1}{2}}mathbf {u} ^{T}{boldsymbol {Sigma }}mathbf {u} {Big )}.}

The spherical normal distribution can be characterised as the unique distribution where components are independent in any orthogonal coordinate system.^[3][4]

Properties

Density function

Bivariate normal joint density

Non-degenerate case

The multivariate normal distribution is said to be "non-degenerate" when the symmetric covariance matrix Σ{displaystyle {boldsymbol {Sigma }}} is positive definite. In this case the distribution has density^[5]

where x{displaystyle {mathbf {x} }} is a real k-dimensional column vector and |Σ|≡det⁡Σ{displaystyle |{boldsymbol {Sigma }}|equiv operatorname {det} {boldsymbol {Sigma }}} is the determinant of Σ{displaystyle {boldsymbol {Sigma }}}. The equation above reduces to that of the univariate normal distribution if Σ{displaystyle {boldsymbol {Sigma }}} is a 1×1{displaystyle 1times 1} matrix (i.e. a single real number).

The circularly symmetric version of the complex normal distribution has a slightly different form.

Each iso-density locus—the locus of points in k-dimensional space each of which gives the same particular value of the density—is an ellipse or its higher-dimensional generalization; hence the multivariate normal is a special case of the elliptical distributions.

The descriptive statistic (x−μ)TΣ−1(x−μ){displaystyle {sqrt {({mathbf {x} }-{boldsymbol {mu }})^{mathrm {T} }{boldsymbol {Sigma }}^{-1}({mathbf {x} }-{boldsymbol {mu }})}}} is known as the Mahalanobis distance, which represents the distance of the test point x{displaystyle {mathbf {x} }} from the mean μ{displaystyle {boldsymbol {mu }}}. Note that in the case when k=1{displaystyle k=1}, the distribution reduces to a univariate normal distribution and the Mahalanobis distance reduces to the absolute value of the standard score. See also Interval below.

Bivariate case

In the 2-dimensional nonsingular case (k = rank(Σ) = 2), the probability density function of a vector [X Y]′ is:

f(x,y)=12πσXσY1−ρ2exp⁡(−12(1−ρ2)[(x−μX)2σX2+(y−μY)2σY2−2ρ(x−μX)(y−μY)σXσY]){displaystyle f(x,y)={frac {1}{2pi sigma _{X}sigma _{Y}{sqrt {1-rho ^{2}}}}}exp left(-{frac {1}{2(1-rho ^{2})}}left[{frac {(x-mu _{X})^{2}}{sigma _{X}^{2}}}+{frac {(y-mu _{Y})^{2}}{sigma _{Y}^{2}}}-{frac {2rho (x-mu _{X})(y-mu _{Y})}{sigma _{X}sigma _{Y}}}right]right)}

where ρ is the correlation between X and Y and
where σX>0{displaystyle sigma _{X}>0} and σY>0{displaystyle sigma _{Y}>0}. In this case,

μ=(μXμY),Σ=(σX2ρσXσYρσXσYσY2).{displaystyle {boldsymbol {mu }}={begin{pmatrix}mu _{X}\mu _{Y}end{pmatrix}},quad {boldsymbol {Sigma }}={begin{pmatrix}sigma _{X}^{2}&rho sigma _{X}sigma _{Y}\rho sigma _{X}sigma _{Y}&sigma _{Y}^{2}end{pmatrix}}.}

In the bivariate case, the first equivalent condition for multivariate normality can be made less restrictive: it is sufficient to verify that countably many distinct linear combinations of X and Y are normal in order to conclude that the vector [X Y]′ is bivariate normal.^[6]

The bivariate iso-density loci plotted in the x,y-plane are ellipses. As the absolute value of the correlation parameter ρ increases, these loci are squeezed toward the following line :

y(x)=sgn⁡(ρ)σYσX(x−μX)+μY.{displaystyle y(x)=operatorname {sgn}(rho ){frac {sigma _{Y}}{sigma _{X}}}(x-mu _{X})+mu _{Y}.}

This is because this expression, with sgn(ρ) replaced by ρ, is the best linear unbiased prediction of Y given a value of X.^[7]

Degenerate case

If the covariance matrix Σ{displaystyle {boldsymbol {Sigma }}} is not full rank, then the multivariate normal distribution is degenerate and does not have a density. More precisely, it does not have a density with respect to k-dimensional Lebesgue measure (which is the usual measure assumed in calculus-level probability courses). Only random vectors whose distributions are absolutely continuous with respect to a measure are said to have densities (with respect to that measure). To talk about densities but avoid dealing with measure-theoretic complications it can be simpler to restrict attention to a subset of rank⁡(Σ){displaystyle operatorname {rank} ({boldsymbol {Sigma }})} of the coordinates of x{displaystyle mathbf {x} } such that the covariance matrix for this subset is positive definite; then the other coordinates may be thought of as an affine function of the selected coordinates.^{[citation needed]}

To talk about densities meaningfully in the singular case, then, we must select a different base measure. Using the disintegration theorem we can define a restriction of Lebesgue measure to the rank⁡(Σ){displaystyle operatorname {rank} ({boldsymbol {Sigma }})}-dimensional affine subspace of Rk{displaystyle mathbb {R} ^{k}} where the Gaussian distribution is supported, i.e. {μ+Σ1/2v:v∈Rk}{displaystyle {{boldsymbol {mu }}+{boldsymbol {Sigma ^{1/2}}}mathbf {v} :mathbf {v} in mathbb {R} ^{k}}}. With respect to this measure the distribution has density:

f(x)=(det∗(2πΣ))−12e−12(x−μ)′Σ+(x−μ){displaystyle f(mathbf {x} )=left(det nolimits ^{*}(2pi {boldsymbol {Sigma }})right)^{-{frac {1}{2}}},e^{-{frac {1}{2}}(mathbf {x} -{boldsymbol {mu }})'{boldsymbol {Sigma }}^{+}(mathbf {x} -{boldsymbol {mu }})}}

where Σ+{displaystyle {boldsymbol {Sigma }}^{+}} is the generalized inverse and det* is the pseudo-determinant.^[8]

Higher moments

The kth-order moments of x are given by

μ1,…,N(x) =def μr1,…,rN(x) =defE⁡[∏j=1NXjrj]{displaystyle mu _{1,ldots ,N}(mathbf {x} ) {stackrel {mathrm {def} }{=}} mu _{r_{1},ldots ,r_{N}}(mathbf {x} ) {stackrel {mathrm {def} }{=}}operatorname {E} left[prod _{j=1}^{N}X_{j}^{r_{j}}right]}

where r₁ + r₂ + ⋯ + r_N = k.

The kth-order central moments are as follows

μ1,…,2λ(x−μ)=∑(σijσkℓ⋯σXZ){displaystyle mu _{1,dots ,2lambda }(mathbf {x} -{boldsymbol {mu }})=sum left(sigma _{ij}sigma _{kell }cdots sigma _{XZ}right)}

where the sum is taken over all allocations of the set {1,…,2λ}{displaystyle left{1,ldots ,2lambda right}} into λ (unordered) pairs. That is, for a kth (= 2λ = 6) central moment, one sums the products of λ = 3 covariances (the expected value μ is taken to be 0 in the interests of parsimony):

E⁡[X1X2X3X4X5X6]=E⁡[X1X2]E⁡[X3X4]E⁡[X5X6]+E⁡[X1X2]E⁡[X3X5]E⁡[X4X6]+E⁡[X1X2]E⁡[X3X6]E⁡[X4X5]+E⁡[X1X3][⁡X2X4]E⁡[X5X6]+E⁡[X1X3]E⁡[X2X5]E⁡[X4X6]+E⁡[X1X3]E⁡[X2X6]E⁡[X4X5]+E⁡[X1X4]E⁡[X2X3]E⁡[X5X6]+E⁡[X1X4]E⁡[X2X5]E⁡[X3X6]+E⁡[X1X4]E⁡[X2X6]E⁡[X3X5]+E⁡[X1X5]E⁡[X2X3]E⁡[X4X6]+E⁡[X1X5]E⁡[X2X4]E⁡[X3X6]+E⁡[X1X5]E⁡[X2X6]E⁡[X3X4]+E⁡[X1X6]E⁡[X2X3]E⁡[X4X5]+E⁡[X1X6]E⁡[X2X4]E⁡[X3X5]+E⁡[X1X6]E⁡[X2X5]E⁡[X3X4].{displaystyle {begin{aligned}&operatorname {E} [X_{1}X_{2}X_{3}X_{4}X_{5}X_{6}]\[8pt]={}&operatorname {E} [X_{1}X_{2}]operatorname {E} [X_{3}X_{4}]operatorname {E} [X_{5}X_{6}]+operatorname {E} [X_{1}X_{2}]operatorname {E} [X_{3}X_{5}]operatorname {E} [X_{4}X_{6}]+operatorname {E} [X_{1}X_{2}]operatorname {E} [X_{3}X_{6}]operatorname {E} [X_{4}X_{5}]\[4pt]&{}+operatorname {E} [X_{1}X_{3}]operatorname {[} X_{2}X_{4}]operatorname {E} [X_{5}X_{6}]+operatorname {E} [X_{1}X_{3}]operatorname {E} [X_{2}X_{5}]operatorname {E} [X_{4}X_{6}]+operatorname {E} [X_{1}X_{3}]operatorname {E} [X_{2}X_{6}]operatorname {E} [X_{4}X_{5}]\[4pt]&{}+operatorname {E} [X_{1}X_{4}]operatorname {E} [X_{2}X_{3}]operatorname {E} [X_{5}X_{6}]+operatorname {E} [X_{1}X_{4}]operatorname {E} [X_{2}X_{5}]operatorname {E} [X_{3}X_{6}]+operatorname {E} [X_{1}X_{4}]operatorname {E} [X_{2}X_{6}]operatorname {E} [X_{3}X_{5}]\[4pt]&{}+operatorname {E} [X_{1}X_{5}]operatorname {E} [X_{2}X_{3}]operatorname {E} [X_{4}X_{6}]+operatorname {E} [X_{1}X_{5}]operatorname {E} [X_{2}X_{4}]operatorname {E} [X_{3}X_{6}]+operatorname {E} [X_{1}X_{5}]operatorname {E} [X_{2}X_{6}]operatorname {E} [X_{3}X_{4}]\[4pt]&{}+operatorname {E} [X_{1}X_{6}]operatorname {E} [X_{2}X_{3}]operatorname {E} [X_{4}X_{5}]+operatorname {E} [X_{1}X_{6}]operatorname {E} [X_{2}X_{4}]operatorname {E} [X_{3}X_{5}]+operatorname {E} [X_{1}X_{6}]operatorname {E} [X_{2}X_{5}]operatorname {E} [X_{3}X_{4}].end{aligned}}}

This yields (2λ−1)!2λ−1(λ−1)!{displaystyle {tfrac {(2lambda -1)!}{2^{lambda -1}(lambda -1)!}}} terms in the sum (15 in the above case), each being the product of λ (in this case 3) covariances. For fourth order moments (four variables) there are three terms. For sixth-order moments there are 3 × 5 = 15 terms, and for eighth-order moments there are 3 × 5 × 7 = 105 terms.

The covariances are then determined by replacing the terms of the list [1,…,2λ]{displaystyle [1,ldots ,2lambda ]} by the corresponding terms of the list consisting of r₁ ones, then r₂ twos, etc.. To illustrate this, examine the following 4th-order central moment case:

E⁡[Xi4]=3σii2E⁡[Xi3Xj]=3σiiσijE⁡[Xi2Xj2]=σiiσjj+2σij2E⁡[Xi2XjXk]=σiiσjk+2σijσikE⁡[XiXjXkXn]=σijσkn+σikσjn+σinσjk.{displaystyle {begin{aligned}operatorname {E} left[X_{i}^{4}right]&=3sigma _{ii}^{2}\[4pt]operatorname {E} left[X_{i}^{3}X_{j}right]&=3sigma _{ii}sigma _{ij}\[4pt]operatorname {E} left[X_{i}^{2}X_{j}^{2}right]&=sigma _{ii}sigma _{jj}+2sigma _{ij}^{2}\[4pt]operatorname {E} left[X_{i}^{2}X_{j}X_{k}right]&=sigma _{ii}sigma _{jk}+2sigma _{ij}sigma _{ik}\[4pt]operatorname {E} left[X_{i}X_{j}X_{k}X_{n}right]&=sigma _{ij}sigma _{kn}+sigma _{ik}sigma _{jn}+sigma _{in}sigma _{jk}.end{aligned}}}

where σij{displaystyle sigma _{ij}} is the covariance of X_i and X_j. With the above method one first finds the general case for a kth moment with k different X variables, E[XiXjXkXn]{displaystyle Eleft[X_{i}X_{j}X_{k}X_{n}right]}, and then one simplifies this accordingly. For example, for E⁡[Xi2XkXn]{displaystyle operatorname {E} [X_{i}^{2}X_{k}X_{n}]}, one lets X_i = X_j and one uses the fact that σii=σi2{displaystyle sigma _{ii}=sigma _{i}^{2}}.

Likelihood function

If the mean and variance matrix are known, a suitable log likelihood function for a single observation x is

ln⁡L=−12[ln⁡(|Σ|)+(x−μ)TΣ−1(x−μ)+kln⁡(2π)]{displaystyle ln L=-{frac {1}{2}}left[ln(|{boldsymbol {Sigma }}|,)+(mathbf {x} -{boldsymbol {mu }})^{rm {T}}{boldsymbol {Sigma }}^{-1}(mathbf {x} -{boldsymbol {mu }})+kln(2pi )right]},

where x is a vector of real numbers (to derive this, simply take the log of the PDF). The circularly symmetric version of the complex case, where z is a vector of complex numbers, would be

ln⁡L=−ln⁡(|Σ|)−(z−μ)†Σ−1(z−μ)−kln⁡(π){displaystyle ln L=-ln(|{boldsymbol {Sigma }}|,)-(mathbf {z} -{boldsymbol {mu }})^{dagger }{boldsymbol {Sigma }}^{-1}(mathbf {z} -{boldsymbol {mu }})-kln(pi )}

i.e. with the conjugate transpose (indicated by †{displaystyle dagger }) replacing the normal transpose (indicated by T{displaystyle {}^{rm {T}}}). This is slightly different than in the real case, because the circularly symmetric version of the complex normal distribution has a slightly different form.

A similar notation is used for multiple linear regression.^[9]

Differential entropy

The differential entropy of the multivariate normal distribution is^[10]

h(f)=−∫−∞∞∫−∞∞⋯∫−∞∞f(x)ln⁡f(x)dx,=12ln⁡(|(2πe)kΣ|)=k2ln⁡(2πe)+12ln⁡(|Σ|)=k2+k2ln⁡(2π)+12ln⁡(|Σ|){displaystyle {begin{aligned}hleft(fright)&=-int _{-infty }^{infty }int _{-infty }^{infty }cdots int _{-infty }^{infty }f(mathbf {x} )ln f(mathbf {x} ),dmathbf {x} ,\&={frac {1}{2}}ln left(left|left(2pi eright)^{k}{boldsymbol {Sigma }}right|right)={frac {k}{2}}ln left(2pi eright)+{frac {1}{2}}ln left(left|{boldsymbol {Sigma }}right|right)={frac {k}{2}}+{frac {k}{2}}ln left(2pi right)+{frac {1}{2}}ln left(left|{boldsymbol {Sigma }}right|right)\end{aligned}}}

where the bars denote the matrix determinant and k is the dimensionality of the vector space.

Kullback–Leibler divergence

The Kullback–Leibler divergence from N0(μ0,Σ0){displaystyle {mathcal {N}}_{0}({boldsymbol {mu }}_{0},{boldsymbol {Sigma }}_{0})} to N1(μ1,Σ1){displaystyle {mathcal {N}}_{1}({boldsymbol {mu }}_{1},{boldsymbol {Sigma }}_{1})}, for non-singular matrices Σ₀ and Σ₁, is:^[11]

DKL(N0‖N1)=12{tr⁡(Σ1−1Σ0)+(μ1−μ0)TΣ1−1(μ1−μ0)−k+ln⁡|Σ1||Σ0|},{displaystyle D_{text{KL}}({mathcal {N}}_{0}|{mathcal {N}}_{1})={1 over 2}left{operatorname {tr} left({boldsymbol {Sigma }}_{1}^{-1}{boldsymbol {Sigma }}_{0}right)+left({boldsymbol {mu }}_{1}-{boldsymbol {mu }}_{0}right)^{rm {T}}{boldsymbol {Sigma }}_{1}^{-1}({boldsymbol {mu }}_{1}-{boldsymbol {mu }}_{0})-k+ln {|{boldsymbol {Sigma }}_{1}| over |{boldsymbol {Sigma }}_{0}|}right},}

where k{displaystyle k} is the dimension of the vector space.

The logarithm must be taken to base e since the two terms following the logarithm are themselves base-e logarithms of expressions that are either factors of the density function or otherwise arise naturally. The equation therefore gives a result measured in nats. Dividing the entire expression above by log_e 2 yields the divergence in bits.

When μ1=μ0{displaystyle {boldsymbol {mu }}_{1}={boldsymbol {mu }}_{0}},

DKL(CN0‖CN1)=tr⁡(Σ1−1Σ0)−k+ln⁡|Σ1||Σ0|.{displaystyle D_{text{KL}}({mathcal {CN}}_{0}|{mathcal {CN}}_{1})=operatorname {tr} left({boldsymbol {Sigma }}_{1}^{-1}{boldsymbol {Sigma }}_{0}right)-k+ln {|{boldsymbol {Sigma }}_{1}| over |{boldsymbol {Sigma }}_{0}|}.}

Mutual information

The mutual information of a distribution is a special case of the Kullback–Leibler divergence in which P{displaystyle P} is the full multivariate distribution and Q{displaystyle Q} is the product of the 1-dimensional marginal distributions. In the notation of the Kullback–Leibler divergence section of this article, Σ1{displaystyle {boldsymbol {Sigma }}_{1}} is a diagonal matrix with the diagonal entries of Σ0{displaystyle {boldsymbol {Sigma }}_{0}}, and μ1=μ0{displaystyle {boldsymbol {mu }}_{1}={boldsymbol {mu }}_{0}}. The resulting formula for mutual information is:

I(X)=−12ln⁡|ρ0|,{displaystyle I({boldsymbol {X}})=-{1 over 2}ln |{boldsymbol {rho }}_{0}|,}

where ρ0{displaystyle {boldsymbol {rho }}_{0}} is the correlation matrix constructed from Σ0{displaystyle {boldsymbol {Sigma }}_{0}}.

In the bivariate case the expression for the mutual information is:

I(x;y)=−12ln⁡(1−ρ2).{displaystyle I(x;y)=-{1 over 2}ln(1-rho ^{2}).}

Cumulative distribution function

The notion of cumulative distribution function (cdf) in dimension 1 can be extended in two ways to the multidimensional case, based on rectangular and ellipsoidal regions.

The first way is to define the cdf F(x){displaystyle F(mathbf {x} )} of a random vector X{displaystyle mathbf {X} } as the probability that all components of X{displaystyle mathbf {X} } are less than or equal to the corresponding values in the vector x{displaystyle mathbf {x} }:^[12]

F(x)=P(X≤x),where X∼N(μ,Σ).{displaystyle F(mathbf {x} )=mathbb {P} (mathbf {X} leq mathbf {x} ),quad {text{where }}mathbf {X} sim {mathcal {N}}({boldsymbol {mu }},,{boldsymbol {Sigma }}).}

Though there is no closed form for F(x){displaystyle F(mathbf {x} )}, there are a number of algorithms that estimate it numerically.^[12][13]

Another way is to define the cdf F(r){displaystyle F(r)} as the probability that a sample lies inside the ellipsoid determined by its Mahalanobis distance r{displaystyle r} from the Gaussian, a direct generalization of the standard deviation
.^[14]
In order to compute the values of this function, closed analytic formulae exist,^[14] as follows.

Interval

The interval for the multivariate normal distribution yields a region consisting of those vectors x satisfying

(x−μ)TΣ−1(x−μ)≤χk2(p).{displaystyle ({mathbf {x} }-{boldsymbol {mu }})^{T}{boldsymbol {Sigma }}^{-1}({mathbf {x} }-{boldsymbol {mu }})leq chi _{k}^{2}(p).}

Here x{displaystyle {mathbf {x} }} is a k{displaystyle k}-dimensional vector, μ{displaystyle {boldsymbol {mu }}} is the known k{displaystyle k}-dimensional mean vector, Σ{displaystyle {boldsymbol {Sigma }}} is the known covariance matrix and χk2(p){displaystyle chi _{k}^{2}(p)} is the quantile function for probability p{displaystyle p} of the chi-squared distribution with k{displaystyle k} degrees of freedom.^[15]
When k=2,{displaystyle k=2,} the expression defines the interior of an ellipse and the chi-squared distribution simplifies to an exponential distribution with mean equal to two.

Complementary cumulative distribution function (tail distribution)

The complementary cumulative distribution function (ccdf) or the tail distribution
is defined as F¯(x)=1−P(X≤x){displaystyle {overline {F}}(mathbf {x} )=1-mathbb {P} (mathbf {X} leq mathbf {x} )}.
When X∼N(μ,Σ){displaystyle mathbf {X} sim {mathcal {N}}({boldsymbol {mu }},,{boldsymbol {Sigma }})}, then
the ccdf can be written as a probability the maximum of dependent Gaussian variables:^[16]

F¯(x)=P(∪i{Xi≥xi})=P(maxiYi≥0),where Y∼N(μ−x,Σ).{displaystyle {overline {F}}(mathbf {x} )=mathbb {P} (cup _{i}{X_{i}geq x_{i}})=mathbb {P} (max _{i}Y_{i}geq 0),quad {text{where }}mathbf {Y} sim {mathcal {N}}({boldsymbol {mu }}-mathbf {x} ,,{boldsymbol {Sigma }}).}

While no simple closed formula exists for computing the ccdf, the maximum of dependent Gaussian variables can
be estimated accurately via the Monte Carlo method.^[16][17]

Joint normality

Normally distributed and independent

If X{displaystyle X} and Y{displaystyle Y} are normally distributed and independent, this implies they are "jointly normally distributed", i.e., the pair (X,Y){displaystyle (X,Y)} must have multivariate normal distribution. However, a pair of jointly normally distributed variables need not be independent (would only be so if uncorrelated, ρ=0{displaystyle rho =0} ).

Two normally distributed random variables need not be jointly bivariate normal

The fact that two random variables X{displaystyle X} and Y{displaystyle Y} both have a normal distribution does not imply that the pair (X,Y){displaystyle (X,Y)} has a joint normal distribution. A simple example is one in which X has a normal distribution with expected value 0 and variance 1, and Y=X{displaystyle Y=X} if |X|>c{displaystyle |X|>c} and Y=−X{displaystyle Y=-X} if |X|<c{displaystyle |X|<c}, where c>0{displaystyle c>0}. There are similar counterexamples for more than two random variables. In general, they sum to a mixture model.

Correlations and independence

In general, random variables may be uncorrelated but statistically dependent. But if a random vector has a multivariate normal distribution then any two or more of its components that are uncorrelated are independent. This implies that any two or more of its components that are pairwise independent are independent. But, as pointed out just above, it is not true that two random variables that are (separately, marginally) normally distributed and uncorrelated are independent.

Conditional distributions

If N-dimensional x is partitioned as follows

x=[x1x2] with sizes [q×1(N−q)×1]{displaystyle mathbf {x} ={begin{bmatrix}mathbf {x} _{1}\mathbf {x} _{2}end{bmatrix}}{text{ with sizes }}{begin{bmatrix}qtimes 1\(N-q)times 1end{bmatrix}}}

and accordingly μ and Σ are partitioned as follows

μ=[μ1μ2] with sizes [q×1(N−q)×1]{displaystyle {boldsymbol {mu }}={begin{bmatrix}{boldsymbol {mu }}_{1}\{boldsymbol {mu }}_{2}end{bmatrix}}{text{ with sizes }}{begin{bmatrix}qtimes 1\(N-q)times 1end{bmatrix}}}

Σ=[Σ11Σ12Σ21Σ22] with sizes [q×qq×(N−q)(N−q)×q(N−q)×(N−q)]{displaystyle {boldsymbol {Sigma }}={begin{bmatrix}{boldsymbol {Sigma }}_{11}&{boldsymbol {Sigma }}_{12}\{boldsymbol {Sigma }}_{21}&{boldsymbol {Sigma }}_{22}end{bmatrix}}{text{ with sizes }}{begin{bmatrix}qtimes q&qtimes (N-q)\(N-q)times q&(N-q)times (N-q)end{bmatrix}}}

then the distribution of x₁ conditional on x₂ = a is multivariate normal (x₁ | x₂ = a) ~ N(μ, Σ) where

μ¯=μ1+Σ12Σ22−1(a−μ2){displaystyle {bar {boldsymbol {mu }}}={boldsymbol {mu }}_{1}+{boldsymbol {Sigma }}_{12}{boldsymbol {Sigma }}_{22}^{-1}left(mathbf {a} -{boldsymbol {mu }}_{2}right)}

and covariance matrix

Σ¯=Σ11−Σ12Σ22−1Σ21.{displaystyle {overline {boldsymbol {Sigma }}}={boldsymbol {Sigma }}_{11}-{boldsymbol {Sigma }}_{12}{boldsymbol {Sigma }}_{22}^{-1}{boldsymbol {Sigma }}_{21}.}^[18]

This matrix is the Schur complement of Σ₂₂ in Σ. This means that to calculate the conditional covariance matrix, one inverts the overall covariance matrix, drops the rows and columns corresponding to the variables being conditioned upon, and then inverts back to get the conditional covariance matrix. Here Σ22−1{displaystyle {boldsymbol {Sigma }}_{22}^{-1}} is the generalized inverse of Σ22{displaystyle {boldsymbol {Sigma }}_{22}}.

Note that knowing that x₂ = a alters the variance, though the new variance does not depend on the specific value of a; perhaps more surprisingly, the mean is shifted by Σ12Σ22−1(a−μ2){displaystyle {boldsymbol {Sigma }}_{12}{boldsymbol {Sigma }}_{22}^{-1}left(mathbf {a} -{boldsymbol {mu }}_{2}right)}; compare this with the situation of not knowing the value of a, in which case x₁ would have distribution
Nq(μ1,Σ11){displaystyle {mathcal {N}}_{q}left({boldsymbol {mu }}_{1},{boldsymbol {Sigma }}_{11}right)}.

An interesting fact derived in order to prove this result, is that the random vectors x2{displaystyle mathbf {x} _{2}} and y1=x1−Σ12Σ22−1x2{displaystyle mathbf {y} _{1}=mathbf {x} _{1}-{boldsymbol {Sigma }}_{12}{boldsymbol {Sigma }}_{22}^{-1}mathbf {x} _{2}} are independent.

The matrix Σ₁₂Σ₂₂⁻¹ is known as the matrix of regression coefficients.

Bivariate case

In the bivariate case where x is partitioned into X₁ and X₂, the conditional distribution of X₁ given X₂ is^[19]

X1∣X2=x2 ∼ N(μ1+σ1σ2ρ(x2−μ2),(1−ρ2)σ12).{displaystyle X_{1}mid X_{2}=x_{2} sim {mathcal {N}}left(mu _{1}+{frac {sigma _{1}}{sigma _{2}}}rho (x_{2}-mu _{2}),,(1-rho ^{2})sigma _{1}^{2}right).}

where ρ{displaystyle rho } is the correlation coefficient between X₁ and X₂.

Bivariate conditional expectation

In the general case

(X1X2)∼N((μ1μ2),(σ12ρσ1σ2ρσ1σ2σ22)){displaystyle {begin{pmatrix}X_{1}\X_{2}end{pmatrix}}sim {mathcal {N}}left({begin{pmatrix}mu _{1}\mu _{2}end{pmatrix}},{begin{pmatrix}sigma _{1}^{2}&rho sigma _{1}sigma _{2}\rho sigma _{1}sigma _{2}&sigma _{2}^{2}end{pmatrix}}right)}

The conditional expectation of X₁ given X₂ is:

E⁡(X1∣X2=x2)=μ1+ρσ1σ2(x2−μ2){displaystyle operatorname {E} (X_{1}mid X_{2}=x_{2})=mu _{1}+rho {frac {sigma _{1}}{sigma _{2}}}(x_{2}-mu _{2})}

Proof: the result is obtained by taking the expectation of the conditional distribution X1∣X2{displaystyle X_{1}mid X_{2}} above.

In the centered case with unit variances

(X1X2)∼N((00),(1ρρ1)){displaystyle {begin{pmatrix}X_{1}\X_{2}end{pmatrix}}sim {mathcal {N}}left({begin{pmatrix}0\0end{pmatrix}},{begin{pmatrix}1&rho \rho &1end{pmatrix}}right)}

The conditional expectation of X₁ given X₂ is

E⁡(X1∣X2=x2)=ρx2{displaystyle operatorname {E} (X_{1}mid X_{2}=x_{2})=rho x_{2}}

and the conditional variance is

var⁡(X1∣X2=x2)=1−ρ2;{displaystyle operatorname {var} (X_{1}mid X_{2}=x_{2})=1-rho ^{2};}

thus the conditional variance does not depend on x₂.

The conditional expectation of X₁ given that X₂ is smaller/bigger than z is (Maddala 1983, p. 367^[20]) :

E⁡(X1∣X2<z)=−ρϕ(z)Φ(z),{displaystyle operatorname {E} (X_{1}mid X_{2}<z)=-rho {phi (z) over Phi (z)},}

E⁡(X1∣X2>z)=ρϕ(z)(1−Φ(z)),{displaystyle operatorname {E} (X_{1}mid X_{2}>z)=rho {phi (z) over (1-Phi (z))},}

where the final ratio here is called the inverse Mills ratio.

Proof: the last two results are obtained using the result E⁡(X1∣X2=x2)=ρx2{displaystyle operatorname {E} (X_{1}mid X_{2}=x_{2})=rho x_{2}}, so that

E⁡(X1∣X2<z)=ρE(X2∣X2<z){displaystyle operatorname {E} (X_{1}mid X_{2}<z)=rho E(X_{2}mid X_{2}<z)} and then using the properties of the expectation of a truncated normal distribution.

Marginal distributions

To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to marginalize out) from the mean vector and the covariance matrix. The proof for this follows from the definitions of multivariate normal distributions and linear algebra.^[21]

Example

Let X = [X₁, X₂, X₃] be multivariate normal random variables with mean vector μ = [μ₁, μ₂, μ₃] and covariance matrix Σ (standard parametrization for multivariate normal distributions). Then the joint distribution of X′ = [X₁, X₃] is multivariate normal with mean vector μ′ = [μ₁, μ₃] and covariance matrix
Σ′=[Σ11Σ13Σ31Σ33]{displaystyle {boldsymbol {Sigma }}'={begin{bmatrix}{boldsymbol {Sigma }}_{11}&{boldsymbol {Sigma }}_{13}\{boldsymbol {Sigma }}_{31}&{boldsymbol {Sigma }}_{33}end{bmatrix}}}.

Affine transformation

If Y = c + BX is an affine transformation of X ∼N(μ,Σ),{displaystyle mathbf {X} sim {mathcal {N}}({boldsymbol {mu }},{boldsymbol {Sigma }}),} where c is an M×1{displaystyle Mtimes 1} vector of constants and B is a constant M×N{displaystyle Mtimes N} matrix, then Y has a multivariate normal distribution with expected value c + Bμ and variance BΣB^T i.e., Y∼N(c+Bμ,BΣBT){displaystyle mathbf {Y} sim {mathcal {N}}left(mathbf {c} +mathbf {B} {boldsymbol {mu }},mathbf {B} {boldsymbol {Sigma }}mathbf {B} ^{rm {T}}right)}. In particular, any subset of the X_i has a marginal distribution that is also multivariate normal.
To see this, consider the following example: to extract the subset (X₁, X₂, X₄)^T, use

B=[10000…001000…000010…0]{displaystyle mathbf {B} ={begin{bmatrix}1&0&0&0&0&ldots &0\0&1&0&0&0&ldots &0\0&0&0&1&0&ldots &0end{bmatrix}}}

which extracts the desired elements directly.

Another corollary is that the distribution of Z = b · X, where b is a constant vector with the same number of elements as X and the dot indicates the dot product, is univariate Gaussian with Z∼N(b⋅μ,bTΣb){displaystyle Zsim {mathcal {N}}left(mathbf {b} cdot {boldsymbol {mu }},mathbf {b} ^{rm {T}}{boldsymbol {Sigma }}mathbf {b} right)}. This result follows by using

B=[b1b2…bn]=bT.{displaystyle mathbf {B} ={begin{bmatrix}b_{1}&b_{2}&ldots &b_{n}end{bmatrix}}=mathbf {b} ^{rm {T}}.}

Observe how the positive-definiteness of Σ implies that the variance of the dot product must be positive.

An affine transformation of X such as 2X is not the same as the sum of two independent realisations of X.

Geometric interpretation

The equidensity contours of a non-singular multivariate normal distribution are ellipsoids (i.e. linear transformations of hyperspheres) centered at the mean.^[22] Hence the multivariate normal distribution is an example of the class of elliptical distributions. The directions of the principal axes of the ellipsoids are given by the eigenvectors of the covariance matrix Σ{displaystyle {boldsymbol {Sigma }}}. The squared relative lengths of the principal axes are given by the corresponding eigenvalues.

If Σ = UΛU^T = UΛ^1/2(UΛ^1/2)^T is an eigendecomposition where the columns of U are unit eigenvectors and Λ is a diagonal matrix of the eigenvalues, then we have

X ∼N(μ,Σ)⟺X ∼μ+UΛ1/2N(0,I)⟺X ∼μ+UN(0,Λ).{displaystyle mathbf {X} sim {mathcal {N}}({boldsymbol {mu }},{boldsymbol {Sigma }})iff mathbf {X} sim {boldsymbol {mu }}+mathbf {U} {boldsymbol {Lambda }}^{1/2}{mathcal {N}}(0,mathbf {I} )iff mathbf {X} sim {boldsymbol {mu }}+mathbf {U} {mathcal {N}}(0,{boldsymbol {Lambda }}).}

Moreover, U can be chosen to be a rotation matrix, as inverting an axis does not have any effect on N(0, Λ), but inverting a column changes the sign of U's determinant. The distribution N(μ, Σ) is in effect N(0, I) scaled by Λ^1/2, rotated by U and translated by μ.

Conversely, any choice of μ, full rank matrix U, and positive diagonal entries Λ_i yields a non-singular multivariate normal distribution. If any Λ_i is zero and U is square, the resulting covariance matrix UΛU^T is singular. Geometrically this means that every contour ellipsoid is infinitely thin and has zero volume in n-dimensional space, as at least one of the principal axes has length of zero; this is the degenerate case.

"The radius around the true mean in a bivariate normal random variable, re-written in polar coordinates (radius and angle), follows a Hoyt distribution."^[23]

Estimation of parameters

The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is straightforward.

In short, the probability density function (pdf) of a multivariate normal is

f(x)=1(2π)k|Σ|exp⁡(−12(x−μ)TΣ−1(x−μ)){displaystyle f(mathbf {x} )={frac {1}{sqrt {(2pi )^{k}|{boldsymbol {Sigma }}|}}}exp left(-{1 over 2}(mathbf {x} -{boldsymbol {mu }})^{rm {T}}{boldsymbol {Sigma }}^{-1}({mathbf {x} }-{boldsymbol {mu }})right)}

and the ML estimator of the covariance matrix from a sample of n observations is

Σ^=1n∑i=1n(xi−x¯)(xi−x¯)T{displaystyle {widehat {boldsymbol {Sigma }}}={1 over n}sum _{i=1}^{n}({mathbf {x} }_{i}-{overline {mathbf {x} }})({mathbf {x} }_{i}-{overline {mathbf {x} }})^{T}}

which is simply the sample covariance matrix. This is a biased estimator whose expectation is

E[Σ^]=n−1nΣ.{displaystyle E[{widehat {boldsymbol {Sigma }}}]={frac {n-1}{n}}{boldsymbol {Sigma }}.}

An unbiased sample covariance is

Σ^=1n−1∑i=1n(xi−x¯)(xi−x¯)T.{displaystyle {widehat {boldsymbol {Sigma }}}={1 over n-1}sum _{i=1}^{n}(mathbf {x} _{i}-{overline {mathbf {x} }})(mathbf {x} _{i}-{overline {mathbf {x} }})^{rm {T}}.}

The Fisher information matrix for estimating the parameters of a multivariate normal distribution has a closed form expression. This can be used, for example, to compute the Cramér–Rao bound for parameter estimation in this setting. See Fisher information for more details.

Bayesian inference

In Bayesian statistics, the conjugate prior of the mean vector is another multivariate normal distribution, and the conjugate prior of the covariance matrix is an inverse-Wishart distribution W−1{displaystyle {mathcal {W}}^{-1}} . Suppose then that n observations have been made

X={x1,…,xn}∼N(μ,Σ){displaystyle mathbf {X} ={mathbf {x} _{1},dots ,mathbf {x} _{n}}sim {mathcal {N}}({boldsymbol {mu }},{boldsymbol {Sigma }})}

and that a conjugate prior has been assigned, where

p(μ,Σ)=p(μ∣Σ) p(Σ),{displaystyle p({boldsymbol {mu }},{boldsymbol {Sigma }})=p({boldsymbol {mu }}mid {boldsymbol {Sigma }}) p({boldsymbol {Sigma }}),}

where

p(μ∣Σ)∼N(μ0,m−1Σ),{displaystyle p({boldsymbol {mu }}mid {boldsymbol {Sigma }})sim {mathcal {N}}({boldsymbol {mu }}_{0},m^{-1}{boldsymbol {Sigma }}),}

and

p(Σ)∼W−1(Ψ,n0).{displaystyle p({boldsymbol {Sigma }})sim {mathcal {W}}^{-1}({boldsymbol {Psi }},n_{0}).}

Then,^{[citation needed]}

p(μ∣Σ,X)∼N(nx¯+mμ0n+m,1n+mΣ),p(Σ∣X)∼W−1(Ψ+nS+nmn+m(x¯−μ0)(x¯−μ0)′,n+n0),{displaystyle {begin{array}{rcl}p({boldsymbol {mu }}mid {boldsymbol {Sigma }},mathbf {X} )&sim &{mathcal {N}}left({frac {n{bar {mathbf {x} }}+m{boldsymbol {mu }}_{0}}{n+m}},{frac {1}{n+m}}{boldsymbol {Sigma }}right),\p({boldsymbol {Sigma }}mid mathbf {X} )&sim &{mathcal {W}}^{-1}left({boldsymbol {Psi }}+nmathbf {S} +{frac {nm}{n+m}}({bar {mathbf {x} }}-{boldsymbol {mu }}_{0})({bar {mathbf {x} }}-{boldsymbol {mu }}_{0})',n+n_{0}right),end{array}}}

where

x¯=n−1∑i=1nxi,S=n−1∑i=1n(xi−x¯)(xi−x¯)′.{displaystyle {begin{aligned}{bar {mathbf {x} }}&=n^{-1}sum _{i=1}^{n}mathbf {x} _{i},\mathbf {S} &=n^{-1}sum _{i=1}^{n}(mathbf {x} _{i}-{bar {mathbf {x} }})(mathbf {x} _{i}-{bar {mathbf {x} }})'.end{aligned}}}

Multivariate normality tests

Multivariate normality tests check a given set of data for similarity to the multivariate normal distribution. The null hypothesis is that the data set is similar to the normal distribution, therefore a sufficiently small p-value indicates non-normal data. Multivariate normality tests include the Cox–Small test^[24]
and Smith and Jain's adaptation^[25] of the Friedman–Rafsky test created by Larry Rafsky and Jerome Friedman.^[26]

Mardia's test^[27] is based on multivariate extensions of skewness and kurtosis measures. For a sample {x₁, ..., x_n} of k-dimensional vectors we compute

Σ^=1n∑j=1n(xj−x¯)(xj−x¯)TA=16n∑i=1n∑j=1n[(xi−x¯)TΣ^−1(xj−x¯)]3B=n8k(k+2){1n∑i=1n[(xi−x¯)TΣ^−1(xi−x¯)]2−k(k+2)}{displaystyle {begin{aligned}&{widehat {boldsymbol {Sigma }}}={1 over n}sum _{j=1}^{n}left(mathbf {x} _{j}-{bar {mathbf {x} }}right)left(mathbf {x} _{j}-{bar {mathbf {x} }}right)^{T}\&A={1 over 6n}sum _{i=1}^{n}sum _{j=1}^{n}left[(mathbf {x} _{i}-{bar {mathbf {x} }})^{T};{widehat {boldsymbol {Sigma }}}^{-1}(mathbf {x} _{j}-{bar {mathbf {x} }})right]^{3}\&B={sqrt {frac {n}{8k(k+2)}}}left{{1 over n}sum _{i=1}^{n}left[(mathbf {x} _{i}-{bar {mathbf {x} }})^{T};{widehat {boldsymbol {Sigma }}}^{-1}(mathbf {x} _{i}-{bar {mathbf {x} }})right]^{2}-k(k+2)right}end{aligned}}}

Under the null hypothesis of multivariate normality, the statistic A will have approximately a chi-squared distribution with 1/6⋅k(k + 1)(k + 2) degrees of freedom, and B will be approximately standard normal N(0,1).

Mardia's kurtosis statistic is skewed and converges very slowly to the limiting normal distribution. For medium size samples (50≤n<400){displaystyle (50leq n<400)}, the parameters of the asymptotic distribution of the kurtosis statistic are modified^[28] For small sample tests (n<50{displaystyle n<50}) empirical critical values are used. Tables of critical values for both statistics are given by Rencher^[29] for k = 2, 3, 4.

Mardia's tests are affine invariant but not consistent. For example, the multivariate skewness test is not consistent against
symmetric non-normal alternatives.^[30]

The BHEP test^[31] computes the norm of the difference between the empirical characteristic function and the theoretical characteristic function of the normal distribution. Calculation of the norm is performed in the L²(μ) space of square-integrable functions with respect to the Gaussian weighting function μβ(t)=(2πβ2)−k/2e−|t|2/(2β2){displaystyle scriptstyle mu _{beta }(mathbf {t} )=(2pi beta ^{2})^{-k/2}e^{-|mathbf {t} |^{2}/(2beta ^{2})}}. The test statistic is

Tβ=∫Rk|1n∑j=1neitTΣ^−1/2(xj−x)¯−e−|t|2/2|2μβ(t)dt=1n2∑i,j=1ne−β22(xi−xj)TΣ^−1(xi−xj)−2n(1+β2)k/2∑i=1ne−β22(1+β2)(xi−x¯)TΣ^−1(xi−x¯)+1(1+2β2)k/2{displaystyle {begin{aligned}T_{beta }&=int _{mathbb {R} ^{k}}left|{1 over n}sum _{j=1}^{n}e^{imathbf {t} ^{T}{widehat {boldsymbol {Sigma }}}^{-1/2}(mathbf {x} _{j}-{bar {mathbf {x} )}}}-e^{-|mathbf {t} |^{2}/2}right|^{2};{boldsymbol {mu }}_{beta }(mathbf {t} ),dmathbf {t} \&={1 over n^{2}}sum _{i,j=1}^{n}e^{-{beta ^{2} over 2}(mathbf {x} _{i}-mathbf {x} _{j})^{T}{widehat {boldsymbol {Sigma }}}^{-1}(mathbf {x} _{i}-mathbf {x} _{j})}-{frac {2}{n(1+beta ^{2})^{k/2}}}sum _{i=1}^{n}e^{-{frac {beta ^{2}}{2(1+beta ^{2})}}(mathbf {x} _{i}-{bar {mathbf {x} }})^{T}{widehat {boldsymbol {Sigma }}}^{-1}(mathbf {x} _{i}-{bar {mathbf {x} }})}+{frac {1}{(1+2beta ^{2})^{k/2}}}end{aligned}}}

The limiting distribution of this test statistic is a weighted sum of chi-squared random variables,^[31] however in practice it is more convenient to compute the sample quantiles using the Monte-Carlo simulations.^{[citation needed]}

A detailed survey of these and other test procedures is available.^[32]

Drawing values from the distribution

A widely used method for drawing (sampling) a random vector x from the N-dimensional multivariate normal distribution with mean vector μ and covariance matrix Σ works as follows:^[33]

1. Find any real matrix A such that A A^T = Σ. When Σ is positive-definite, the Cholesky decomposition is typically used, and the extended form of this decomposition can always be used (as the covariance matrix may be only positive semi-definite) in both cases a suitable matrix A is obtained. An alternative is to use the matrix A = UΛ^½ obtained from a spectral decomposition Σ = UΛU^-1 of Σ. The former approach is more computationally straightforward but the matrices A change for different orderings of the elements of the random vector, while the latter approach gives matrices that are related by simple re-orderings. In theory both approaches give equally good ways of determining a suitable matrix A, but there are differences in computation time.


2. Let z = (z₁, …, z_N)^T be a vector whose components are N independent standard normal variates (which can be generated, for example, by using the Box–Muller transform).


3. Let x be μ + Az. This has the desired distribution due to the affine transformation property.

See also

• 
  Chi distribution, the pdf of the 2-norm (or Euclidean norm) of a multivariate normally distributed vector (centered at zero).


• 
  Complex normal distribution, an application of bivariate normal distribution


• 
  Copula, for the definition of the Gaussian or normal copula model.


• 
  Multivariate t-distribution, which is another widely used spherically symmetric multivariate distribution.


• 
  Multivariate stable distribution extension of the multivariate normal distribution, when the index (exponent in the characteristic function) is between zero and two.


• Mahalanobis distance


• Wishart distribution


• Matrix normal distribution

References

Literature