Example: The Multivariate Normal distribution Recall the univariate normal distribution 2 1 1 2 2 x fx e the bivariate normal distribution 1 2 2 21 2 2 2 1, 21 xxxxxxyy xxyy xy fxy e The k-variate Normal distributionis given by: 1 1 2 1 /2 1/2 1, k 2 k fx x f e x x μ xμ where 1 2 k x x x x 1 2 k μ 11 12 1 12 22 2 12 k k kk kk Example: The.
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[edit]
The multivariate normal distribution of a k-dimensional random vector X=(X1,â¦,Xk){displaystyle mathbf {X} =(X_{1},ldots ,X_{k})} can be written in the following notation:
or to make it explicitly known that X is k-dimensional,
with k-dimensional mean vector
and kÃk{displaystyle ktimes k}covariance matrix
such that 1â¤i,jâ¤k.{displaystyle 1leq i,jleq k.} The inverse of the covariance matrix is called the precision matrix, denoted by Q=Σâ1{displaystyle {boldsymbol {Q}}={boldsymbol {Sigma }}^{-1}}.
Definitions[edit]Standard normal random vector[edit]
A real random vectorX=(X1,â¦,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 Xnâ¼N(0,1){displaystyle X_{n}sim {mathcal {N}}(0,1)} for all n{displaystyle n}.[1]:p. 454
Centered normal random vector[edit]
A real random vector X=(X1,â¦,Xk)T{displaystyle mathbf {X} =(X_{1},ldots ,X_{k})^{mathrm {T} }} is called a centered normal random vector if there exists a deterministic kÃâ{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 â{displaystyle ell } components.[1]:p. 454
Normal random vector[edit]
A real random vector X=(X1,â¦,Xk)T{displaystyle mathbf {X} =(X_{1},ldots ,X_{k})^{mathrm {T} }} is called a normal random vector if there exists a random â{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 kÃâ{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:
Xâ¼N(μ,Σ)âºthere exist μâRk,AâRkÃâ such that X=AZ+μ for Znâ¼N(0,1),i.i.d.{displaystyle mathbf {X} sim {mathcal {N}}(mathbf {mu } ,{boldsymbol {Sigma }})quad iff quad {text{there exist }}mathbf {mu } in mathbb {R} ^{k},{boldsymbol {A}}in mathbb {R} ^{ktimes ell }{text{ such that }}mathbf {X} ={boldsymbol {A}}mathbf {Z} +mathbf {mu } {text{ for }}Z_{n}sim {mathcal {N}}(0,1),{text{i.i.d.}}}
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[edit]
The following definitions are equivalent to the definition given above. A random vector X=(X1,â¦,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.
The spherical normal distribution can be characterised as the unique distribution where components are independent in any orthogonal coordinate system.[3][4]
Properties[edit]Density function[edit]
Bivariate normal joint density
Non-degenerate case[edit]
The multivariate normal distribution is said to be 'non-degenerate' when the symmetric covariance matrixΣ{displaystyle {boldsymbol {Sigma }}} is positive definite. In this case the distribution has density[5]
fX(x1,â¦,xk)=expâ¡(â12(xâμ)TΣâ1(xâμ))(2Ï)k|Σ|{displaystyle f_{mathbf {X} }(x_{1},ldots ,x_{k})={frac {exp left(-{frac {1}{2}}({mathbf {x} }-{boldsymbol {mu }})^{mathrm {T} }{boldsymbol {Sigma }}^{-1}({mathbf {x} }-{boldsymbol {mu }})right)}{sqrt {(2pi )^{k}|{boldsymbol {Sigma }}|}}}}
where x{displaystyle {mathbf {x} }} is a real k-dimensional column vector and |Σ|â¡detΣ{displaystyle |{boldsymbol {Sigma }}|equiv 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 1Ã1{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 locusâthe locus of points in k-dimensional space each of which gives the same particular value of the densityâis an ellipse or its higher-dimensional generalization; hence the multivariate normal is a special case of the elliptical distributions.
The descriptive statistic (xâμ)TΣâ1(xâμ){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[edit]
In the 2-dimensional nonsingular case (k = rank(Σ) = 2), the probability density function of a vector [XY]Ⲡis:
where Ï is the correlation between X and Y andwhere ÏX>0{displaystyle sigma _{X}>0} and ÏY>0{displaystyle sigma _{Y}>0}. In this case,
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]â² 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 Ï increases, these loci are squeezed toward the following line :
This is because this expression, with sgn(Ï) (where sgn is the Sign function) replaced by Ï, is the best linear unbiased prediction of Y given a value of X.[7]
Degenerate case[edit]
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 rankâ¡(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 rankâ¡(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:vâRk}{displaystyle {{boldsymbol {mu }}+{boldsymbol {Sigma ^{1/2}}}mathbf {v} :mathbf {v} in mathbb {R} ^{k}}}. With respect to this measure the distribution has density:
where Σ+{displaystyle {boldsymbol {Sigma }}^{+}} is the generalized inverse and det* is the pseudo-determinant.[8]
Higher moments[edit]
The kth-order moments of x are given by
where r1 + r2 + ⯠+ rN = k.
The kth-order central moments are as follows
where the sum is taken over all allocations of the set {1,â¦,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):
This yields (2λâ1)!2λâ1(λâ1)!{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 à 5 = 15 terms, and for eighth-order moments there are 3 à 5 à 7 = 105 terms.
The covariances are then determined by replacing the terms of the list [1,â¦,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:
where Ïij{displaystyle sigma _{ij}} is the covariance of Xi and Xj. 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 Eâ¡[Xi2XkXn]{displaystyle operatorname {E} [X_{i}^{2}X_{k}X_{n}]}, one lets Xi = Xj and one uses the fact that Ïii=Ïi2{displaystyle sigma _{ii}=sigma _{i}^{2}}.
Likelihood function[edit]
If the mean and variance matrix are known, a suitable log likelihood function for a single observation x is
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
i.e. with the conjugate transpose (indicated by â {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[edit]
The differential entropy of the multivariate normal distribution is[10]
where the bars denote the matrix determinant and k is the dimensionality of the vector space.
KullbackâLeibler divergence[edit]
The KullbackâLeibler 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]
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 loge 2 yields the divergence in bits.
When μ1=μ0{displaystyle {boldsymbol {mu }}_{1}={boldsymbol {mu }}_{0}},
Mutual information[edit]
The mutual information of a distribution is a special case of the KullbackâLeibler 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 KullbackâLeibler 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:
where Ï0{displaystyle {boldsymbol {rho }}_{0}} is the correlation matrix constructed from Σ0{displaystyle {boldsymbol {Sigma }}_{0}}.[citation needed]
In the bivariate case the expression for the mutual information is:
Cumulative distribution function[edit]
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]
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 distancer{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[edit]
The interval for the multivariate normal distribution yields a region consisting of those vectors x satisfying
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 Ïk2(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)[edit]
The complementary cumulative distribution function (ccdf) or the tail distribution is defined as F¯(x)=1âP(Xâ¤x){displaystyle {overline {F}}(mathbf {x} )=1-mathbb {P} (mathbf {X} leq mathbf {x} )}. When Xâ¼N(μ,Σ){displaystyle mathbf {X} sim {mathcal {N}}({boldsymbol {mu }},{boldsymbol {Sigma }})}, thenthe ccdf can be written as a probability the maximum of dependent Gaussian variables:[16]
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[edit]Normally distributed and independent[edit]
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, Ï=0{displaystyle rho =0} ).
Two normally distributed random variables need not be jointly bivariate normal[edit]
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=âX{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[edit]
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[edit]
If N-dimensional x is partitioned as follows
and accordingly μ and Σ are partitioned as follows
then the distribution of x1 conditional on x2 = a is multivariate normal (x1 | x2 = a) ~ N(μ, Σ) where
and covariance matrix
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 Σ22â1{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 Σ12Σ22â1(aâμ2){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 distributionNq(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=x1âΣ12Σ22â1x2{displaystyle mathbf {y} _{1}=mathbf {x} _{1}-{boldsymbol {Sigma }}_{12}{boldsymbol {Sigma }}_{22}^{-1}mathbf {x} _{2}} are independent.
The matrix Σ12Σ22â1 is known as the matrix of regression coefficients.
Bivariate case[edit]
In the bivariate case where x is partitioned into X1 and X2, the conditional distribution of X1 given X2 is[19]
where Ï{displaystyle rho } is the correlation coefficient between X1 and X2.
Bivariate conditional expectation[edit]In the general case[edit]
The conditional expectation of X1 given X2 is:
Proof: the result is obtained by taking the expectation of the conditional distribution X1â£X2{displaystyle X_{1}mid X_{2}} above.
In the centered case with unit variances[edit]
The conditional expectation of X1 given X2 is
and the conditional variance is
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]) :
where the final ratio here is called the inverse Mills ratio.
Proof: the last two results are obtained using the result Eâ¡(X1â£X2=x2)=Ïx2{displaystyle operatorname {E} (X_{1}mid X_{2}=x_{2})=rho x_{2}}, so that
Marginal distributions[edit]
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 Xâ² = [X1, X3] is multivariate normal with mean vector μⲠ= [μ1, μ3] and covariance matrixΣâ²=[Σ11Σ13Σ31Σ33]{displaystyle {boldsymbol {Sigma }}'={begin{bmatrix}{boldsymbol {Sigma }}_{11}&{boldsymbol {Sigma }}_{13}{boldsymbol {Sigma }}_{31}&{boldsymbol {Sigma }}_{33}end{bmatrix}}}.
Affine transformation[edit]
If Y = c + BX is an affine transformation of Xâ¼N(μ,Σ),{displaystyle mathbf {X} sim {mathcal {N}}({boldsymbol {mu }},{boldsymbol {Sigma }}),} where c is an MÃ1{displaystyle Mtimes 1} vector of constants and B is a constant MÃN{displaystyle Mtimes N} matrix, then Y has a multivariate normal distribution with expected value c + Bμ and variance BΣBT i.e., Yâ¼N(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 Xi 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
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 Zâ¼N(bâ
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
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[edit]
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 = UÎUT = UÎ1/2(UÎ1/2)T is an eigendecomposition where the columns of U are unit eigenvectors and Î is a diagonal matrix of the eigenvalues, then we have
Moreover, U can be chosen to be a rotation matrix, as inverting an axis does not have any effect on N(0, Î), but inverting a column changes the sign of U's determinant. The distribution N(μ, Σ) is in effect N(0, I) scaled by Î1/2, rotated by U and translated by μ.
Conversely, any choice of μ, full rank matrix U, and positive diagonal entries Îi yields a non-singular multivariate normal distribution. If any Îi is zero and U is square, the resulting covariance matrix UÎUT 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[edit]
The derivation of the maximum-likelihoodestimator of the covariance matrix of a multivariate normal distribution is straightforward.
In short, the probability density function (pdf) of a multivariate normal is
and the ML estimator of the covariance matrix from a sample of n observations is
Bivariate Normal Distribution Density In R
which is simply the sample covariance matrix. This is a biased estimator whose expectation is
An unbiased sample covariance is
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 CramérâRao bound for parameter estimation in this setting. See Fisher information for more details.
Bayesian inference[edit]
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 distributionWâ1{displaystyle {mathcal {W}}^{-1}} . Suppose then that n observations have been made
and that a conjugate prior has been assigned, where
where
and
Then,[citation needed]
where
Multivariate normality tests[edit]
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 CoxâSmall test[24]and Smith and Jain's adaptation[25] of the FriedmanâRafsky 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, .., xn} of k-dimensional vectors we compute
Under the null hypothesis of multivariate normality, the statistic A will have approximately a chi-squared distribution with 1/6â
k(k + 1)(k + 2) degrees of freedom, and B will be approximately standard normalN(0,1).
Mardia's kurtosis statistic is skewed and converges very slowly to the limiting normal distribution. For medium size samples (50â¤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 againstsymmetric 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)=(2Ïβ2)âk/2eâ|t|2/(2β2){displaystyle scriptstyle mu _{beta }(mathbf {t} )=(2pi beta ^{2})^{-k/2}e^{-|mathbf {t} |^{2}/(2beta ^{2})}}. The test statistic is
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[edit]
A widely used method for drawing (sampling) a random vector x from the N-dimensional multivariate normal distribution with mean vector μ and covariance matrixΣ works as follows:[33]Amar bangla software free download for pc.
See also[edit]
References[edit]
Standard Bivariate Normal Distribution PdfLiterature[edit]
Bivariate Normal Distribution Pdf Python
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Multivariate_normal_distribution&oldid=916889037'
Comments are closed.
|
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |