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Statistical reasoning in dependent pgeneralized elliptically contoured distributions and beyond
 WolfDieter Richter^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s4048801700743
© The Author(s) 2017
 Received: 27 December 2016
 Accepted: 4 August 2017
 Published: 20 September 2017
Abstract
First, likelihood ratio statistics for checking the hypothesis of equal variances of twodimensional Gaussian vectors are derived both under the standard \(\left (\sigma ^{2}_{1},\sigma ^{2}_{2},\varrho \right)\)parametrization and under the geometric (a,b,α)parametrization where a ^{2} and b ^{2} are the variances of the principle components and α is an angle of rotation. Then, the likelihood ratio statistics for checking the hypothesis of equal scaling parameters of principle components of ppower exponentially distributed twodimensional vectors are considered both under independence and under rotational or correlation type dependence. Moreover, the role semiinner products play when establishing various likelihood equations is demonstrated. Finally, the dependent pgeneralized polar method and the dependent pgeneralized rejectionacceptance method for simulating starshaped distributed vectors are presented.
Keywords
 Modeling with correlation and variances of Euclidean coordinates
 Modeling with rotation and variances of principle components
 Geometric parametrization
 Likelihood ratio
 pgeneralized Fisher distribution
 Semiinner product
 Dependent pgeneralized polar method
 Dependent pgeneralized rejectionacceptance simulation
 Starshaped distributions
PACS
 02.50.r
 02.50.Ng
 02.70.Rr
 02.70.Uv
Mathematics Subject Classification
 62F03
 62F10
 62H15
 62E10
Introduction
One of the classical statistical problems deals with comparing variances. Results for quite arbitrary pairs of dependent random variables are due to Morgan (1939), Pitman (1939), Tiku and Balakrishnan (1986), McCulloch (1987), Wilcox (1990) and Mudholkar et al. (2003) and are recently reviewed in Wilcox (2015). For a survey of various practical applications in Snedecor and Cochran (1967), Lord and Novick (1968), Games et al. (1972), Levy (1976) and Rothstein et al. (1981), see again Wilcox (2015). A closely related but nevertheless rather different study in Richter (2016) compares scaling parameters of twodimensional axesaligned pgeneralized elliptically contoured distributions. Such distributions show independence if the density generating function is that of the pgeneralized Gaussian law and l _{ p }dependence if the density generating function is of another type, but they show no rotational or correlation type dependence. The present paper is aimed now to study correlation type dependence modeling within the family of ppower exponential distribution laws. It is well known that two jointly Gaussian distributed random variables are independent if and only if they are uncorrelated. The density level sets of such a vector are axesaligned ellipses. If the components of a twodimensional Gaussian vector are not independent then the vector may be constructed by rotating through its distribution center an axesaligned elliptically contoured distributed Gaussian vector that has heteroscedastic components. The correlation coefficient may be expressed in such situation in terms of the angle of rotation and the ratio of variances, see Dietrich et al. (2013). This type of dependence between two random variables is called here a rotational or correlation type dependence. Basic facts on modeling twodimensional Gaussian vectors with correlation and variances of Euclidean coordinates on the one hand and with rotation and variances of principle components on the other hand will be summarized in the presented paper. Considering these two models side by side demonstrates different aspects of ’standard’ modeling with the more stochastically interpretable parameters and of ’flexible’ modeling with the more geometrically motivated parameters.
It is outlined in Wilcox (2015) that “seemingly the bestknown technique for testing \(H_{0}: \sigma _{1}^{2}=\sigma _{2}^{2}\) is a method derived by Morgan (1939) and Pitman (1939). Letting U=X+Y and V=X−Y, if the null hypothesis is true, then ϱ _{ UV }, Pearsons correlation between U and V, is zero. So testing H _{0} can be accomplished by testing ϱ _{ UV }=0.” We do not consider here the test problem in the same full generality as in Wilcox (2015) where the joint distribution of X and Y is not basically restricted to belong to the families of pgeneralized elliptically contoured or starshaped distributions. While it is proved in Wilcox (2015) that certain heteroscedastic consistent estimators perform well in certain cases of heavy tailed distributions, here we use case sensitive estimators depending on the given value of the shapetail parameter p, see Sections 3 and 4 and recognize the consequences drawn in Section 5. Note that cases of heavier and lighter than Gaussian distribution tails are observed here in dependence of whether p∈(0,2) or p>2, respectively. A study demonstrating far and narrow tail effects when sampling from those distribution classes can be seen in Richter (2015a).
In the case of a twodimensional Gaussian distribution Φ _{ μ,Σ }, it turns out that the class of distributions satisfying H _{0} is the union of the following two subsets. The elements of the first one are the spherical Gaussian distributions and the second one contains all elliptically contoured Gaussian distributions having the lines y=+(−)x as the main axes of their density level ellipses. Any number from the interval (−1,1) is attained by the correlation coefficient of a suitably chosen element from the latter subset. Thus, H _{0} covers two quite different cases of correlation and uncorrelation. We modify here the null hypothesis in a way that one of these two subsets is not included.
One of the likelihood equations needed to be solved for constructing the likelihood ratio statistic for testing the just mentioned modified hypothesis is formulated here on using a socalled semiinner product in the sample space. This rises the question whether this analytical tool plays also a role in estimating location. We give a positive answer to this question in the case of axesaligned ppower exponential distributions.
The paper is structured as follows. Gaussian correlation models and likelihood ratio tests for checking equality of variances of two dependent random variables are studied in Section 2. The content of this section is of some interest of its own although it might be partly known to the reader. Testing equality of scaling parameters of axesaligned ppower exponential distributions is dealt with in Section 3. The more general results are presented in Sections 46. Section 4 deals with testing equality of scaling parameters of principal components of general, i.e. arbitrarily rotated, pgeneralized elliptically contoured ppower exponential distributions. Derivations are omitted in the sections on Gaussian and axesaligned pgeneralized elliptically contoured distributions. They can be considered being standard and follow also from proving the more general results in Section 4. Throughout Sections 24, we restrict our consideration to the case of known expectations. Practical examples of this type are given in Richter (2016). Section 5 gives a new geometricanalytical insight into estimating the location parameter of the ppower exponential, or pgeneralized Gaussian or Laplace, law using semiinner products in the sample space. Differently from the situation of statistics in Gaussian sample distributions, many statistical questions in pgeneralized Gaussian and more general starshaped sample distributions cannot yet fully be answered in a theoretical way. For intermittent empirical studies, and much beyond it, methods for simulating such distributions are needed. Generalizing the methods in Kalke and Richter (2013), Section 6 presents corresponding direct and acceptancerejection methods and indicates how to extend to the dependent pgeneralized multivariate case the classical and the rejecting polar methods in Box and Muller (1958) and Marsaglia and Bray (1964), respectively.
Likelihood ratio tests for scaling parameters in twodimensional Gaussian distributions
Testing equality of scaling parameters can be interpreted in Gaussian models at least in two different ways. We deal here with equality of variances of the marginal variables or Euclidean coordinates if the Gaussian density is given in the classical \(\left (\sigma ^{2}_{1},\sigma ^{2}_{2},\varrho \right)\) variancescorrelation parametrization, and with equality of variances of principal components if the Gaussian density is given in the geometric (a,b,α)parametrization from Dietrich et al. (2013) where a ^{2} and b ^{2} are the variances of the principle components and α is an angle of rotation.
2.1 The common \(\left (\sigma ^{2}_{1},\sigma ^{2}_{2},\varrho \right)\)parametrization
2.1.1 The case of an unknown correlation coefficient
Here, \({\Sigma _{x}^{2}}/{\Sigma _{y}^{2}}\) is the ratio of two dependent Chisquared distributed random variables. The distributions of all statistics considered here and in later sections may be simulated using the methods presented in Section 6. Alternatively, the geometric measure representation in Richter (2014) may be used to establish the exact distributions of several of these statistics, or at least to derive suitable approximations.
2.1.2 The case of a known correlation coefficient
2.2 The geometric (a,b,α)parametrization
2.2.1 The case of an unknown α
If H _{0} is true then the correlation or rotational dependence of the Euclidean coordinates is zero. Correspondingly, no restricted under H _{0} estimator of the angle of rotation α has any effect onto the statistic Q ^{∗}.
2.2.2 The case of a known α
The plugin version of this statistic where, for unknown \(\alpha, \alpha =\hat \alpha =\text {mle}\left (\alpha \right) \), is just the statistic from the previous section. Differently from this situation, the likelihood ratio statistic in Section 2.1.1 using both the unrestricted and the restricted maximum likelihood estimators of α, ist not such an immediate plugin version of the statistic considered in Section 2.1.2.
Likelihood ratio test for scaling parameters in axesaligned pgeneralized elliptically contoured distributions
Simulating quantiles F _{30,30,q }(2),q=0.05,q=0.95
Simulation sample size N:  200  400  800  2000  F _{30,30,q }(2) 

5percentile  0.541  0.523  0.546  0.545  0.543 
95percentile  1.933  1.984  1.864  1.858  1.841=1/0.543 
Tests for equal scaling parameters in correlational dependent pgeneralized elliptically contoured distributions
This section is aimed to generalize the results presented in Section 3 for the case that two random variables may be rotational or correlation type dependent. To this end, we start in Section 4.1 with a pgeneralization of the (a,b,α)representation of the Gaussian law. Section 4.2 is aimed to give a geometric explanation of correlation in the particular case of a twodimensional Gaussian distribution. Roughly spoken, correlation is interpreted by rotation under heteroscedasticity. Section 4.3 presents a test for checking homoscedasticity of principal components.
4.1 The geometric ((a,b),p,α)parametrization
4.2 Geometry of variance homogeneity

either \(\alpha \in \left \{\frac {\pi }{4}, \frac {3\pi }{4}\right \}\) with arbitrary a,b

or \(\alpha \notin \left \{\frac {\pi }{4}, \frac {3\pi }{4}\right \}\) and a=b.
Thus, if σ _{1}=σ _{2}=σ, say, then \(\Psi =2\sigma ^{2}\left (\begin {array}{cc} 1+\varrho & 0 \\ 0 & 1\varrho \\ \end {array} \right)\).
4.3 Testing homoscedasticity of principal components
Example 1
Remark Since the purpose is to test whether H _{0} or not, when there is a correlation between two groups, one might like to consider testing the significance of correlation structure prior to testing H _{0} or not. In the present situation where is no rotational correlation, this would mean to test whether the shapescale parameter satisfies \(\tilde H_{0}: p=2\) or not. Searching the literature the author was not aware of a significance test for this hypothesis, see for example in GonzálezFarías et al. (2009), Yu et al. (2012), Purczynski and BednarzOkrzynska (2014) and Pascal et al. (2017).
The semiinner product [.,.]_{ p } appears also in estimating location
Example 2
We consider now two cases excluded so far.
Example 3
Example 4
Simulation of starshaped distributed random vectors
6.1 Preliminary remarks
For the distribution considered in Section 4.1, K can be chosen as a rotated through the origin axesaligned pgeneralized ellipsoid, \(K=D^{T}(\alpha)B_{(a_{1},a_{2}),p}\), and O _{ S } means the corresponding stargeneralized surface content measure.
6.2 Dependent pgeneralized acceptancerejection method
Proof
Example 5
For simulating a random vector following a pgeneralized elliptically contoured distribution law, put \( A=O B_{a,p}\ \text {with}\ B_{a,p}=\left \{x\in {\mathcal {R}}^{d}: \sum \limits _{i=1}^{d}\frac {x_{i}}{a_{i}}^{p}\leq 1\right \},\ \text {and}\ C_{i} \geq a_{i}>0, i=1,\ldots,d, a=(a_{1},\ldots,a_{d})^{T}.\)
Example 6
For simulating norm or antinorm contoured distributed vectors one can chose the acceptance region \(A= \left \{x\in {\mathcal {R}}^{d}: x\leq 1\right \}\), where . is an arbitrary norm or antinorm and the constants C _{ i }>0 are chosen such that A⊂O[0_{(d)},C _{(d)}].
Example 7
If A=P is a starshaped polyhedron having the origin as an interior point, one can check whether a point belongs to A using the various representations of the Minkowski functional of A given in Richter and Schicker ( 2016b ).
Example 8
If A is as described inNolan (2016), check the condition given there.
Step 3. It is well known that if A is a starshaped subset of \({\mathcal {R}}^{d}\) having the origin as an interior point then the Minkowski functional \(h_{A}(x)=\inf \left \{\lambda >0: x\in \lambda A\right \}, x\in {\mathcal {R}}^{d}\) is well defined. A normalization of the stopping element based upon this functional is used in the following lemma.
Step 4.Lemma 2 The random element \(\phantom {\dot {i}\!}X_{\partial A}=\xi _{\tau ^{A}}/h_{A}(\xi _{\tau ^{A}})\) follows the stargeneralized uniform distribution on \(S=\partial A=\left \{x\in {\mathcal {R}}^{d}: h_{A}(x)= 1\right \}\), X _{ ∂ A }∼ω _{ S } for short.
Proof
Example 5
Example 6
Remark 1
Let NonNegSim denote the set of all nonnegative random variables for which there is known a simulation method. Extensive overviews of simulation algorithms for nonuniform random variables are given in Rubinstein (1981) and Devroye (1986). If R∈NonNegSim is independent of X _{ ∂ A } where X _{ ∂ A } is a stargeneralized uniformly on the star sphere ∂ A distributed random vector then the random vector R X _{ ∂ A } follows a starshaped distribution centered at the origin, Φ _{ A } say.
As to summarize, Steps 14 together constitute an acceptancerejection algorithm for simulating random vectors following a starshaped distribution law.
Example 9
Remark 2
6.3 Dependent pgeneralized polar method
cf. Definition 4 in the same paper.
Step 1 Start the algorithm by generating a nonnegative random number R according to the density h _{0}.Step 2 Generate random numbers Φ _{1},…,Φ _{ d−2} from [0,π) and Φ _{ d−1} from [0,2π) following the densities h _{1},…,h _{ d−2} and h _{ d−1}, respectively.Step 3 Carry out transformation (7).Step 4 Return Y=O R(U _{1},…,U _{ d })^{ T }+ν.
This algorithm generates a random vector Y following the pgeneralized elliptically contoured distribution law Φ _{ g,a,p,ν,O }, see Theorem 4 and Remark 11 in Richter (2014). The particular case d=2,a _{1}=a _{2}=1 has been dealt with in Kalke and Richter (2013). Finally, we notice that \(X\sim \Phi _{g,a,p,0_{d},I_{d}}=\Phi _{g,B_{a,p},0_{d}}\).
Discussion
Comparing mvvc with pcvr models led to some new aspects in testing equality of variances or scaling parameters. Effects of rotational dependence are outlined. A new geometric interpretation of certain likelihood equations is given in terms of a semiinner product. Based upon the present results for the more specific models dealt with in Sections 25, it could be of some interest to reconsider in the future the more general model in Wilcox (2015) and to possibly draw some new conclusions for this model. Our results might further stimulate a comparison of simulation methods, e.g. for particular cases being in the intersection of the work in Nolan (2016) and in Richter and Schicker (2016a, b). To this end, one would particularly have to determine the Minkowski functionals of the sets considered in Nolan (2016) and then to compare the approximative simulation method there with the exact method presented in Richter and Schicker (2016a,b). Challenging questions are opened for deriving new exact statistical distributions, e.g. of Σ _{ X }/Σ _{ Y }, from dependent sample distributions, and to compare these results with corresponding simulation results. As another open problem it remains to combine rotational and l _{ p }dependence. Consequences the latter notion has for the derivation of exact distributions of certain statistics have been studied in Müller and Richter (2015, 2016a, b). There, the effects caused by the deviation of a density generating function from that of the ppower exponential law are studied in various situations.
Declarations
Acknowledgements
The author is grateful to the Associate Editor for his valuable comments leading to an improvement of the paper.
Competing interests
The author declares that he has no competing interest.
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