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Statistical reasoning in dependent pgeneralized elliptically contoured distributions and beyond
Journal of Statistical Distributions and Applications volume 4, Article number: 21 (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.
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
In this section, we consider the marginal variables variancescorrelation (mvvc) model. Likelihood ratio tests with respect to the equality of two variances will be given separately for the cases of a known and an unknown correlation coefficient. Let (X _{ i },Y _{ i })^{T},i=1,…,n be independent Gaussian random vectors following the density φ _{ μ,Σ }(.,.)=φ(.,.σ _{1},σ _{2},ϱ) where μ=(μ _{1},μ _{2})^{T} is a vector from \(\mathbb R^{2}\) and \(\Sigma =\left (\begin {array}{cc} \sigma _{1}^{2} & \varrho \sigma _{1}\sigma _{2} \\ \varrho \sigma _{1}\sigma _{2} & \sigma _{2}^{2} \\ \end {array} \right)\)is a positive definite matrix, and let (x _{ i },y _{ i })^{T},i=1,…,n be a corresponding concrete sample. We introduce the likelihood function
and its restriction to the case of equal variances, \(\tilde L\left (\sigma, \varrho \right)=L\left (\sigma, \sigma, \varrho \right)\).
2.1.1 The case of an unknown correlation coefficient
We intend now to decide between the two hypotheses
using the likelihood ratio test statistic \(Q=\tilde L\left (\tilde \sigma,\tilde \varrho \right)/L\left (\hat \sigma _{1},\hat \sigma _{2},\hat \varrho \right)\) where \(\left (\hat \sigma _{1},\hat \sigma _{2},\hat \varrho \right)=mle\left (\sigma _{1},\sigma _{2},\varrho \right)\) and \(\left (\tilde \sigma,\tilde \varrho \right)=mle_{H_{0}}\left (\sigma,\varrho \right)\) are maximum likelihood and restricted to H _{0} such estimators, respectively. Standard calculations show that Q allows the representation
where
Let α∈(0,1). According to the likelihood ratio rule, H _{0} will be rejected if Q<t _{ α } where t _{ α } is chosen from the interval (0,1) in a way such that
A restatement of this size αtest is based upon the following alternative representation of the likelihood ratio,
wher
an
Rewording the corresponding likelihood ratio decision rule, it is then a size (α _{1}+α _{2})test to reject H _{0} if
where λ _{ q },q∈(0,1), is suitably chosen from (0,∞) such that
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
Let ϱ=ϱ _{0} be a known number and put L(σ _{1},σ _{2})=L(σ _{1},σ _{2},ϱ _{0}) and \(\tilde L\left (\sigma \right)=L\left (\sigma,\sigma,\varrho _{0}\right)\). The likelihood ratio \(Q=\sup \limits _{\sigma } \tilde L(\sigma)/\sup \limits _{\sigma _{1},\sigma _{2}} L\left (\sigma _{1},\sigma _{2}\right)\) allows the representation
Let α∈(0,1). The likelihood ratio decision rule leads to rejecting H _{0} if Q<t _{ α },α∈(0,1) satisfies \(P\left (Q<t_{\alpha }\right)_{H_{0}}=\alpha \) or, equivalently, if Q ^{1/n}/2<z _{ α } for a suitably chosen z _{ α }=z(t _{ α })∈(0,1/2), that is, if
2.2 The geometric (a,b,α)parametrization
We consider now the principal components variancesrotation (pcvr) model. For simplicity, we assume that μ _{1}=μ _{2}=0. The geometric parametrization of the Gaussian density is then
see Dietrich et al. (2013). Here,
and
with \( \gamma =\frac {1}{2}\arctan \left (2\varrho \sigma _{1}\sigma _{2}/\left (\sigma _{1}^{2}\sigma _{2}^{2}\right)\right) \). We put arctan(+(−)∞)=+(−)π/2 and remark that a ^{2} and b ^{2} are the variances of principal components of the related Gaussian random vector. The Euclidean coordinates of such a vector are correlated if ϱ≠0 and may then also be called rotational dependent because then α≠0.
For testing equality of variances of principle components
we introduce the likelihood function
and its restriction to \(H_{0}, \tilde L^{*}\left (a,\alpha \right)= L^{*}\left (a, a, \alpha \right). \)
2.2.1 The case of an unknown α
The likelihood ratio statistic \(Q^{*}=\max \limits _{a,\alpha }\tilde L^{*}\left (a,\alpha \right)_{H_{0}}/\max \limits _{a,b,\alpha } L^{*}\left (a,b,\alpha \right)\) in case α is to be estimated, allows the representation
where
and the maximum likelihood estimator \(\hat \alpha = \text {mle}\left (\alpha \right)\) is
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
If the angle of rotation α is known, the likelihood ratio allows the representation
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
The present section is aimed to shortly summarize some results from the axesaligned or independence case. To start with, we recall that the univariate ppower exponential distribution has the density
which is also called pgeneralized Gaussian or Laplace density, p>0. The parameter p controls both the shape of the density and the tail behaviour of the distribution and may therefore be called a shapetail parameter. Note that C _{ p }=p ^{1−1/p}/(2Γ(1/p)) and the first and second order moments of a correspondingly distributed random variable X are
Moreover, such random variable X allows the stochastic representation
where X _{0} follows the standard ppower exponential density, i.e. X _{0}∼f _{ p }(.;0,1). Because of this representation, σ is called a scaling parameter. Note that EX−μ^{p}=σ ^{p}. Two independent such variables follow the joint product density
having the distribution center (μ _{1},μ _{2})^{T}∈R ^{2} and whose level sets are axesaligned pgeneralized ellipses. Note that the axesaligned pgeneralized elliptically contoured ppower exponential densities introduced this way should not be confused with functions of the type f _{(X,Y)}(x,y)=C exp{−Q(x,y)^{p/2}} with Q being a quadratic form. The latter type of densities has been considered in Kuwana and Kariya (1991), Gómez et al. (1998), GómezVillegas et al. (2011) and Dang et al. (2015) and may also be called elliptically contoured ppower exponential densities. The corresponding type of distributions may be considered as a particular Kotz type distribution within the broad family of elliptically contoured distributions, see Fang et al. (1990) and Nadarajah (2003). Testing
in the model of the present section means checking equality of scaling parameters. Let (X _{ i },Y _{ i }),i=1,…,n be independent random vectors following the density f(.,.σ _{1},σ _{2}) and put X _{(n)}=(X _{1},…,X _{ n })^{T}, Y _{(n)}=(Y _{1},…,Y _{ n })^{T}. We still assume that the expectations μ _{1} and μ _{2} are known. In case of a true hypothesis H _{0}, the test statistic
follows the pgeneralized Fisher distribution with (n,n) d.f., \( T_{H_{0}}\,\sim \, F_{n,n}(p). \) The latter distribution was derived in Richter (2009). It can be considered as the distribution of the ratio of independent pgeneralized Chisquared distributed variables that were introduced in Richter (2007). The density of the pgeneralized Fisher distribution with (n,n) degrees of freedom is according to Richter (2009)
see Fig. 1. With the notation
the statistic T can alternatively be represented as
The decision rule according to which one rejects H _{0} if \(T<F_{n,n,\alpha _{2}}(p)\) or \(T>F_{n,n,1\alpha _{1}}(p)\) performs an exact size (α _{1}+α _{2})test. This test turns out to be the corresponding likelihood ratio test. Figures 2, 3 and 4 deal with the performance of this test showing histograms of simulation results for the test statistic T, under H _{0}. To this end, a random vector (X,Y)^{T} following the distribution \(\Phi _{(1,1),p,(0,0),I_{2}}\) was simulated n×Ntimes, and the value of the statistic T was calculated Ntimes based upon this sample. The choices of the values of n and p allows direct comparisons with Fig. 1.
Figure 5 shows the influence an increasing simulation sample size N has onto the accuracy of the estimation of the density of the test statistic if the null hypothesis is true. In the case n=30,p=2 and for four different values of the simulation sample size N, Table 1 presents the correspondingly calculated percentiles of orders 5 and 95, respectively, and the exact Fisher quantiles F _{30,30,q }=F _{30,30,q }(2),q∈{0.05, 0.95}.
The likelihood ratio test can be equivalently reformulated as to reject H _{0} if, for a suitably chosen c, the likelihood ratio Q satisfies Q<c. Let
and \(\tilde L(\sigma)=L(\sigma,\sigma)\), and denote unrestricted and restricted under H _{0} mle’s of σ _{1},σ _{2} and σ _{1}(=σ _{2}=σ, say) by \(\hat \sigma _{1}, \hat \sigma _{2}\) and \(\tilde \sigma \), respectively. The likelihood ratio statistic \( Q= {\tilde L(\tilde \sigma)}/{L(\widehat \sigma _{1}, \widehat \sigma _{2})}\) allows the representations
According to the general geometric measuretheoretical methodology of investigation in Richter (2014) and papers referred to there, the restricted distribution function of T if H _{0} is true is
where
is a cone with vertex in 0∈R ^{2n} and z_{ p }=z_{(1,1),p }. A geometric measure representation of the standardized ppower exponential law applies to show that this distribution is the pgeneralized Fisher or F _{ n,n }(p)distribution. As mentioned before, this method may also be used to derive exact distributions of other statistics. Differently from what was considered so far, throughout the following section the vectors (X _{ i },Y _{ i })^{T} are allowed to be rotated through (μ _{1},μ _{2})^{T}, in other words, the variables X _{ i },Y _{ i } are allowed to be rotational dependent or correlation type dependent, i=1,2,….
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
Let a random vector follow a rotational dependent pgeneralized elliptically contoured ppower exponential distribution, \((X,Y)^{T} \sim \Phi _{(a,b),p,(\mu _{1},\mu _{2}),D(\alpha)}\) where a,b,p are positive parameters, (μ _{1},μ _{2})^{T}∈R ^{2} and \(D(\alpha)=\left (\begin {array}{cc} \cos \alpha & \sin \alpha \\ \sin \alpha & \cos \alpha \end {array}\right)\) with 0≤α<2π is an orthogonal matrix causing a clockwise rotation around the origin through an angle of size α. Such random vector has according to Richter (2014) and Richter (2015a), (33), the density
where the functional
is a norm if p≥1 and an antinorm if 0<p≤1. For the latter notion, see Moszyńska and Richter (2012). The level sets of the density f _{(X,Y)} are pgeneralized ellipses being not necessarily axesaligned but centered at the point (μ _{1},μ _{2})^{T}.
Moreover, the stochastic representation
holds true where \( R\geq 0\ \text {and}\ U\sim \omega _{E_{(a,b),p}}\) are independent, R follows the density
and U follows the E _{(a,b),p }stargeneralized uniform distribution on the Borel σfield of the pgeneralized axesaligned ellipse having main axes of lengths 2a,2b,
Thus,
where \(\mathfrak U\) denotes the E _{(a,b),p }generalized arclength measure. Let us finally remark that another definition of a bivariate pgeneralized error density is given in Taguchi (1978).
4.2 Geometry of variance homogeneity
In this section, we exploit the fact that under heteroscedasticity a rotation causes a particular type of dependence, and give a new geometric interpretation of the hypothesis of variance homogeneity. To this end, we restrict our consideration once again to the Gaussian case. Let us assume that (X,Y)^{T} is an anticlockwise rotated axesaligned Gaussian vector
where
Then,
According to Dietrich et al. (2013), one can represent the parameters σ _{1},σ _{2},ϱ in terms of the parameters a,b,α as follows:
and
wher
and
Let us try now to geometrically understand the hypothesis \( H_{0}: \sigma _{1}^{2}=\sigma _{2}^{2}\) that is often supposed to hold in the statistical literature. It follows from the representations
that hypothesis H _{0} means that

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.
If a=b then ϱ=0 and Σ=σ ^{2} I _{2n } thus the density level sets of (ξ,η)^{T} and (X,Y)^{T} are Euclidean circles. If \(\alpha \in \left \{\frac {\pi }{4},\frac {3\pi }{4}\right \}\) then these level sets are arbitrary ellipses with main axes belonging to the lines \(\left \{(x,y)^{T}\in \mathbb R^{2}: y=x\right \}\) and \(\left \{(x,y)^{T}\in \mathbb R^{2}: y=x\right \}\), and the correlation attains any value from the interval (−1,1). Thus, H _{0} is not focussing, or is wavering, with respect to the parameters a,b,α and the shape of the density level ellipses and might therefore not always being primarily of interest, from this geometric point of view. If one presumes just the hypothesis
then the sample distribution and with it the distributions of all statistics derived from this sample vector are the same as in the axesaligned and homoscedastic case.
Let us finally consider the following well known particular case of homoscedasticity. If (X,Y)^{T}∼Φ _{ μ,Σ } then the random vector
has the covariance matrix
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
We are well motivated now for testing equality of scaling parameters in the ((a,b),p,α)parameterized model by checking the hypothesis H _{0}:a=b vs. the alternative H _{ A }:a≠b. Throughout this section, let
Let us further be given a concrete sample \(\mathfrak x_{i}=(x_{i},y_{i})^{T}, i=1,\ldots, n \) from independent identically and according to Φ _{(a,b),p,(0,0),D(α)} distributed random vectors. Then, the likelihood function \(\phantom {\dot {i}\!}L(a,b,\alpha)\) satisfies the equation
Let us consider the first two of the three likelihood equations. The partial derivatives of lnL with respect to a and b attain the value zero if \(a=\hat a(\hat \alpha)\) and \(b=\hat b(\hat \alpha)\), respectively, where
what ever the value of α is. The resulting equation
will be used later for constructing the likelihood ratio statistic. An angle \(\hat \alpha \) solves the third likelihood equation if
An ndimensional vectoralgebraic reformulation of this equation is
where [.,.]_{ p } denotes a semiinnerproduct defined by
For the theory and applications of semiinner products we refer to Lumer (1961), Giles (1967), Dragomir (2004) and Horváth et al. (2015). We just mention here that, for all x,y,z from R ^{n},a∈R,
In general, a semiinner product is not symmetric and nonlinear in the second argument. With the notations \(\xi _{i}(\alpha)=\left (\mathfrak x_{1}^{T}\theta _{i}(\alpha),\ldots,\mathfrak x_{n}^{T}\theta _{i}(\alpha)\right)^{T}, i=1,2\) and x _{(n)}=(x _{1},…,x _{ n }),y _{(n)}=(y _{1},…,y _{ n }), \(\hat \alpha \) solves the equation
or
The Hessian matrix of lnL(a,b,α) at the critical point \((a,b,\alpha)=(\hat a,\hat b,\hat \alpha)\) is
with
and
where
Obviously, Δ _{1}<0 and Δ _{2}>0 where
Let Δ _{3}= det(H M). If Δ _{3}<0 then \(\phantom {\dot {i}\!}L(a,b,\alpha)\) attains a local maximum at the point \(=(a,b,\alpha)=(\hat a,\hat b,\hat \alpha)\), see, e.g., Arens et al. (2013), Section 24.6. Under this assumption, \((\hat a,\hat b,\hat \alpha)=\text {mle} (a, b, \alpha).\) Note that Δ _{3}<0 if and only if
Thus, for finding mle (a,b,α), one has to solve (2) under the constraint (3). If p=2 then the semiinner product [.,.]_{ p } is symmetric, thus \(\hat \alpha \) satisfies either the equation \([ \xi _{1}(\hat \alpha),\xi _{2}(\hat \alpha) ]_{2}=0\), or \(\xi _{1}(\hat \alpha)_{2}^{2}=\xi _{2}(\hat \alpha)^{2}\). The first and second equations mean that
respectively, where ∠(ξ,η) denotes the angle between the vectors ξ and η, and arctan(+(−)∞)=+(−)π/2. We consider now the the H _{0}restricted likelihood function
and put
The partial derivative of \(\tilde L\) with respect to a attains the value zero if \(a=\tilde a(\alpha)\) where
what ever the value of α is. Thus, for suitable choice of \(\tilde \alpha,\) the maximum value of the restricted likelihood function \(\tilde L\) can be represented as
An angle \(\tilde \alpha \) solves the second restricted likelihood equation iff it satisfies the equation
which can be reformulated as
If p=2 then every \(\tilde \alpha \in [0,\frac {\pi }{2})\) solves this equation. Our test statistic
satisfies the representation
The likelihood ratio decision rule means to reject H _{0} if for some suitably chosen t∈(0,1) there holds Q<t. We remark that the present statistic becomes the same as that in the axesaligned pgeneralized elliptically contoured case in Section 3 if \(\hat \alpha =\tilde \alpha \in \left \{0,\frac {\pi }{2}\right \}\) and μ _{1}=μ _{2}=0. Moreover, the present decision rule can be equivalently reformulated then as to reject H _{0} if \(T_{p}=\frac {Y_{[1]}_{p}^{p}}{Y_{[2]}_{p}^{p}}\) attains sufficiently small or large values where
Example 1
∙ If α=0 then θ _{1}(α)=(1,0)^{T} and θ _{2}(α)=(0,1). ∙ If α is known (then \(\hat \alpha = \tilde \alpha =\alpha \)) then the considered decision rule means in other words to reject H _{0} for large values of \(\sqrt {R_{p}}+1/\sqrt {R_{p}}\) where
Note that, for i=1,…,n,
where I _{2} denotes the 2×2unit matrix. The statistic R _{ p } has therefore independently of the actual value of the angle of rotation α the same pgeneralized Fisher distribution as the likelihood ratio statistic in Section 3. Thus, in this case, rotational dependence is without influence onto the null distribution of the likelihood ratio statistic for proving \(H_{0}: \sigma _{1}^{2}=\sigma _{2}^{2}\), or not.
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
Many authors were dealing with estimating parameters of the ppower exponential distribution. Without aiming completeness, and without going into any details, we refer to Stacy and Mihram (1965), Harter (1967), Rahman and Gokhale (1996), Varanasi and Aazhang (1989), Do and Vetterli (1988), Mineo and Ruggieri (2005), GonzálezFarías et al. (2009), Saatci and Akan (2010). It is well known from the GaussMarkov theorem that orthogonal projections play a fundamental role in estimating parameters in the theory of linear models. The notion of an orthogonal projection is closely connected with that of a scalar product. If the standard Gaussian distribution is the sample distribution in R ^{n} then it is natural to use the Euclidean norm for several statistical calculations. This norm is generated by the Euclidean scalar product in R ^{n}×R ^{n}. If the density of the sample vector X _{(n)} is, for some p≥1,
then it is natural to work with the norm ._{ p } which is not generated by an inner product if p≠2,
It is known, however, that this norm is generated by the semiinner product [.,.]_{ p } considered in Section 4, \(x_{p}=[x,x]_{p}^{1/2}\). The present section is aimed to verify that this semiinner product plays also a role in estimating the location parameter of a ppower exponential distribution. Let \(L(\mu)=f_{X_{(n)}}(x)\). Maximizing L with respect to μ is equivalent to minimizing the function
Let x _{(1)}≤…≤x _{(n)} be the ordered values of the concrete sample vector x=(x _{1},…,x _{ n })^{T}. Given μ, there exists a natural number n _{1} such that
Thus, f ^{′}(μ)=p(f _{1}(μ)−f _{2}(μ))where
If p∈(1,∞), the functions f _{ i },i=1,2 are monotonously increasing/decreasing if i=1/i=2, respectively. Moreover, these functions are continuous, and satisfy f _{1}(x _{(1)})=0,f _{1}(x _{(n)})>0 and f _{2}(x _{(n)})=0,f _{2}(x _{(1)})>0. Thus there exists a uniquely determined \(\hat \mu \) such that \((\hat \mu, f_{1}(\hat \mu))=(\hat \mu, f_{2}(\hat \mu))\) is the intersection point of the curves {(μ,f _{1}(μ)):μ∈[x _{(1)},x _{(n)}]} and {(μ,f _{2}(μ)):μ∈[x _{(1)},x _{(n)}]}. Based upon a bisection algorithm, \(\hat \mu \) can be numerically calculated and is the solution of the equation \(f'(\mu)=0_{\mu =\hat \mu }, \) i.e.
Example 2
If p=2 then (4) reads a
thus \(\hat \mu =\bar {x}_{n}\).
We consider now two cases excluded so far.
Example 3
In the case p=1,
thus \(f'(\hat \mu)=0\) iff \( \natural \left \{x_{i}<\hat \mu \right \} =\natural \left \{x_{i}>\hat \mu \right \}\) where ♮{…} means the number the event written between the brackets occurs. For odd \(n, \hat \mu =x_{[n/2]+1}\), and for even n, every \(\hat \mu \) from [x _{[n/2]},x _{[n/2]+1}] satisfies this condition, thus \(\hat \mu \) is the sample median.
Example 4
In the case p=∞, we first define the notion ._{ ∞ }.(a) Let f(μ)= max{x _{1}−μ,…,x _{ n }−μ} and i ^{∗} such that \(\phantom {\dot {i}\!}f(\mu)=x_{i^{*}}\mu .\) Then
where
By definition,
(b) If for some i ^{∗} there holds \(x_{i^{*}}\hat \mu =\max \left \{x_{1}\hat \mu ,\ldots,x_{n}\hat \mu \right \}\) then \(\hat \mu \) is maximum likelihood estimator of μ. The number \(\hat \sigma =\max x_{i}\hat \mu \) is the smallest number satisfying \(\hat \sigma \leq x_{i}\hat \mu \leq \hat \sigma,\forall i\), thus \(x_{(n)}\hat \sigma \leq \hat \mu \leq x_{(1)}+\hat \sigma. \) The smallest possible \(\hat \sigma \) satisfies \(2\hat \sigma \geq x_{(n)}x_{(1)}\), thus \(\hat \sigma =(x_{(n)}x_{(1)})/2. \) It follows that \(\hat \mu =(x_{(n)}+x_{(1)})/2 =\text {midrange }\left \{x_{1},\ldots,x_{n}\right \}\).
With
it follows in the general setting that the uniquely determined solution \(\hat \mu \) of the equation (4) is a relative maximum point of the likelihood function L, thus \(\hat \mu =\text {mle}(\mu)\). This means that
or, equivalently,
Thus, on the one hand, \(\hat \mu \) solves the oscillating fixed point equation
On the other hand, it follows that \(\hat \mu =mle(\mu)\) satisfies the equation
which means that \(\hat \mu =\bar x_{n}\) if p=2. Under suitable assumptions upon the convergence of \(\hat \mu \) and the limit \(\mu ^{*}=\lim \limits _{n\rightarrow \infty }\hat \mu \), it follows
thus \(\lim \limits _{n\rightarrow \infty } n^{2/p}[1_{n},x\hat \mu 1_{n}]_{p}=0\) can be reformulated as
Simulation of starshaped distributed random vectors
6.1 Preliminary remarks
It may be of interest to determine exact distributions of the statistics dealt with in Sections 25. To this end, one might use various analytical tools like, e.g., a geometric measure representation as a starting point of explicit analytical derivations. As an alternative to such derivations, we present here simulation methods which allow to generate stochastic approximations of statistical distributions. Let (X _{1,j },Y _{1,j })^{T},…,(X _{ n,j },Y _{ n,j })^{T}, j=1,…,N be independent samples of independent random vectors following the rotational dependent pgeneralized elliptically contoured density f _{(X,Y)} defined in Section 4.1., and let further
be a sample of i.i.d. copies of a real valued statistic T. For sufficiently large N, the probability P(T<t) can be stochastically approximated by the relative frequency \(\frac {1}{N}\sum \limits _{i=1}^{N} I_{(\infty,t)}(T_{i})\). To this end, we present an acceptancerejection method for simulating random vectors (X _{ i,j },Y _{ i,j })^{T} in Section 6.2, and a generalized polar method in Section 6.3. This will be done even under the much more general assumption that (X,Y)^{T} follows an arbitrary starshaped distribution. This class includes that of pgeneralized elliptically contoured distributions. For approaches to general distribution classes see Fernández et al. (1995), Arnold et al. (2008), Kamiya et al. (2008), Sarabia and GómezDéniz (2008), Balkema and Nolde (2010). A geometric representation of starshaped distributions is given in Richter (2014). We refer to the letter paper for main notions and recall that (X,Y)^{T} allows the stochastic representation \( X\overset {d}{=}R\cdot U \) where R and U are stochastically independent, R is a nonnegative random variable, and the singular random vector U follows the stargeneralized uniform distribution ω _{ S } on the Borel σfield \(\mathfrak B(S)\) of the starsphere S being the topological boundary of a suitably defined star body K,
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
General aspects of acceptancerejection or simply rejection methods are studied in Kalke and Richter (2013) and applied there to the pgeneralized rejecting polar method. Platonically generalized uniformly distributed and polyhedral starshaped distributed random vectors are generated this way in Richter and Schicker (2014, 2016a) as well as Richter and Schicker 2016b, respectively. If the starspheres are represented in a certain analytical way, Nolan (2016) aims to exploit the geometric measure representation approximatively in a sense, not yet explicitly defined. Here, we demonstrate how to generate in four steps starshaped distributed vectors. Step 1. To start with, let C _{ i },i=1,…,d be positive constants, C _{(d)}=(C _{1},…,C _{ d })^{T} and \(0_{(d)}=(0,\ldots,0)^{T}\in \mathcal R^{d}\). We denote by
an axesaligned ddimensional rectangle and by \(O\in \mathcal R^{d\times d}\) an orthogonal matrix. Using the further notation
we assume that the random vectors ξ _{ n },n=1,2,… follow the uniform distribution on O(0_{(d)},C _{(d)}), and are independent. Because the vector O ^{−1} ξ _{ n } follows the product measure of uniform distributions on univariate intervals, \(U_{(0,C_{1})}\times \ldots \times U_{(0,C_{d})}\), it can immediately be simulated. Step 2. Let the acceptance region
be a star body having the origin as an interior point. According to Remark A.1 in Kalke and Richter (2013), the stopping time
is almost surely finite if P(ξ _{1}∈A)>0. The following lemma says that the stopping element
is uniformly distributed in the acceptance region A, \(\xi _{\tau _{A}}\sim U_{A}\).
Lemma 1 The stopping element \(\phantom {\dot {i}\!}\xi _{\tau ^{A}}\) satisfies the equation
Proof
It follows from
that
□
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
Let \(\tilde M\in {\mathcal {B}}^{d}\cap \partial A\) and \(sector(\tilde M)=\left \{\lambda x: x\in \tilde M, 0\leq \lambda \leq 1\right \}\). Then
□
Example 5
, continued. The Minkowski functional of the set B _{ a,p }=O ^{−1} A is \(h_{O^{1}A}(x)=x_{a,p}=\left (\sum \limits _{i=1}^{d} x_{i}^{p}\right)^{1/p}\) and, with S=∂ B _{ a,p },
Example 6
, continued. For arbitrary norm or antinorm ., h _{ A }(x)=x, and for all \(\tilde M\in \mathcal B^{d}\cap S\), S={x:x=1},
where K ^{∗} is the unit ball of the norm .^{∗} being dual to the norm . in the first case, and the antipolar set of K={x:x≤1} in the second one. Moreover, N(𝜗) is the outer/inner normal vector to S at the point (𝜗,x _{ d }(𝜗)),N(𝜗)=(grad x _{ d }(𝜗),−1)^{T}.
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
In case of a distribution having a density generating function, g say, the cumulative distribution function of R=R(g) is
thus R(g) can be simulated accordingly. To this end, let U be uniformly distributed on (0,1), then \(F_{R(g)}^{1}(U)\overset {d}{=}R(g)\).
Remark 2
If a density generating function g satisfies the equation
then it is called a density generator. Methods of estimating a density generator are described in Liebscher and Richter (2017).
6.3 Dependent pgeneralized polar method
The classical polar method is due to Box and Muller (1958). If the acceptance rate of the algorithm described in the previous section is not large enough, or for some other reason, one might seek for a direct stargeneralization of the polar method. We just mention here that there are different particular methods for directly generating the stargeneralized uniform distribution on a star sphere. For the pgeneralized polar method, e.g., such method has been established in Kalke and Richter (2013) and applied in Richter (2015a). The independent coordinate representation of general twodimensional norm contoured distributions which is the basis for a normgeneralization of the polar method is proved in Richter (2015b). Below, we describe an algorithm for an (a,p,O)generalization of the polar method of Box and Muller where a=(a _{1},…,a _{ d })^{T},a _{ i }>0,i=1,…,d;p>0 and O is an orthogonal d×dmatrix. To this end, we assume that a random vector X=(X _{1},…,X _{ d })^{T} follows an axesaligned pgeneralized elliptically contoured distribution having density generating function g and vector of scaling parameters a, \(X\sim \Phi _{g, B_{a,p}}\). Let further Y=O X+ν denote a transformation vector, with an orthogonal d×dmatrix O and ν∈R ^{d}, and put
and, for ϕ _{ i }∈[0,π), i=1,…,d−2,ϕ _{ d−1}∈[0,2π),
where the generalized trigonometric functions \(\sin _{(a_{i},a_{i+1};p)}(\phi _{i}), \cos _{(a_{i},a_{i+1};p)}(\phi _{i})\) and the normalizing functions \(N_{(a_{i},a_{i+1};p)}(\phi _{i})\) are defined in Richter (2014). With suitably chosen constants C _{ i }, the functions h _{ i } are the densities of independent random variables R,Φ _{1},…,Φ _{ n−1} jointly satisfying the stochastic representation X=R U where
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.
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Richter, W. Statistical reasoning in dependent pgeneralized elliptically contoured distributions and beyond. J Stat Distrib App 4, 21 (2017) doi:10.1186/s4048801700743
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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