In this paper we propose an idea to use a certain property of multivariate probability distributions, that we call the conditional quantile reproducibility, to decrease the amount of observations required to construct a statistical estimate of a n-dimensional conditional quantile of the distribution. For the class of probability distributions, satisfying to this property, we present several results, proving that in many cases the reproducibility property allows us to restore the n-dimensional conditional quantile by solving a certain type of Pfa an di erential equation. The equation is constructed from functions, derived only from the 2-dimensional marginal distributions of the initial distribution.
Many of the widely used multivariate probability distributions satisfy to a certain property of the multivariate conditional quantiles which we call \quantile reproducibility". Among them are the multivariate Gaussian distribution, Student's distribution, Logistic distribution, Pareto distribution, Gamma distribution, Clayton's Copula distribution and others (see [9, 8, 6, 5]). This article gives several results related to the quantile reproducibility property of the multivariate distributions. Particularly we prove that when the distribution satis es to the quantile reproducibility property (we also say it \has reproducible conditional quantiles"), the solution of a Pfa an di erential equation of certain form, that can be constructed by only knowing 2-dimensional marginal distributions of the original distribution, is equal to a distribution's multivariate conditional quantile. In other words by solving the Pfa an di erential equation we build a multivariate quantile from bivariate functions, describing the probability distribution.
The outline of this article is as follows. In Section 2 we introduce the multivariate conditional quantiles and discuss some of their applications. In Section 3 we give the de nition of the quantile reproducibility for multivariate distributions and talk on its geometrical interpretation.
In Section 4 we introduce the Pfa an di erential equation for the distribution which we refer to as the \quantile equation". We also show that the solution of the maximum possible dimension for the quantile equation of a distribution with reproducible quantiles equals a conditional quantile of the maximum dimension. Section 5 gives several concrete examples of multivariate distributions with reproducible quantiles along with the corresponding quantile equations. In Section 6 we give an intermediate version of the quantile reproducibility property and prove that, when it is satis ed, one can nd the solutions of the quantile equation of intermediate dimensions. In Section 7 we illustrate this theorem by giving an example of the distribution with this version of quantile reproducibility and nding the solutions of the quantile equation for it. In Section 8 we discuss how the quantile reprodicibilty property could be used in statistical estimation. For the case when it is known that the distribution has reproducible conditional quantiles, we propose a technique to build the multivariate quantile estimate by only using bivariate observations. We also show that this technique allows us to reduce the required number of observations when compared to a traditional approach to multivariate quantile estimation. 2
Conditional Quantiles and Their Applications First, we will give the de nition of the multivariate conditional quantile. Consider a system of random variables X1; X2; : : : ; Xn: Suppose we have the conditional cumulative distribution function Fij1:::^i:::n(xijx1; : : : xbi; : : : ; xn) (the sign ^ over the element means that the element is omitted), which is continuous and increases monotonically in xi for any xed vector (x1; : : : xbi; : : : ; xn) 2 Rn 1.
The conditional quantile qi(jp1):::^i:::n(x1; : : : xbi; : : : ; xn) of level p 2 [0; 1] for a random variable Xi given X1; : : : ; Xci; : : : ; Xn is de ned by the equation Fij1:::^i:::n(q(p) ij1:::^i:::n(x1; : : : xbi; : : : ; xn)j x1; : : : xbi; : : : ; xn) p for any (x1; : : : xbi; : : : ; xn) 2 Rn 1.
The conditional median mij1:::^i:::n(x1; : : : xbi; : : : ; xn) for a random variable Xi given X1; : : : ; Xci; : : : ; Xn is a conditional quantile of level p = 12 . Another way to de ne a conditional quantile is to give a certain point x (x1; : : : ; xn) 2 Rn, lying on the quantile surface: = F1j 2:::n q1(xj2:)::n(x2; : : : ; xn)j x2; : : : ; xn F1j 2:::n(x1j x2; : : : ; xn): The conditional quantile is then said to be going through x . In this case the level of the quantile is given by p = F1j 2:::n(x1j x2; : : : ; xn): One of the several applications of the conditional quantiles is using them as random variable estimates. Consider the following situation. We have a set of random variables X1; X2; : : : ; Xn. We observe the rst n 1 of them, and the last one needs to be estimated. We also know the conditional distribution function Fnj1:::n 1(xn j x1; : : : ; xn 1) which is supposed to be monotonic in xn. Now, having de ned a loss function (u) = ( u; 1)u; u 0; u > 0:
2 (0; 1): we want to get the estimate for the last random variable Xn, so that it would minimize the conditional loss. fb(x1; : : : ; xn 1) : min Mf f( ) (Xn
f (x1; : : : ; xn 1)) j x1; : : : ; xn 1g Rather simple calculations will lead us to the following formula for the Xn estimate.
Fnj1:::n 1(f^(x1; : : : ; xn 1) j x1; : : : ; xn 1) That is any value to satisfy to this equation would minimize the loss function. As long as the F is monotonic in xn, there is only one such value, which is the conditional quantile ( ) fb(x1; : : : ; xn 1) = Fnj11:::n 1( j x1; : : : ; xn 1) = qnj1:::n 1(x1; : : : ; xn 1): Further on, it is also known (see [10]), that the estimate, minimizing the conditional loss, will also minimize the expected loss (the bayesian risk): Mf 3 PfX1 PfXi (Xn fb(X1; : : : ; Xn 1))g = min Mf f( ) (Xn f (X1; : : : ; Xn 1))
Reproducible Conditional Quantiles Consider a vector of random variables X = (X1; : : : ; Xn) with the cumulative distribution function F1:::n(x1; : : : ; xn), the multivariate strictly positive density By xing a point x = (x1; : : : ; xn) 2 Rn, we get a family of conditional quantiles, going through the selected point, which act as level surfaces (or curves) of the conditional CDF: F1j2:::n q1(xj2:)::n(x2; : : : ; xn) j x2; : : : ; xn F1j2:::n(x1 j x2; : : : ; xn); q1(xj2:)::n(x2; : : : ; xn) = x1;
(xi ;xj )(xj ) j xj Fijj qijj
Fijj (xi j xj ); qi(jxji ;xj )(xj ) = xi ; i 6= j:
De nition 1. (see [8]) We say that the multivariate probability distribution
F1:::n(x1; : : : ; xn) has reproducible conditional quantiles if the system of
identities
(
For a geometric interpretation, consider the curves, parametrized by the \small" conditional quantiles (k = 2; n).
k(x ; t) = fq1(xjk1;xk)(t); : : : ; qk(xk1jk1;xk)(t); t; qk(x+k1+jk1;xk)(t); : : : ; qn(xjkn;xk)(t)g; The quantile reproducibility would mean that these curves lie on the \big" quantile surface:
(x )(x2; : : : ; xn) = fq1(xj2:)::n(x2; : : : ; xn); x2; : : : ; xng: 4
Quantile Pfa an Di erential Equations and Their Relation to Quantile Reproducibility It can be shown that for the class of multivariate probability distributions with reproducible conditional quantiles we can construct the \big" (n 1)-variate conditional quantile as the solution of a Pfa an di erential equation of special form. The equation itself is based on the functions, derived from the conditional quantiles of dimension 1, corresponding to the 2-dimensional marginal distributions of the initial distribution. In other words, using the property of quantile reproducibility, we can virtually shift from bivariate functions, characterizing the probability distribution, to its multivariate characteristic.
Again we consider a random vector X = (X1; : : : ; Xn) with cumulative distribution function F1:::n(x1; : : : ; xn) and strictly positive density We denote by ei the basis vectors in Rn. The point over the quantile means dif(xi ;xj )(xj ) is the derivative of the one-dimensional quantile ferentiation, that is q_ijj (xi ;xj )(xj ) with respect to xj , going through the point (xi ; xj ) and taken at qijj xj = xj .
To simplify the notation let us expand the determinant along the rst row (xi ;xj )(xj ).
Each of the cofactors A1k will depend on a set of quantile derivatives q_ijj
Now we replace the point x = (x1; : : : ; xn) with x = (x1; : : : ; xn), which is
variable in Rn, and consider the di erential 1-form
Next we construct a Pfa an di erential equation for the form
! =
! =
n
X A1k(x1; : : : ; xn)dxk:
k=1
n
X A1k(x1; : : : ; xn)dxk = 0:
k=1
We call it the quantile equation.
! =
n
X A1i(x1; : : : ; xn)dxi = 0
i=1
(
is completely integrable. The solution of (
A well known result, the Frobenius theorem (see [1] p. 97), gives a necessary
and su cient condition of the complete integrability of the Pfa an di erential
equation. It states that the equation (
Theorem 1. If the probability distribution F1:::n(x1; : : : ; xn) with a joint PDF
positive on Rn has reproducible conditional quantiles (
Tabl.1. Quantile equations for some distributions Densities 1. (x; m; [ ij ]) 1{Gaussian distribution, 2{Cauchy distribution, 3{Logistic distribution, 4{Pareto distribution 6
The Darboux Class for Quantile Di erential Equations and Its Relation to Reproducibility Now what if the quantile di erential equation is not completely integrable? Are there cases when we can say something about the solutions of this equation? To answer this question, let us rst remind the reader about one of the characteristics of the di erential 1-forms (and as a consequence of the Pfa an di erential equations) - the Darboux class. As the Darboux theorem (see [3]) states, the class gives the maximum dimension of the integral manifold of the corresponding Pfa an di erential equation.
De nition 2. If the di erential 1-form ! satis es the equality ! ^ (d!)r 6= 0; but
! ^ (d!)r+1 = 0; then we say that the Darboux class of the di erential form equals 2r + 1. As we have already mentioned (see section 4), the equality d! ^ ! = 0 gives the criterion for complete integrability of a Pfa an equation (Frobenius theorem). In this case r = 0, so the Darboux class of the di erential form ! is equal to 1.
Theorem (Darboux's theorem). If the Darboux class of the di erential 1-form
1 ! equals 2r + 1, then by a smooth local change of coordinates the Pfa an
equation
(
y3(x1; :::; xn) = C3 = const; : : : ;
y2r+1(x1; :::; xn) = C2r+1 = const:
As noted above, if the condition (
Let us now sum up. For the given quantile equation we can calculate the Darboux class of the corresponding di erential 1-form. From this value we obtain the maximum dimension of the integral manifold. But still what is the integral manifold itself? We will now show, that for multivariate probability distributions with certain type of quantile reproducibility, the solutions can be given explicitly. To do this, we will rst establish one useful property of matrix determinants (see [5]).
Let us consider the determinant of order n, where n 1 k, of the form2 dx1 1 : : : q_1jk q_1jk+1
: : : q_1jn 1 dx2 q_2j1 : : : : : : : : : : : : : : : : : : : : : q_2jk q_2jk+1 q_kjk+1
: : : : : : q_2jn 1 : : : q_kjn 1 dxk q_kj1 : : : 1 dxk+1 q_k+1j1
: : : q_k+1jk 1 : : : : : : : : : : : : : : : : : : : : : q_k+1jn 1 : : : dxn 1 q_n 1j1
: : : q_n 1jk q_n 1jk+1 : : : 1 dxn q_nj1 : : : q_njk q_njk+1
: : : q_njn 1 : We will denote the cofactor of the element dxi in the rst row of M by A(dxi). Lemma 1. The following expansion for the determinant M is true: M = The proof of this lemma is given in [5].
2The given expansion holds for determinants of general form. and the determinant S = (x0)(x1; : : : ; xk) = k = n q_1(xjk0)(xk) q_2(xjk0)(xk) : : : : : :
: : : q_2(xj10)(x1) : : : q_k(xj10)(x1) : : : 1 6= 0;
Suppose x0 := (x01; : : : ; x0n). For all the natural numbers s = 1; n i = s + 1; n we will denote: 1 and (x0) (x10;:::;xs0;xi0)(x1; : : : ; xs): qij1:::s(x1; : : : ; xs) = qij1:::s Next, for a random vector X = (X1; : : : ; Xn) we will suppose that k < n 1 of its variables are xed. Without loss of generality we can think that these are the rst k variables x1; : : : ; xk.
Theorem 2. If the probability distribution F1:::n(x1; :::; xn) has reproducible k-dimensional conditional quantiles, that is qi(jx10:)::k(x1; q2(x10)(x1); q3(x10)(x1); : : : ; qk(x10)(x1)) = qi(x10)(x1)
j j j j qij1:::k(q1(x20)(x2); x2; q3(x20)(x2); : : : ; qk(x20)(x2)) = qi(x20)(x2) (x0)
j j j j
: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
qij1:::k(q1(xk0)(xk); q2(xk0)(xk); : : : ; qk(x01)jk(xk); xk) = qi(xk0)(xk)
(x0)
j j j
(
Proof. Let us limit our consideration to the (k + 1)-dimensional marginal probability distribution F1:::ki(x1; : : : ; xk; xi): Then the k-dimensional conditional quantile qi(jx10:)::k(x1; : : : ; xk);
i = k + 1; n;
will be the \big" quantile of the probability distribution with the corresponding
conditional distribution function Fij1:::k(xi j x1; : : : ; xk): So this distribution has
a reproducible \big" conditional quantile. From the condition (
: : : dxk q_1(xjk0)(xk) q_2(xjk0)(xk) : : : that is wi(x1; : : : ; xk; qi(jx:0::)k(x1; : : : ; xk)) Let us now consider the quantile equation for the initial distribution. w(x1; : : : ; xn) = = dx1
: : : q_n(xjn0) 1(xn 1) = 0: We can apply Lemma 1 to expand the left part of the equation: 0; i = k + 1; n:
(
Note 1. If the conditions of the theorem are satis ed, the quantile equation
(
When the Darboux class of the quantile equation is equal to 2(n k) 1, the
surface (
Note 2. If we add to (
Example of the Distribution with an Intermediate Darboux Class To illustrate theorem 2, let us consider a mixture of two 5-dimensional Cauchy distributions with density Now let us calculate the Darboux class of the form ! to determine the maximum dimension of the solution of the quantile equation. The calculations show that d! 6= 0; d! ^ ! 6= 0 almost surely in R5; d! ^ d! 0: So the Darboux class of the form ! is almost surely equal to 3 and the maximum dimension of the solution of the quantile equation (11) is also almost surely equal to 3.
The integral manifold of the maximum possible dimension, going through the point x = (x1; : : : ; x5) is given by the equalities x4 = x4 s
1 + x21 + 3x22 + 4x23 1 + (x1)2 + 3(x2)2 + 4(x3)2 ; x5 = x5 s
1 + x21 + 3x22 + 4x23 1 + (x1)2 + 3(x2)2 + 4(x3)2 ; which exactly match the two 3-dimensional conditional quantiles going through x : q_4(xj12)3 (x1; x2) and q_5(xj12)3 (x1; x2). That is the solution is S (x ) = n x1; x2; x3; q4(xj12)3 (x1; x2; x3) ; q5(xj12)3 (x1; x2; x3) o :
Finally it is easy to verify that the 3-dimensional conditional quantiles satisfy to the quantile reproducibility property. So, according to theorem 2, S (x ) should be the solution of the quantile equation. Since the Darboux class of the form ! is equal to 3, then, as it is stated in note 1, this is the solution of the maximum possible dimension.
It is also easy to show, that the condition of note 2 is satis ed, so S (x ) is a part of the \big" conditional quantile surface S (x ) n x1; x2; x3; x4; q5(xj12)34 (x1; x2; x3; x4) o : 8
Statistical Application of Quantile Reproducibility The theorem 1 talks on how to obtain the n 1-dimensional quantile from a set of 1-dimensional quantile derivatives, which obviously can be calculated from bivariate marginal densities of the distribution. This logically leads to an idea to try to build a statistical estimate of the \big" conditional quantile of the distribution from a number of estimates of its bivariate densities. The outline of the algorithm would be as follows:
First estimate 1-dimensional quantiles of the distribution and their derivatives from a number of bivariate observations (see for example [2]). Then using these quantile derivative estimates build the Pfa an quantile equation 2 so that it's completely integrable and its solution approximates the \big" conditional quantile.
Solve the quantile equation numerically and obtain the n-dimensional quantile estimate.
Obviously for this algorithm one only needs a set of 2-dimensional observations all of which can be made independently. With the direct approach to estimate the n 1-dimensional qunatile one would need to observe the entire vector of n dimensions.
Now let us roughly compare the number of observations required to build the n 1-dimensional quantile estimate when using the traditional direct approach and the algorithm described above. For convenience we will assume that the quantile and the quantile derivative estimation in both cases is done by rst estimating the corresponding distribution densities and deriving conditional densities from the estimates.
With the traditional approach we consider a sample of observations n(x(11); : : : ; x(n1)); : : : ; (x(1rn); : : : ; x(nrn))o for a random vector X = (X1; : : : ; Xn) with the density f1:::n(x1; : : : ; xn). From the observations we construct the density estimate f^1:::n(x1; : : : ; xn). To simplify calculations we'll suppose that the histogram method is used. For each variable Xi we divide the sample range into m intervals. This way we get mn n-dimensional parallelepipeds. If, in order to get a good estimate in each parallelepiped we need k observations, then the total amount of observations required to estimate the density of X would be rn = k mn.
If we use the algorithm proposed above we don't need to estimate the n-variate
density f1:::n(x1; : : : ; xn). Instead we construct estimates for the marginal
densities fij (xi; xj ); i 6= j; i; j = 1; n, using the observations
n(xi(
This work was partially supported by a grant of RFBR (project 16-01-00184 A).