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Proc. R. Soc. A (2010) 466, 2079–2096
doi:10.1098/rspa.2009.0482
Published online 15 February 2010
Reciprocal symmetry, unimodality and
Khintchine’s theorem
BY YOGENDRA P. CHAUBEY1 , GOVIND S. MUDHOLKAR2
AND
M. C. JONES3, *
1 Department
of Mathematics and Statistics, Concordia University, Montreal,
Quebec, Canada H3G 1M8
2 Department of Statistics and Biostatistics, University of Rochester,
Rochester, NY 14627, USA
3 Department of Mathematics and Statistics, The Open University,
Milton Keynes MK7 6AA, UK
The symmetric distributions on the real line and their multi-variate extensions
play a central role in statistical theory and many of its applications. Furthermore,
data in practice often consist of non-negative measurements. Reciprocally symmetric
distributions defined on the positive real line may be considered analogous to symmetric
distributions on the real line. Hence, it is useful to investigate reciprocal symmetry in
general, and Mudholkar and Wang’s notion of R-symmetry in particular. In this paper, we
shall explore a number of interesting results and interplays involving reciprocal symmetry,
unimodality and Khintchine’s theorem with particular emphasis on R-symmetry. They
bear on the important practical analogies between the Gaussian and inverse Gaussian
distributions.
Keywords: Cauchy–Schlömilch transformation; Gaussian–inverse Gaussian analogies;
Khintchine’s theorem; log-symmetry; R-symmetry; unimodal distributions
1. Introduction
R-symmetry and log-symmetry, which together fall under the rubric of reciprocal
symmetry, are to non-negative random variables what ordinary symmetry is to
arbitrary random variables. Inverse Gaussian (IG) distributions are as central to
the R-symmetric distributions as the lognormal distribution is to log-symmetric
distributions and the Gaussian (G) distribution is to the symmetric distributions.
Data from homogeneous, non-mixture, populations can usually be assumed to be
unimodal and the concept of unimodality of density was characterized in the 1930s
by Khintchine. These three concepts, reciprocal symmetry, G–IG analogies and
Khintchine’s theorem on unimodality, play basic roles in statistical modelling,
analysis and theory, and inter-relations between these three are the subject of
this paper. We begin with an extended introduction, looking at the definitions,
roles and practical importance of each of the three concepts in turn.
*Author for correspondence ([email protected]).
Received 15 September 2009
Accepted 14 January 2010
2079
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(a) Reciprocal symmetry
Let Y be a random variable following an absolutely continuous distribution
function F with probability density function (pdf) f on the positive half-line R+ .
There are two natural and complementary concepts of reciprocal symmetry of f .
First, there is R-symmetry (Mudholkar & Wang 2007), which means that
q
,
(1.1)
f (qy) = f
y
for all y > 0 and some q > 0. Second, there is log-symmetry (Seshadri 1965),
so-called because it corresponds to ordinary symmetry of the distribution of
log Y . In density terms,
f (d/y)
,
(1.2)
yf (dy) =
y
for all y > 0 and some d > 0. While the importance of log-symmetry is obvious
through its transformational link with ordinary symmetry, the importance of
R-symmetry emerges as a driver of the extraordinary analogies between G and IG
distributions described in §1b to follow. Note that our use of the term ‘reciprocal
symmetry’ differs from its use in theoretical physics.
An initial comparison of these two notions of reciprocal symmetry was given
by Jones (2008). Before continuing to concentrate, in this paper, on differences
between R- and log-symmetry, it is worth recalling conditions under which R- and
log-symmetric distributions coincide; call these doubly symmetric distributions.
Trivially, this cannot happen if q = d. It is also easy to see that the lognormal
distribution corresponding to log Y ∼ N (m, s2 ) is both R-symmetric about q =
2
em−s and log-symmetric about d = em . Jones & Arnold (2008) showed that
absolutely continuous doubly symmetric distributions consist of a subset of those
distributions that have the same moments as the lognormal distribution. Write
k = d/q. Then, a doubly symmetric density f ∗ takes a piecewise form derived in
a closely related context by Pakes (1996), namely
f ∗ (qy) ∝
∞
k 2i(i−1) y 2i−1 j(k 4(i−1) y 2 )I (k −2i < y ≤ k 2−2i ),
(1.3)
i=−∞
together with the additional requirement that j(u) = j(1/(k 4 u)), 1/k 4 < u ≤ 1.
This class includes the lognormal, the Askey/Berg densities (e.g. Berg 1998) and
another example explicitly constructed by Jones and Arnold, but does not include
the famous class of Stieltjes densities (e.g. Heyde 1983).
(b) G–IG analogies
The basic paradigm for statistical theory and methods concerns questions
about location and variability. The IG distribution is a distribution for nonnegative random variables with two parameters, one being the expectation and
the other a measure of dispersion. This is a first respect in which it is similar to
the G distribution. The IG distribution has its origins in the studies of Brownian
motion by Schrödinger (1915) and by Smoluchowski (1915). However, its inverse
relation—in terms of Laplace transforms—to the G distribution was discovered
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by Tweedie (1945) who gave it the current name. The G distribution has a
stronger following because of its physical explanations and numerous analytical
derivations summarized, for example, in Rao (1965). However, the IG distribution
has similar, numerous derivations and explanations, for example, by Schrödinger,
Smoluchowski, Tweedie, Wald, Huff and Halphen. These are summarized in
the historical survey chapter of Seshadri (1993). There is also a more recent
maximum entropy characterization of the IG distribution by Mudholkar &
Tian (2002), which is analogous to Shannon’s (1949) famous maximum entropy
characterization of the G distribution.
The G–IG analogies, in terms of both distributional and inferential properties,
are remarkable and manifold. Tweedie gave the earliest of these analogies,
which were highlighted in Folks & Chhikara (1976), a paper that made the
IG distribution better known and broadly used. The G–IG analogies have been
substantially extended and tabulated. For summaries, see tables 6 and 7 of
Mudholkar & Natarajan (2002), appendix B of Mudholkar & Wang (2007) and
table 2 of Mudholkar et al. (2009); the last two run to some 40 items. Mudholkar &
Natarajan (2002) also defined a concept of IG symmetry; this was mathematically
well defined but intuitively opaque. This led, in Mudholkar & Wang (2007), to
the alternative notion of R-symmetry, which offers some physical transparency.
Moreover, it then turns out that the R-symmetric distribution closely connected
to the IG distribution through which properties and consequences most readily
flow is the root reciprocal inverse Gaussian (RRIG) distribution. This is the
distribution of one over the square root of an IG random variate; its density
is given in equation (1.9).
The lognormal, the best known of the log-symmetric distributions for nonnegative data that is used in a variety of applications, has the advantages of
familiarity and simple transformation. On the other hand, the pivotal RRIG and
the related IG distributions, in view of the G–IG analogies, offer simple methods
for direct inference on population means without confounding by variability
aspects.
(c) Khintchine’s theorem
A simplified version of Khintchine’s formula for representing the distribution
function of a unimodal distribution with mode at 0 is given by
∞
Vz (x) dH (z),
(1.4)
F (x) =
−∞
where Vz (x) is the distribution function of the uniform distribution on (−z, z),
z > 0 and H is a distribution function. Khintchine (1938) identified the
distribution function H as
H (x) = F (x) − xf (x).
(1.5)
In the case that f (x) = F (x) exists and is continuous everywhere, the density
h(x) is given by
(1.6)
h(x) = −xf (x).
This leads to the following characterization of symmetric unimodal (with mode
at 0) distributions (Dharmadhikari & Joag-Dev 1988, p. 5).
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Theorem 1.1 (Shepp 1962). If X is symmetric and unimodal (with mode at 0),
then F belongs to the closed convex hull of the set of all uniform distributions
on symmetric intervals (−a, a) with a > 0. Equivalently, if F is symmetric and
unimodal, then there exists a symmetric random variable Z such that F is
the distribution function of UZ , where U is uniform (0,1), independent of Z .
Alternatively, F is the distribution function of VZ , where Z is non-negative and
V is uniform on (−1, 1).
The basic characterization of unimodality owing to Khintchine (1938) received
enhanced attention when Gnedenko & Kolmogorov (1949) in their famous
monograph on the limit distributions of sums of independent random variables
used a false theorem owing to Lapin which claimed convolutions of real-valued
unimodal random variables to be unimodal. K. L. Chung, in his translation of
the monograph, highlighted the error via a counterexample and stated Wintner’s
(1938) result that convolutions of symmetric unimodal random variables are
symmetric unimodal (see also Feller 1971, p. 168).
The concept of unimodality of real-valued random variables, because of its
importance and ubiquity in statistical theory and applications, has received
considerable attention in the literature. It has been extensively studied, e.g. Laha
(1961), Medgyessy (1967), Sun (1967) and Wolfe (1971). It has also been variously
generalized, e.g. to alpha unimodality of Olshen & Savage (1970), generalized
unimodality of Ghosh (1974), multi-variate A-unimodality of Anderson (1955)
and S-unimodality of Das Gupta (1976). (For other technical details, see
Dharmadhikari & Joag-Dev (1988) and Bertin et al. (1997)).
Wintner’s (1938) result has been used by Birnbaum (1948) to analyse the
notion of peakedness. The result, when read to say that the area under a
symmetric unimodal curve over a symmetric interval decreases monotonically as
the interval is translated, yields monotonic power functions that are crucial for
setting sample sizes. Anderson’s (1955) theorem is a multi-variate generalization
of the earlier-mentioned reading of Wintner’s result (see also Sherman 1955).
It is used in Das Gupta et al. (1964), and later by many others, to study
power functions of multi-variate tests (e.g. Mudholkar 1965). Mudholkar (1966)
replaced the symmetry in Anderson’s (1955) result by a general group invariance
to obtain G-majorization and G-monotonicity related to majorization properties
(Marshall & Olkin 1979). Vitale (1990) offers a further generalization of
Mudholkar (1966). This result, among other applications, yields numerous
probability inequalities (Mudholkar 1969; Tong 1980; Dalal & Mudholkar 1988;
Dharmadhikari & Joag-Dev 1988) and is used for analysing multi-variate
peakedness (Mudholkar 1972).
(d) Further background and outline
Our starting point is the following:
Theorem 1.2 (Mudholkar & Wang 2007). Let f be the pdf of a unimodal random
variable that is R-symmetric about 1, then f belongs to the closed convex hull of
the set of all uniform distributions on R-symmetric intervals (1/a, a) with a > 1,
or equivalently, f is the pdf of UZ + (1 − U )/Z , where Z > 1 and U is uniform
on (0,1), independent of Z .
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Theorem 1.2 for R-symmetric unimodal random variables may be seen in
parallel to Shepp’s representation of a symmetric unimodal random variable on
the whole real line. Mudholkar & Wang (2007) proved theorem 1.2 by establishing
the following Khintchine-type representation of the distribution function of an
R-symmetric random variable unimodal around 1,
∞
Wa (x) dQ(a),
(1.7)
F (x) =
1
where Wa (x) represents the distribution function of a random variable distributed
uniformly on (1/a, a), a > 1 and Q is a distribution function defined on the
interval [1, ∞). Mudholkar & Wang (2009) used theorem 1.2 to establish the
monotonicity of the power function of the test of significance for the IG mean.
In the remainder of this paper, we treat the canonical cases in which Y follows
distributions with q = 1 or d = 1. In this case, unimodal R-symmetric distributions
will have their modes at 1. The general cases can be reconstructed from these
by considering the distributions of qY and dY , respectively. That is, all results
remain valid if q = 1 or d = 1 provided the distributions are rescaled appropriately.
For example, in theorem 1.2, if f is R-symmetric about q, then it is the pdf of
VZ + (q − V )/Z , where V is uniform on (0, q).
Two particularly important R-symmetric distributions, mentioned earlier, that
will be considered from time to time in what follows are the lognormal distribution
which, when q = 1, has density
1
1
2
2
(1.8)
exp − (log x + m ) x > 0
fL (x) = √
2m
2pm
and the RRIG distribution (Mudholkar & Wang 2007) with density
l
1 2
2l
exp −
x−
x >0
fR (x) =
p
2
x
(1.9)
when q = 1; here, m, l > 0.
In §2, we present an alternative form of theorem 1.2, identify the distribution
Q for a given R-symmetric pdf f and study its consequences. Section 3 gives
a one-to-one correspondence between a symmetric family of distributions (on
R around 0) and an R-symmetric family (defined on R+ with mode at 1)
and explores further relations with log-symmetry and R-symmetry. Conclusions
follow in §4.
2. A Khintchine-type theorem and its consequences
(a) A Khintchine-type theorem for R-symmetric distributions
Theorem 2.1 is very similar to theorem 1.2, which is theorem 5.4 of Mudholkar &
Wang (2007). The only difference is that in theorem 1.2, Z has support (1, ∞)
while in theorem 2.1 Z has support (0, 1). The theorem provides a striking and
attractive property of R-symmetry.
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Theorem 2.1. Let f be the pdf of a unimodal random variable that is
R-symmetric about 1, then f belongs to the convex hull of all symmetric densities
on R-symmetric intervals [a, 1/a] with 0 < a < 1 or, equivalently, f is the pdf of
X = U /Z + (1 − U )Z , where Z is defined on (0, 1) and distributed independently
of the uniform (0, 1) random variable U .
Similar to equation (1.7), the distribution function F (x) of an R-symmetric
unimodal random variable can be expressed as
1
(2.1)
F (x) = Wz (x)dQ(z),
0
where Q is a distribution function defined on (0, 1) and Wz (x) is the distribution
function of a uniform (z, 1/z), 0 < z < 1, random variable. The following theorem
addresses the 1–1 correspondence between the distributions of X and Z .
Theorem 2.2. Let X be a non-negative random variable that is R-symmetric
and unimodal about 1, with distribution function F and pdf f , then the random
variable Z is uniquely defined by its distribution function Q and pdf q (assuming
that f exists) given by
1
1
Q(z) = F (z) + 1 − F
+
− z f (z), 0 < z < 1
(2.2)
z
z
and
1
q(z) =
− z f (z),
z
0 < z < 1.
(2.3)
Proof. Consider some distribution Q on (0, 1). Then, we can use the
representation in theorem 2.1 to write
⎧ x
(x − z)
⎪
⎪
dQ(z),
if x < 1,
⎪
⎨ 0 (1/z) − z
F (x) =
1/x
⎪
⎪
(1/z) − x
⎪1 −
⎩
dQ(z), if x ≥ 1.
(1/z) − z
0
Also,
f (x) =
min(x,1/x)
0
1
dQ(z).
(1/z) − z
(2.4)
We find that the Radon–Nikodym derivative of Q with respect to f is 1/x − x.
Therefore, for 0 < x < 1,
x 1
Q(x) =
− z df (z).
z
0
Integrating by parts and noting that f (1/x) = f (x) and that xf (x) → 0 as x → ∞,
we get equation (2.2) for all continuity points x on (0, 1). At the remaining points,
we determine Q by right continuity, proving that F determines Q uniquely. In
particular, if f (z) = F (z) exists and is continuous almost everywhere, then Q
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has the density q given by equation (2.3). It can be directly checked that q
is a density for any unimodal R-symmetric f ; its non-negativity corresponds
to the unimodality of f and the demonstration of its unit integral uses the
R-symmetry of f .
Alternative approach. A derivation of theorem 2.1 and formula (2.2) that
parallels the development of Khintchine’s theorem by Jones (2002) provides
a useful geometric explanation of various aspects of unimodal R-symmetric
distributions. Let X ∼ f , f being the pdf of an R-symmetric unimodal random
variable with mode at 1, so that f (0) = f (∞) = 0, and define random variable
Y by
(X , Y ) ∼ fX ,Y (x, y) = I (0 ≤ y ≤ f (x)),
where I (E) is the indicator function of the event E. Furthermore, let
f (x) = f (x)I (0 < x < 1) and fr (x) = f (x)I (1 < x < ∞). Then, we have the
following lemma:
Lemma 2.3. (i) The conditional distribution of X given Y is given by
1
X |Y = y ∼ U f−1 (y), −1
f (y)
(2.5)
and (ii)
U
(2.6)
+ (1 − U )Z , where Z = f−1 (Y ).
Z
Proof. Note that owing to unimodality of f , it is increasing for 0 < x < 1 and
decreasing for 1 < x < ∞. Thus, fr and f are invertible and we have fr−1 (y) =
1/f−1 (y), 0 < y < f (1). From this, it is easy to see that equation (2.5) holds and
thus, unconditionally, that equation (2.6) holds.
X=
Lemma 2.4. For 0 < y < 1, the distribution function of Y is given by
1
1
−1
−1
FY (y) = F (f (y)) + y
− f (y) + 1 − F
.
f−1 (y)
f−1 (y)
(2.7)
Proof. For 0 < y < f (1), we have
y ∞
I (0 ≤ z ≤ f (x)) dx dz
FY (y) =
0 0
∞
=
min{y, f (x)} dx
0
=
f −1 (y)
0
f (x) dx +
1/f −1 (y)
f−1 (y)
y dx +
∞
1/f−1 (y)
Equation (2.7) now easily follows from the above equation.
≤ z) = P(f−1 (Y ) ≤ z) = FY (f (z)),
f (x) dx.
using lemma 2.4, equations
As Q(z) = P(Z
(2.2) and (2.3) follow.
The following examples provide the densities of Z corresponding to the familiar
R-symmetric lognormal and RRIG distributions.
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12
µ=2
µ = 1.5
µ=1
µ = 0.5
µ = 0.1
µ = 0.01
10
qL(z)
8
6
4
2
0
0
0.2
0.4
0.6
0.8
1.0
z
Figure 1. Pdfs qL for m = 0.01, 0.1, 0.5, 1, 1.5, 2.
Example 2.5. For the lognormal distribution, fL (x) = (− log x)fL (x)/(mx) and
hence, for 0 < z < 1
1 1 1 − z 2 (− log z)fL (z)
qL (z) =
2
mz
1
1 1
2
2
2
=
(2.8)
1 − z (− log z) exp − (log z + m ) .
2m
2pm3 z 2
This pdf is plotted for different values of m in figure 1, which shows that the
density of Z is more concentrated near 0 for large values of m and near 1 for small
values of m.
Example 2.6. For the RRIG distribution, fR (x) = l 1 − x 4 fR (x)/x 3 and so for
0<z <1
1
qR (z) = l 4 (1 − z 2 )(1 − z 4 )fR (z)
z
2 1
2l3 1
l
z−
(1 − z 2 )(1 − z 4 ) exp −
.
(2.9)
=
p z4
2
z
It is interesting to note that if Z ∼ qR , then Y = l(Z − 1/Z )2 follows the c2
distribution on three degrees of freedom. Figure 2 gives plots of qR (z) for various
values of l which shows that Z is more concentrated near 0 for small values of l
and near 1 for large values of l.
(b) A link between R-symmetric and log-symmetric random variables
Theorems 1.2 and 2.1 are both analogues of Khintchine’s theorem for
R-symmetric random variables that differ only in the support of the random
variable Z ; in theorem 2.1, the support is (0, 1), whereas in theorem 1.2 it is
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qR(z)
15
l = 0.01
l = 0.1
l = 0.5
l=1
l = 1.5
l=2
10
5
0
0
0.2
0.4
0.6
0.8
1.0
z
Figure 2. Pdfs qR for l = 0.01, 0.1, 0.5, 1, 1.5, 2.
[1, ∞). However, these two can be combined to express a unimodal R-symmetric
Y in terms of a uniform (0, 1) random variable U and a log-symmetric random
variable Zm as at equation (2.10).
The link between the two versions of reciprocal symmetry relies on the fact
that if f is R-symmetric, then f (y) = −f (1/y)/y 2 , which leads to the following:
Theorem 2.7. An R-symmetric random variable Y can be expressed as
1
1
Y = U max Zm ,
+ (1 − U ) min Zm ,
(2.10)
Zm
Zm
in terms of a uniform (0,1) random variable U and an independent log-symmetric
random variable Zm , where Zm is Z with probability 1/2 and Z1 = 1/Z with
probability 1/2.
Proof. Now, while the density of Z in theorem 2.2 is given by equation (2.3),
which is on support 0 < z < 1, the density of Z1 ≡ 1/Z can readily be shown to
have the same form, just on a different support:
1
− z1 f (z1 ), 1 < z1 .
q1 (z1 ) =
z1
If we now define Zm as in the statement of the theorem, its density is
1 1
− zm f (zm ), 0 < zm ,
qm (zm ) =
2 zm
(2.11)
and Zm has the complementary property of log-symmetry! This can be easily
verified as 1/Zm has the density given by
1
1
1 1
1 1
− 2 =
− zm f − zm f (zm ) = qm (zm ).
2 zm
zm
zm
2 zm
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Concretely, combining theorems 1.2 and 2.1, an R-symmetric random variable,
Y , can be written in terms of a uniform random variable, U , and the above
log-symmetric random variable, Zm , via equation (2.10).
Remark 2.8. If f (.) is smooth and well behaved in the sense that f (1) = 0,
f (1) < 0 and f (1) is finite, then the densities associated with Zm are not
unimodal. For example,
1
1
1
− z f (z) − 1 + 2 f (z)
qm (z) =
2
z
z
so that qm (1) = 0 and
qm (z) =
1
2
1
1
2f (z)
− z f (z) − 2 1 + 2 f (z) +
z
z
z3
so that qm (1) = −2f (1) > 0: qm has an antimode at z = 1.
Remark 2.9. Now, define Ym = log Zm . By the definition of log-symmetry, the
distribution of Ym , with density
pm (x) = 12 (1 − e2x )f (ex ),
x ∈ R,
(2.12)
is symmetric. However, a similar analysis shows that, under the same
assumptions, pm also has an antimode at its centre, x = 0. It seems that typically
qm and pm are bimodal densities.
(c) Distributions for Z specified
We have seen that a smooth R-symmetric unimodal random variable X can
be expressed in terms of a uniform random variable U on (0,1) and a random
variable Z . In this section, we explore some examples of distributions for Z and
their consequences for the distribution of X . They are somewhat instructive in the
sense that a distribution for Z on the interval (0, 1) will produce an R-symmetric
distribution, but not necessarily a smooth one.
Example 2.10. If Z is uniform on (0,1), then the distribution of X is given by
⎧
⎨− 12 log(1 − x 2 ),
if 0 < x < 1,
(2.13)
fU (x) =
⎩log x − 1 log(x 2 − 1), if x ≥ 1.
2
Of course, this density tends to ∞ as x → 1. Few other choices for q are so
accommodating and most result in rather nasty special function densities for f .
This goes, for example, in general, for beta densities. (We shall not write out any
such rebarbative formulae.)
Example 2.11. Formula (2.1) can be written as
−(1/2) log(1−M 2 (x)) q( 1 − e−2w ) dw,
f (x) =
(2.14)
0
where M (x) = min(x, 1/x). Candidates for simple results are therefore densities
that can be written as simple functions of (1 − z 2 ). For example, truncate a
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p=6
p=5
p=4
p=3
p=2
p=1
fp(x)
0.6
0.4
0.2
0
0
2
4
6
8
10
x
Figure 3. Pdfs fp for p = 1, 2, 3, 4, 5, 6.
symmetric beta distribution on [−1, 1], with parameter p > 0 say, on to support
[0, 1]. This has density
1
22(p−1) B(p, p)
(1 − z 2 )p−1 =
2
(1 − z 2 )p−1 ,
B((1/2), p)
0 < z < 1.
The resulting unimodal R-symmetric density for p = 1 is
fp (x) =
1
{1 − (1 − M 2 (x))p−1 },
(p − 1)B((1/2), p)
(2.15)
where M (x) is again min(x, 1/x). (As p → 1, fp (x) → fU (x) above.) For integer
p > 1, fp (1) is finite and fp is p − 2 times continuously differentiable at 1.
Figure 3 gives plots of fp (x) for various values of p. This demonstrates the
degree of smoothness of f at its mode, and it is seen that as p → 1 the density
tends to ∞ for x → 1.
(d) Unique representation of R-symmetry
It is tempting to consider random variables X as in theorem 2.1 by replacing U
by a random variable V that is not uniform. One such is to take V ∼ beta(n, 1)
for n a positive integer. This is the analogue in the ordinary unimodal case of the
‘a-unimodality’ as defined by Olshen & Savage (1970); see also Dharmadhikari &
Joag-Dev (1988) and Bertin et al. (1997). The pdf of X as an extension of equation
(2.4) in this case is
M (x)
(x − z)n−1
q(z) dz.
(2.16)
f (x) = n
(1/z − z)n
0
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However, we can show that only the uniform distribution for U preserves the
R-symmetry. To prove this, note that for Z ∼ q and V ∼ h, for arbitrary densities
q and h on (0, 1), the density of X = V /Z + (1 − V )Z is given by
f (x) =
M (x)
0
1
h
1/z − z
x −z
q(z) dz.
1/z − z
For f to be R-symmetric, we would also need
M (x)
1/x − z
1
1
h
q(z) dz
=
f
1/z − z
1/z − z
x
0
(2.17)
(2.18)
to equal f (x). This is clearly so if h is uniform and clearly not so otherwise,
as the following simple argument shows. Fix on any non-uniform h, such as
beta(n, 1), then for equation (2.17) to hold, the appropriate choice of q depends
on f . On the other hand, any q that satisfies equality of the right-hand sides
of equations (2.17) and (2.18) would not depend on f . Thus, it shows that
(unimodal) R-symmetric X s cannot satisfy X = V /Z + (1 − V )Z , V and Z
defined on (0, 1) and independent, for any V other than uniform.
3. Symmetric distributions associated with R-symmetric distributions
(a) A general representation of R-symmetric distributions
Following Boros & Moll (2004, §13.2), Baker (2008) offers the following result:
∞
∞ b 2
f
cx −
f (u 2 ) du, c > 0, b ≥ 0,
(3.1)
dx =
c
cx
0
0
as a version of the classical, but not so well-known, Cauchy–Schlömilch transformation (for which see Amdeberhan et al. submitted). The original transformation was used to evaluate seemingly intractable integrals; the version at equation
(3.1) is useful for applications in probability and statistics. Thus, it follows that
if g(u), u ≥ 0 is a pdf, termed the mother pdf by Baker (2008), then f (x) =
cg(|cx − b/cx|) is also a pdf√
for x ≥ 0, referred to as the daughter pdf. Moreover,
f is R-symmetric about q = b/c. Note that there is a one-to-one correspondence
between the daughter pdf f and the mother pdf g, f being obtained from g by
shifting and redistributing its probability mass. The following theorem clarifies
the correspondence between the symmetric distributions on the real line and
the R-symmetric distributions with non-negative real support. Without
√ loss of
generality, set the scale parameter c = 1 and the centre of R-symmetry b = 1.
Theorem 3.1. If g(x) is the pdf of a symmetric real-valued random variable X
defined on R, i.e. g(x) = g(−x) for all x ∈ R, then
1
, x > 0,
(3.2)
f (x) = 2g x −
x
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R-symmetry of probability distributions
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is an R-symmetric density and conversely, any R-symmetric density f gives rise
to an ordinary symmetric g on R through
g(x) = f (x + 1 + x 2 ).
(3.3)
(This neat form actually corresponds to a rescaling of g relative to equation (3.2).)
Proof. Let g be the density of an ordinary symmetric distribution on R, then
R-symmetry of f is obvious. The fact that f is necessarily a density can be derived
from equation (3.1). Explicitly,
∞
∞ ∞
1
1
1
f (x) dx =
g x−
g
dx +
− w dw
2
x
w
0
0
0 w
∞ 1
1
=
g x−
1 + 2 dx
x
x
0
∞
g(z) dz = 1.
=
−∞
Here, the substitutions w = 1/x and then z = x − 1/x were used. The converse
follows similarly.
Remark 3.2. The density f defined in equation (3.2) is unimodal if and only if
g is unimodal.
Below are given some examples of R-symmetric densities obtained from some
familiar choices of g.
Example 3.3. Normal g gives rise, in this way, to the RRIG f , equation (1.9).
Example 3.4. A suitably scaled t density on n degrees of freedom has density
proportional to (2 + x 2 )−(n+1)/2 , and gives rise to a certain scaled ‘generalized F ’
R-symmetric density with density proportional to x n+1 /(1 + x 4 )(n+1)/2 (this is the
density of a suitably scaled Fn/2+1,n/2 random variable raised to the 1/4 power).
Example 3.5. The symmetric version of the hyperbolic distribution of
Barndorff-Nielsen (1977) on R suitably scaled has density
1
g(x) =
exp(−x 4 + x 2 )
(3.4)
4K1 (x)
for x > 0 and K1 a Bessel function (see also Barndorff-Nielsen & Blaesild 1983).
Its R-symmetric counterpart from equation (3.2) is the positive hyperbolic
distribution with its two parameters equal to x (see also Barndorff-Nielsen &
Blaesild 1983); it has density
1
1
exp −x x +
(3.5)
f (x) =
2K1 (x)
x
on R+ .
Conversely, some examples of not-so-familiar symmetric unimodal densities
obtained from R-symmetric unimodal densities follow.
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Y. P. Chaubey et al.
p=6
p=5
p=4
p=3
p=2
p=1
gp(x)
0.6
0.4
0.2
0
−3
−2
−1
0
x
1
2
3
Figure 4. Pdfs gp for p = 1, 2, 3, 4, 5, 6.
Example 3.6. Lognormal f gives rise, in this way, to the novel symmetric
unimodal density
1
e−m/2
−1
2
exp − {sinh (x)} .
(3.6)
g(x) = √
2m
2pm
Example 3.7. Density (2.13) yields another remarkable novel symmetric
unimodal density, albeit one with an infinite spike at the origin:
g(x) = − 12 log 2|x|( 1 + x 2 − |x|) .
(3.7)
Example 3.8. Better behaved novel symmetric unimodal densities arise in
similar fashion from equation (2.15):
p−1 1
2
gp (x) =
.
(3.8)
1 − 2|x|( 1 + x − |x|)
(p − 1)B((1/2), p)
Figure 4 gives plots of gp (x) for various values of p that clearly show symmetric
densities, which have fatter tails as p becomes larger. Density (3.8) tends to
density (3.7) as p → 1; so density (3.7) is also shown in figure 4.
Remark 3.9. Note that the equivalent transformation of variables relating Y on
R with X on R+ through Y = X − (1/X ) (Jones 2007) is quite different. It relates
to log-symmetric distributions in the sense that if X follows a log-symmetric
distribution on R+ , then Y follows an ordinary symmetric distribution on
R. This—along with various aspects of the development above—is because
X − 1/X = 2 sinh(log X ).
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R-symmetry of probability distributions
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(b) A Khintchine-type match-up between R-symmetric and log-symmetric
distributions
We can now link the general formulation (3.2) with a general form for q. It is
the case that
1
1
f (x) = 2 1 + 2 g x −
,
(3.9)
x
x
so that
2 1
4
.
q(z) = 3 1 − z g z −
z
z
(3.10)
For any such distribution, if Z ∼ q, then Y = 1/Z − Z follows the distribution
with density (y) ≡ −2yg (y), y > 0.
Now, once more let U ∼ U (0, 1) and, independently, Y ∼ (y), y > 0. Then,
R-symmetric (R) and log-symmetric (L) random variables can be represented in
terms of Y as follows.
— We know from theorem 2.1 that for unimodal R-symmetric distributions,
the associated random variable R can be written as
1 1
(3.11)
− Z = { Y 2 + 4 + (2U − 1)Y }.
R=Z +U
Z
2
— It is also the case, however, that Khintchine’s theorem for ordinary
symmetric unimodal distributions (random variable X , say) on the real line
can be written as X = (2U − 1)Y . But a log-symmetric random variable L
is of the form L = eX . It follows that for log-symmetric distributions based
on unimodal g, the associated random variable L can be written as
L = exp{(2U − 1)Y }.
(3.12)
This seemingly provides an intriguing match-up between unimodal R and L
based on unimodal g in terms of different functions of U and Y .
4. Conclusions
The results of §§2 and 3 have given a number of insights into the theoretical role
and consequences of R-symmetry. Section 2 considered in detail a Khintchine–
type theorem for R-symmetric distributions and various important specific
distributions associated with it. Section 3 considered the equivalence between
R-symmetric distributions and distributions composed of ordinary symmetric
distributions whose scale is transformed by the Cauchy–Schlömilch device. This
also provided another intriguing parallel between R- and log-symmetry.
The above are not direct practical consequences of our work but underlie some
important practical questions, most notably that of the role of IG distributions
in statistical practice. In §1b, we stressed the many G–IG analogies that make
the IG distribution such an attractive option for modelling non-negative data.
These analogies have until now retained an air of mystery: they are true, but
why are they true? The central role of the IG distribution in R-symmetry,
through the RRIG distribution, provides an answer; in particular, our one-line
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example 3.3 makes the R-symmetry/Cauchy–Schlömilch link between RRIG and
G distributions, which seems to be the real driver of these analogies. We do
not have space to expand on this in detail here, but note, as just one example,
that the maximum entropy characterization of the RRIG distributions follows
directly by combining Shannon’s characterization of the G distribution with the
Cauchy–Schlömilch transformation.
The research of the first author was partially supported from the author’s Discovery grant from
the Natural Sciences and Engineering Research Council of Canada. The authors are very grateful
to the referees for their fair and helpful remarks.
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