## Abstract

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.

## 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.

### (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 . There are two natural and complementary concepts of reciprocal symmetry of *f*. First, there is R-symmetry (Mudholkar & Wang 2007), which means that
1.1
for all *y*>0 and some *θ*>0. Second, there is log-symmetry (Seshadri 1965), so-called because it corresponds to ordinary symmetry of the distribution of . In density terms,
1.2
for all *y*>0 and some *δ*>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 §1*b* 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 *θ*=*δ*. It is also easy to see that the lognormal distribution corresponding to is both R-symmetric about *θ*=*e*^{μ−σ2} and log-symmetric about *δ*=*e*^{μ}. 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*=*δ*/*θ*. Then, a doubly symmetric density *f** takes a piecewise form derived in a closely related context by Pakes (1996), namely
1.3
together with the additional requirement that *ψ*(*u*)=*ψ*(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 non-negative 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 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 non-negative 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
1.4
where *V*_{z}(*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
1.5
In the case that *f*′(*x*)=*F*′′(*x*) exists and is continuous everywhere, the density *h*(*x*) is given by
1.6

This leads to the following characterization of symmetric unimodal (with mode at 0) distributions (Dharmadhikari & Joag-Dev 1988, p. 5).

### 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 V Z′, 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.*

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,
1.7
where *W*_{a}(*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 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 *θ*=1 or *δ*=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 *θY* and *δY* , respectively. That is, all results remain valid if *θ*≠1 or *δ*≠1 provided the distributions are rescaled appropriately. For example, in theorem 1.2, if *f* is R-symmetric about *θ*, then it is the pdf of *V* *Z*+(*θ*−*V*)/*Z*, where *V* is uniform on (0,*θ*).

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 *θ*=1, has density
1.8
and the RRIG distribution (Mudholkar & Wang 2007) with density
1.9
when *θ*=1; here, *μ*,*λ*>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 around 0) and an R-symmetric family (defined on 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 while in theorem 2.1 *Z* has support (0,1). The theorem provides a striking and attractive property of R-symmetry.

### 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
2.1
where *Q* is a distribution function defined on (0,1) and *W*_{z}(*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*
2.2
*and*
2.3

### Proof.

Consider some distribution *Q* on (0,1). Then, we can use the representation in theorem 2.1 to write
Also,
2.4
We find that the Radon–Nikodym derivative of *Q* with respect to *f* is 1/*x*−*x*. Therefore, for 0<*x*<1,
Integrating by parts and noting that *f*(1/*x*)=*f*(*x*) and that *xf*(*x*)→0 as , 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* 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 and define random variable *Y* by
where *I*(*E*) is the indicator function of the event *E*. Furthermore, let *f*_{ℓ}(*x*)=*f*(*x*)*I*(0<*x*<1) and Then, we have the following lemma:

### Lemma 2.3.

(i) *The conditional distribution of X given Y is given by*
2.5

*and*(ii) 2.6

### Proof.

Note that owing to unimodality of *f*, it is increasing for 0<*x*<1 and decreasing for . Thus, *f*_{r} and *f*_{ℓ} are invertible and we have 0<*y*<*f*(1). From this, it is easy to see that equation (2.5) holds and thus, unconditionally, that equation (2.6) holds. ■

### Lemma 2.4.

*For* 0<*y*<1, *the distribution function of Y is given by*
2.7

### Proof.

For 0<*y*<*f*(1), we have
Equation (2.7) now easily follows from the above equation. ■

As , using lemma 2.4, equations (2.2) and (2.3) follow.

The following examples provide the densities of *Z* corresponding to the familiar R-symmetric lognormal and RRIG distributions.

### Example 2.5.

For the lognormal distribution, and hence, for 0<*z*<1
2.8
This pdf is plotted for different values of *μ* in figure 1, which shows that the density of *Z* is more concentrated near 0 for large values of *μ* and near 1 for small values of *μ*.

### Example 2.6.

For the RRIG distribution, and so for 0<*z*<1
2.9
It is interesting to note that if *Z*∼*q*_{R}, then *Y* =*λ*(*Z*−1/*Z*)^{2} follows the *χ*^{2} distribution on three degrees of freedom. Figure 2 gives plots of *q*_{R}(*z*) for various values of *λ* which shows that *Z* is more concentrated near 0 for small values of *λ* and near 1 for large values of *λ*.

### (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 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 *Z*_{m} 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*
2.10
*in terms of a uniform (0,1) random variable U and an independent log-symmetric random variable Z*_{m}*, where Z*_{m} *is Z with probability 1/2 and Z*_{1}*=*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 *Z*_{1}≡1/*Z* can readily be shown to have the same form, just on a different support:
If we now define *Z*_{m} as in the statement of the theorem, its density is
2.11
and *Z*_{m} has the complementary property of log-symmetry! This can be easily verified as 1/*Z*_{m} has the density given by
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, *Z*_{m}, 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 *Z*_{m} are not unimodal. For example,
so that *q*′_{m}(1)=0 and
so that *q*′′_{m}(1)=−2*f*′′(1)>0: *q*_{m} has an antimode at *z*=1.

### Remark 2.9.

Now, define . By the definition of log-symmetry, the distribution of *Y*_{m}, with density
2.12
is symmetric. However, a similar analysis shows that, under the same assumptions, *p*_{m} also has an antimode at its centre, *x*=0. It seems that typically *q*_{m} and *p*_{m} 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
2.13
Of course, this density tends to as . 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
2.14
where . Candidates for simple results are therefore densities that can be written as simple functions of (1−*z*^{2}). For example, truncate a symmetric beta distribution on [−1,1], with parameter *p*>0 say, on to support [0,1]. This has density

The resulting unimodal R-symmetric density for *p*≠1 is
2.15
where *M*(*x*) is again . (As *p*→1, *f*_{p}(*x*)→*f*_{U}(*x*) above.) For integer *p*>1, *f*_{p}(1) is finite and *f*_{p} is *p*−2 times continuously differentiable at 1.

Figure 3 gives plots of *f*_{p}(*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 ‘α-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
2.16
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
2.17
For *f* to be R-symmetric, we would also need
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:
3.1
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 . 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 .

### Theorem 3.1.

*If g(x) is the pdf of a symmetric real-valued random variable X defined on* *, i.e. g(x)=g(−x) for all* *then*
3.2
*is an R-symmetric density and conversely, any R-symmetric density f gives rise to an ordinary symmetric g on* *through*
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 then R-symmetry of *f* is obvious. The fact that *f* is necessarily a density can be derived from equation (3.1). Explicitly,
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 *ν* degrees of freedom has density proportional to (2+*x*^{2})^{−(ν+1)/2}, and gives rise to a certain scaled ‘generalized *F*’ R-symmetric density with density proportional to *x*^{ν+1}/(1+*x*^{4})^{(ν+1)/2} (this is the density of a suitably scaled *F*_{ν/2+1,ν/2} random variable raised to the 1/4 power).

### Example 3.5.

The symmetric version of the hyperbolic distribution of Barndorff-Nielsen (1977) on suitably scaled has density
3.4
for *ξ*>0 and *K*_{1} 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 *ξ* (see also Barndorff-Nielsen & Blaesild 1983); it has density
3.5
on .

Conversely, some examples of not-so-familiar symmetric unimodal densities obtained from R-symmetric unimodal densities follow.

### Example 3.6.

Lognormal *f* gives rise, in this way, to the novel symmetric unimodal density
3.6

### Example 3.7.

Density (2.13) yields another remarkable novel symmetric unimodal density, albeit one with an infinite spike at the origin: 3.7

### Example 3.8.

Better behaved novel symmetric unimodal densities arise in similar fashion from equation (2.15):
3.8
Figure 4 gives plots of *g*_{p}(*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 with *X* on 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 , then *Y* follows an ordinary symmetric distribution on . This—along with various aspects of the development above—is because .

### (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
3.9
so that
3.10
For any such distribution, if *Z*∼*q*, then *Y* =1/*Z*−*Z* follows the distribution with density ℓ(*y*)≡−2*yg*′(*y*), *y*>0.

Now, once more let *U*∼*U*(0,1) and, independently, *Y* ∼ℓ(*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 3.11— 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*=(2*U*−1)*Y*. But a log-symmetric random variable*L*is of the form*L*=*e*^{X}. It follows that for log-symmetric distributions based on unimodal*g*, the associated random variable*L*can be written as 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 §1*b*, 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 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.

## Acknowledgements

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.

## Footnotes

- Received September 15, 2009.
- Accepted January 14, 2010.

- © 2010 The Royal Society