Scale Mixtures of Normal Distributions

Scale Mixtures of Normal Distributions
Author(s): D. F. Andrews and C. L. Mallows
Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 36, No. 1
(1974), pp. 99-102
Published by: Blackwell Publishing for the Royal Statistical Society
Stable URL: http://www.jstor.org/stable/2984774
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1974] 99
Scale Mixtures of Normal Distributions
By D. F. ANDREWS and C. L. MALLOWS
Bell Laboratories
and University of Toronto
[Received April 1973]
Bell Laboratories
SUMMARY
This paper presents necessary and sufficient conditions under which a random
variable X may be generated as the ratio ZI V where Z and V are independent
and Z has a standard normal distribution. This representation is useful in
Monte Carlo calculations. It is established that when 7 V2 is exponential,
X is double exponential; and that when WV has the asymptotic distribution
of the Kolmogorov distance statistic, X is logistic.
Keywords: SCALE MIXTURES; MONTE CARLO; LOGISTIC; STABLE
1. INTRODUCTION
RECENTLY Relles (1970) and Andrews et al. (1972) have exploited some properties
of the normal distribution in Monte Carlo evaluations. These techniques are useful
when sampling from (symmetric) distributions which can be generated as the ratio
X = Z/ V where Z has a standard normal distribution and V is positive and is independent
of Z.
Given X, it is of interes to know whether such a representation exists and to
determine ways of finding the distribution of the denominator V. Beale and Mallows
(1959) studied this problem; the conditions presented here extend their results and
involve the density function and its derivatives. In some cases they may be easier
to check than the characteristic-function results presented by Beale and Mallows.
2. THE FUNDAMENTAL RELATION
Let the distribution function (d.f.) of V be Gv. It is easy to see that if X = Z/ V
then for all non-zero x, X necessarily has a density fxy(x) given by
fx(x) = (27)A v exp {- 2V2 X2} dGv(v) (2.1)
and fx(O) exists if and only if E(V) is finite. We now show that the density of X is
essentially the Laplace transform of a function related to the d.f. Gv.
Lemma. If h(y) = fx(Jy), h is the Laplace transform of H where
H(t)=
{(ufT)l dGv{(2u)}.
Proof. Put v = (2u)' in (2.1).
In many cases tables of Laplace transforms may be used to find H and hence Gv.
Where Gv is differentiable, we have
H'(t) = (2,7T)- G'{(2t)%}, G'(v) = (27T)- H'(-v2).

100 ANDREWS AND MALLOWS - Scale Mixtures of Distributions [No. 1,
Also, wheneverfx(x) has the representation (2.1), the Laplace transform of G can
be obtained directly:
by using the relation
2 fx(x) exp {- 2t2 X-2} dx = fexp {- vt} dG(v)
exp { -(a2 u2 + b2 u2)} du = (7T/2a2)' exp {- ab i }. (2.2)
3. NECESSARY AND SUFFICIENT CONDITIONS
The following theorem establishes necessary and sufficient conditions for this
representation to exist.
Theorem. If X has a density function fx symmetric about 0 then there exist
independent random variables V, Z with Z standard normal such that Z/ V = X if
and only if the derivatives offx satisfy
()dy) fx(Y) >O for y> 0. (3.1)
Proof. The necessity follows from differentiating (2.1) (with x = y2) with respect
toy.
The sufficiency follows from Bernstein's theorem (see Feller, 1966, p. 416, or
Widder, 1946, p. 161), according to which under (3.1) fx(y1) is a Laplace transform
h(y) =fx(yl) =
exp { -yt} dox(t),
where oc(t) is non-decreasing and the integral converges for 0< y < oo. Letting
A(w) = f
rw2/2
V1(27JT)
dc(x{v2), w> O
the density may be written
fx(x) =
(27T)-i v exp {- Iv2 x2} dA(v).
Now A is non-decreasing. It remains to show that
lim {A(u)-A(u-1)} = 1
u->00
so that G(v) = A(v) - A(O) is a proper d.f. Let
f(u, x) =
u
(27T)-i v exp {- v2 x2} dA(v)
so that
ff(u, x) dx = A(u) -A(u-1).
Since the integrand of f(u, x), is positive for all v >0, we have
0 < tfl, wet
r fx() x>

1974] ANDREWS AND MALLOWS - Scale Mixtures of Distributions 101
Furthermore, f(u, x) converges monotonically to f at all x as u -+ co. Hence by the
monotone convergence theorem
and the theorem is proved.
lim ff(u x) dx = f(x) dx = 1
4. EXAMPLES
4.1. The t-distribution density is a special case of the Pearson Type VII density
fX(X) = {aB(,M - 12)}-1 (1 + X2/a2)-m.
Clearly h(y) =fx(yl) has derivatives of alternating sign. The inverse transform
of h(y) is
7T-I{F(m - 2)}-1 a2m-1 tm-i exp {- a2 t}
which when equated to (2r)-1 GV{(2t)I} yields
G'(v)= {M(m- )}-' (2a2 V2)m-1 a2 v exp {--a2 v2}
so that ja2 V2 has a gamma distribution with shape parameter m - 1. Thus aV is X
with 2m -I degrees of freedom.
When m = 1 so that X is Cauchy, V is X with one degree of freedom, i.e. halfnormal.
4.2. The Laplace or double exponential distribution has density
fx(x) = 1exp{-IxI}.
A simple induction will establish that h(y) =fx(y4) has derivatives of alternating
sign. The inverse transform of h(y) is
4(-rt3)-exp {- (4t)-1}
which when equated to (2a)-' GV{(2t)I} yields
G'(v) - V-3 exp {-(2v2)-l}.
That is, (2V2)-' has an exponential distribution.
4.3. The logistic distribution has density
so that
fx(x)
h(y)
Now h(y) is the transform of
whence
00
exp {-x} [1 + exp {-x}]-2
00
E (- l)k-l k exp {-ky}.
k=1
z (- l)k-1 k2(47rt3)-'exp {- k2/4t},
k=1
GV(V) = 2 E (-l)k-l k2 V-3 exp {k2/2v2}.
k=1

102 ANDREWS AND MALLOWS - Scale Mixtures of Distributions [No. 1,
Thus here (2 V)-1 has the asymptotic distribution of the Kolmogorov distance statistic.
The result, that 2ZK is logistic when Z and K are independent, normal and asymptotic
Kolmogorov respectively, can be established directly using the integral (2.2) but we
know of no probabilistic interpretation of this relation.
To use this result in Monte Carlo work we have to be able to generate pseudorandom
variables having the asymptotic Kolmogorov distribution. One possibility
is to use the relation found by Watson (1961), namely
2K2- E W/j2,
j=1
where W1, W2,... are independent exponential variables. This and some other ideas
are currently under consideration.
4.4. One case where the present methods are of little help is when X is stable (and
symmetric) with exponent a < 2. It is known (Feller, 1966, p. 172) that such a variable
can be represented as X = Z/ V where 1/V2 is (positive) stable with exponent -x.
However, if aO 1 it is not clear how to prove this by the present method. (Also we
do not know how to generate pseudo-random positive stable variables.)
REFERENCES
ANDREWS, D. F., BICKEL, P. J., HAMPEL, F. R., HUBER, P. J., ROGERS, W. H. and TUKEY, J. W.
(1972). Robust Estimates of Location: Survey and Advances. Princeton: Princeton University
Press.
BEALE, E. M. L. and MALLOWS, C. L. (1959). Scale mixing of symmetric distributions with zero
means. Ann. Math. Statist., 40, 1145-1151.
FELLER, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. II. New York:
Wiley.
RELLES, D. A. (1970). Variance reduction techniques for Monte Carlo sampling from Student
distributions. Technometrics, 12, 499-515.
WATSON, G. S. (1961). Goodness-of-fit tests on a circle. Biometrika, 48, 109-114.
WIDDER, D. V. (1946). The Laplace Transform. Princeton: Princeton University Press.