# Statistical analysis of mixture distribution - Smith A.F.M

ISBN 0-470-90763-4

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(a) p is unimodal if 0 ^ A ^ A0, where

A0 = [2(<T4 - CT2 + 1 )3'2 - (2<76 - 3(74 - 3<72 + 2)]1/2/cr,

(b) For A > A0, p is bimodal if and only if n lies in the open interval (zr,, 7i2), where

"r' = l + .'7 ^exp{-i>.f + i[(y,-A)/a]2}, i=l,2,

IX — yt

and {yt,y2) are the roots of

(ct2 - 1 )y3 - A(ff2 - 2)y2 - A2y + Act2 = 0,

with 0 <yx <yz < A.

(c) Otherwise, p is unimodal.

Learning about the components of a mixture

Figure 5.5.1 Regions of bimodality for Example 5.5.1. Bimodality obtains to the right of the dotted curves. Reproduced by permission of the Chief Editor, Scandinavian Actuarial Journal, from Robertson and Fryer (1969)

Figure 5.5.1, from Robertson and Fryer (1969), shows the bimodal range, in terms of n and A, for several values of a. Given er, the region to the right of the dotted curve corresponds to bimodality.

Interesting special results are as follows:

(d) If A ^ 2min(l,rr), p is unimodal.

(e) If A ^ 3 min(l,<7), then (7r,,7r2) exist such that p is bimodal for <n < n2.

(f) If a = 1 (equal variances) there is a critical A„ such that p is bimodal for A > A„: for example, A0 5 = 2 and A0 9 = 3.3

(g) The separation of the modes is less than |p, — p2\.

Clearly, in the current example, less separation is required for bitangentiality than for bimodality. Figure 5.5.2 displays bitangentiality regions obtained from numerical calculations by Robertson and Fryer (1969). They work from the characterization that bitangentiality is equivalent to the existence of four points of inflexion on p(x| tj/), as opposed to just two, and provide a fairly full analysis of the phenomenon. Note that the bimodality regions are, of course, subsets of those for bitangentiality. Note also the following special remarks:

(a) For a2 < 5 — ^/24, there is a range of n for which bitangentiality occurs even

at A = 0.

(b) For <r=l, the critical A„ for bitangentiality is quite stable, as n varies. Wessels (1964) also gives quite a detailed study of this example.

Statistical analysis of Jinite mixture distributions

v

Figure 5.5.2 Regions of bitangentiality for Example 5.5.1. Bitangentiality obtains to the right of the dotted curves. Reproduced by permission of the Chief Editor, Scandinavian Actuarial Journal, from Robertson and Fryer (1969)

Example 5.5.2 Mixture of two multivariate normals Let

p(x) = 7t01,(x|/i„S1) + (1 -7r)4>d(x|/i2,?2),

where (f)Jx |/i. I) denotes the density function for a (/-dimensional normal random vector with mean and covariance matrix X. The modality of the density surface depends on the univariate sections. These can be represented by mixture densities of the form

q{y) = it(f)(y\nl,ol) + (i -n)(l){y\p2yo2), -co <.x< oo,

where Pj = c'nj, 7=1,2

o) = cT I;-c, 7=1,2,

and c is an arbitrary unit vector.

Define

s2/-\ -A*i)Tc(cTSiC + cTI2c)

<5 (c) =

2(cTI1c)(cTX2c)

I hen. from result (e) in Example 5.5.1, there is no bimodality, for any n, if

sup<52(c) < 27/8.

Learning about the components of a mixture ,,,

lOj

Calculation of the extreme c is not trivial. For the special case I, = I, = I. the modes will fall on the section through //, and /i,. Furthermore,

c’lc ’

C'Xl

corresponding to the familiar linear discriminant function. Konstantellos (1980) considers Example 5.5.2 in more detail.

Example 5.5.3 Mixture of two von Mises densities Let

p{x) = 7ri M{x; 0, Ki) + (1 - n j )M(x; 0o, K2), 0 < x < 2n,

0^do^Tt,KltK2>0% where M(x; 0, K) = [In/0(K)]~1 exp [K cos(x — 0)]

and I0(K) is the zero-order modified Bessel function of the first kind (Mardia, 1972, p. 57). Note that 7t, unsubscripted, is used to represent 3.14159.... Detailed conditions for determining whether there is unimodality or bimodality are given by Mardia and Sutton (1975). When 0o = n. bimodality obtains for an intermediate range of 7i,, for any Kj,/C2. For other l)0, the situation is more complex.

Mardia (1972, Sections 3.6, 5.1, and 5.4.3) uses a concatenation of k von Mises densities on (0,2n/k) as a /c-modal density on the circle; bimodal distributions on the sphere are discussed in Mardia (1972, Section 8.5).

In Section 2.2, we mentioned, as alternatives to mixture densities, other parametric densities which may manifest multimodality.

Example 5.5.4 The quartic exponential density Let

p(x) = C(a )exp [ — (a,x + a2x2 + a3x3 + a4x4)], a4 > 0, — oo < x < x ,

where C(af) is the normalizing constant. As mentioned in Section 2.2, the stationarity equation for p(x) is a cubic which has one or three real roots, depending on whether p(x) is unimodal or bimodal. Transformation lo

y = x + a3/4a4 gives

piy) OC exp [ - ([I! y + li2y2 + /*4/)].

where /i, = (8a,a4 — 4a2a3a4 + a3)/8a4,

P2 = (8«2«4 - 3a3)/8a4,

Pi = «4-

IM Statistical analysis of finite mixture distributions

From the theory of cubic equations (Korn and Korn, 1968, p. 23), bimodality follows if and only if < 0. where 6 = 21p2Ji4 + 8 Note that this implies fl2 < 0.

The special form

p(x) oc exp { - [/?(* - p)2 + y(x - p)4]}, y > 0,

provides a three-parameter density which is symmetric about p. Bimodality, with modes at ± v ( - P/2y). occurs if and only if fl < 0 (see Matz, 1978).

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