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

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Ojj = cov(Xh Xj) = [varlX, + cXj) var(A",) c2 var(A^)]/2c.

Day (1969) gives a more natural generalization of the method and points out that, unless d= 1, not all third- and fourth-order moments need be considered in order to define estimates. He needs only all first- and second-order central moments, all marginal third-order central moments, and a final equation which is obtained using a function of the third- and fourth-order central moments which is invariant under rotation of the sample space.

For the simple case in which the common covariance matrix takes the form X = ° moment estimates can be found with the help of moments along the direction of the first principal component (Cooper and Cooper, 1964). More recent developments are presented in Fukunaga and Flick (1983).

No extension of the univariate analysis has been made in the case of unequal covariance matrices. For such examples the method will become more unreliable

because of the potential instability of high-order sample moments (Martin, 1936; Day, 1969).

We conclude this subsection with a survey of some miscellaneous topics and references.

Learning about the parameters of a mixture

l )

John (1970a) considers mixtures of two components of various discrete distributional types.

Falls (1970) combines graphical and moment methods for mixtures of two-parameter Weibulls, but appears to assume that the moment estimators are asymptotically efficient.

As usual, mixtures of uniform densities do not fit into the general pattern. Gupta and Miyawaki (1978) consider three cases of mixtures of Un(0 0) and Un(0,1), 0 < 0 < 1:

(a) 7rj, n2 known: first moment used;

(b) 0 known: first moment used;

(c) 7tj, n2, 9 unknown: first two moments used.

Explicit solution is possible and the asymptotic theory is described.

Direct use of fractional moments appears in Joffe (1964), where the following model is proposed for the grain size distribution of mine dust:

p(x\i//) = nAlexp{-6l -/*) + (! - kM2 exp(- 02 yjx\ x > x0,

where At, A2 are normalizing constants and x0 is the known minimum grain size. Moment equations for the first three half-moments are solved numerically for 71, 0,, and 92\ for details see Johnson and Kotz (1970a, Section 18.10) and Everitt and Hand (1981, Section 3.2.1).

Cohen (1965) estimates a two-component Poisson mixture using the first two sample moments along with a third equation based on the frequency in the zero cell. He also uses estimation based on factorial moments for two specific problems: a mixture of two Poisson distributions with missing zero-cell frequencies and a mixture of a Poisson with a binomial Bi{N,9), with N known (Everitt and Hand, 1981, Section 4.4).

We have commented on the phenomenon of non-real or non-feasible parameter estimates. For two-component mixtures of the general type of Example 4.2.6 (see later), Ord (1972, Section 4.6) chooses estimators which fit the moment equations as well as possible, in a well-defined sense, subject to their lying in the parameter space. He then shows that the problem can be given a linear

programming formulation.

Brownie, Habicht, and Robson (1983) consider a semi-non-parametric

mixture of a normal component with an arbitrary second component, under the assumption that the probability mass associated with the latter is concentrated completely on one side or other of the normal mean. An interesting procedure for estimating the mixing weight is developed from moment equations generated by the indicator function t( ) introduced in Example 4.2.1.

4.2.3 Mixtures of k densities

The method of solving the moment equations (4.2.6) generalizes to a wide range of mixture problems.

Statistical analysis of finite mixture distributions

Example 4.2.6 A general class of k component mixtures Suppose

P(*IĞA)= Z njf(xıj)

J= i

and r,(x) is such that r0(x) = 1 and

= 0s, s = 0, l,...,2/c- 1.

Suppose also that

n

= n "1 Z ls(xi)-1= 1

Then we have a set of moment equations in 0;- (j = 1, , Ac), given by

k

? = ois, s = 0,1,..., 2k 1,

j=i

which can be expressed as

Pn = m, (4.2.8)

where Psj = dSj, s = 0,...,2k1,7= l,...,fc.

Techniques for dealing with sets of non-linear equations such as (4.2.8) are well known (Blischke, 1964; Medgyessy, 1977, pp. 172, 175). Provided the 0/s are all different, n can be solved in terms of the {6j} and therefore eliminated. Furthermore, the required {0;} are the roots of

0k + Dk_l0k-x+--- + Dl6 + Do = 0,

where D = (D0 Dk_ X)T satisfies

m0 mx \ jmk

m, m2 mk \D=_ mk +1

I

I

-1 w* m2k-2l \m2 k-

See also Rennie (1974), Lingappaiah (1975), and Cornell (1962).

Specific cases to which the above analysis applies include the following:

(a) Mixture of exponentials: tfx) = xi/T(s + 1), s ^ 0.

(b) Mixture of binomials, Bi(N,0), with N fixed:

*(*-I)* (x s-F 1)

M - N<W-!)<*-,+!) 01 ,4-29)

See Muench (1936, 1938), Rider (1962), Blischke (1962, 1964, 1965), and Johnson and Kotz (1969, Section 3.11).

Learning about the parameters of a mixture

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