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

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Statistical analysis of finite mixture distributions

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x

Figure 2.2.1 Plot of densities of the lognormal (2.2.4) and the normal mixture (2.2.5)

The figure stimulates the following two remarks, one rather positive and one rather negative, so far as the treatment of data from (2.2.5) is concerned:

(i) The density can be well estimated using the two-parameter lognormal (2.2.4) instead of the five-parameter normal mixture (2.2.5). hi) It will be very difficult to identify the ‘correct’ model; see Chapter 5 for a further discussion of related problems.

(b) Quartic exponential density Consider the density

p(x) = c(ot)exp[ — (otjX + a2x2 + a3x3 + a4x4)],

a4 > 0, — oo < x < oo, where c(or) is a normalizing constant. This density clearly has one or three

Applications of finite mixture models 3j

stationary points, depending on the nature of the roots of the cubic equation

ai + 2a2x + 3a3x2 + 4a4x3 = 0,

so that the curve can be unimodal or bimodal (see Section 5.5).

Since the distribution is a member of the exponential family, the likelihood, in terms of a, is log-concave, so that there is a unique global maximum. Furthermore, if denotes the maximum likelihood estimator of the jth moment of X, with sample moment mJt we have

= j= 1,...,4. (2.2.6)

Numerical solution of these equations (Matz, 1978) provides maximum likelihood estimates of the unknown parameters. For the symmetric case, equations (2.2.6) for j = 1, 2, 4 are solved.

Early work on the quartic exponential is due to Fisher (1921), OToole (1933a, 1933b), Aroian (1948), Dallaville, Orr, and Blocker (1951), and Greenwood and Hartley (1962, p. 465). Dallaville, Orr, and Blocker (1951) used a method of least-

— Porometers estimated by method of moments

-—Parameters estimated by maximum likelihood

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10

Crude birth rate

Figure 2.2.2 Quartic exponential fits to data for annual crude birth rates of 59 countries. Reproduced by permission of the American Statistical Association from Cobb, Koppstein, and Chen, 1983

Statistical analysis of finite mixture distributions

squares fitting (cf. criterion 4.7.1) but forgot the dependence on a of the normalizing constant c(ar). Aroian (1948) used a moment-based method of Generating consistent, asymptotically normal estimators of <x which can be calculated explicitly. The estimators are, however, generally thought to be inferior to those from maximum likelihood. Theoretical and empirical evidence for this is given by Cobb, Koppstein, and Chen (1983), from which Figure 2.2.2 is taken. Cobb, Koppstein, and Chen (1983) also describe a generalization of the quartic exponential density which, while retaining exponential-family membership, allows up to q modes, where q is arbitrary. A similar approach, which leads to a log-concave likelihood with grouped data as well, is described by Burridge (1982).

(c) Gram-Charlier expansions Let

p{z) = [ 1 + I J(P! )H3(z) -I- 2^(/?2 - 3)H4(z) + • • ]0(z),

/?, > 0, — oo < z < oo, (2.2.7)

3.0 3.4 3.8 4-2 4.6 5.0 5.4 5.8 6.2 6.6 7.0

#2

Hffure 2.2.3 plane showing regions of unimodal curves and regions of curves

composed entirely of non-negative ordinates. Reproduced by permission of Biometrika

Trustees from Barton and Dennis, 1952

1. Edgeworth curves: boundary of unimodal region - Gram Charlier curves: boundary of unimodal region

3. Edgeworth curves: boundary of positive definite region

4. Gram Charlier curves: boundary of positive definite region

Applications of finite mixture models

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where /?! and /J2 are the traditional measures of skewness and kurtosis, Hj{z) is the /th-order Hermite polynomial, and 0(z) denotes the standard normal p.d.f. Equation (2.2.7) gives the leading terms in the Gram-Charlier expansion for the density function of a standardized variable; see Johnson and Kotz (1970a, Section 12.4), where the modification known as the Edgeworth expansion is also given. For a non-standardized variable one should use

a" lp((x- p)/o).

In practice, truncated versions of the expansions are used, although the resulting function is usually no longer a probability density function. For truncation corresponding to the terms specified by (2.2.7), Figure 2.2.3 (Johnson and Kotz, 1970a, Section 12.4; Barton and Dennis, 1952) indicates the ranges of

mmol/1

Figure 2.2.4 Predictive densities from two models fitted to histogram data on blood chloride levels. Reproduced by permission of the Institute of Statisticians from Nay lor an

Smith, 1983

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Statistical analysis of finite mixture distributions

(Pl'Pi) for which non-negativity and/or multimodality is guaranteed. Draper and Tierney (1972) add a small correction.

(d) Approximation of a contaminated normal density by a Huber density

Naylor and Smith (1983) analyse a sample of blood chloride levels from a population which is assumed to contain a proportion, n, of healthy individuals and a small proportion. 1 - n, of unhealthy ones. The component distributions are assumed to be N(p, cr2) and N(p,/.2o2), respectively, where 2 > 1. An alternative to this four-parameter model is the three-parameter Huber density, defined by

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