## A mixture of three exponential distributions

#### Source:

A total of 1000 observations was generated by computer to follow the mixture distribution

1/3 E(1) + 1/3 E(4) + 1/3 E(16)

where E(m
) denotes an exponential distribution with mean m
.

#### Remarks:

Exponential distributions are fitted by constraining gamma distributions to have unit
coefficient of variation. Mixtures of exponential distributions are always unimodal
and skewed, and components with large mean completely overlap the components with
small mean. As a result, fitting a mixture of exponential distributions is often a frustrating
experience. In many applications, it may be impossible to estimate all of the proportions
and all of the means; in this example it was possible, but all of the estimates have rather large standard errors, despite the large sample size. MIX does not
plot exponential mixtures very elegantly.

Fitting Gamma components
Proportions and their standard errors
.47942 .36121 .15937
.08481 .06844 .05163
Means and their standard errors
.9770 4.8888 22.0166
.2148 1.4076 4.5749
Sigmas (FIXED COEF. OF VAR. = 1.0000)
.9770 4.8888 22.0166
Degrees of freedom = 25 - 1 + 0 - 0 - 5 - 0 = 19
Chi-squared = 21.4570 (P = .3121)

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