## Two overlapped components with positive skewness

#### Source:

This histogram was artificially constructed by shifting the right-hand component of
the previous histogram 5 units to the left.

### Analysis 1:

Gamma components, unconstrained.

#### Remarks:

This is an excellent fit. The estimates are very close to their true values.

Fitting Gamma components
Proportions and their standard errors
.65054 .34946
.04478 .04478
Means and their standard errors
3.9019 10.7319
.2746 .3286
Sigmas and their standard errors
1.9337 2.0720
.2605 .2311
Degrees of freedom = 16 - 1 + 0 - 0 - 5 - 0 = 10
Chi-squared = 3.18914 (P = .9766)

### Analysis 2:

Normal components, with the two standard deviations constrained to be equal.

#### Remarks:

A poor fit, but the estimates themselves are excellent.

Fitting Normal components
Proportions and their standard errors
.63254 .36746
.03175 .03175
Means and their standard errors
3.7311 10.6590
.1604 .2182
Sigmas (EQUAL) and standard error
1.8396 1.8396
.0963
Degrees of freedom = 16 - 1 + 0 - 0 - 4 - 0 = 11
Chi-squared = 17.7339 (P = .0880)

### Analysis 3:

Normal components, unconstrained.

#### Remarks:

Information about skewness is lost in the overlap. The result is an excellent fit
to the mixture distribution, at least according to the goodness-of-fit test, but
the estimates are badly biased. In particular, the proportion and the standard deviation
of the first component are under-estimated while those of the second component are over-estimated.

Fitting Normal components
Proportions and their standard errors
.55082 .44918
.04397 .04397
Means and their standard errors
3.3582 9.8720
.1615 .4059
Sigmas and their standard errors
1.3787 2.5906
.1250 .3028
Degrees of freedom = 16 - 1 + 0 - 0 - 5 - 0 = 10
Chi-squared = 4.58181 (P = .9173)

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