4.4.4 Example: Mixed Normal Distribution

4.4.4  Example: Mixed Normal Distribution

Consider the data of Exhibit 4.4.

Exhibit 4.4: Data for the example of this section.

We treat the data as a realization of a sample for a mixed normal distribution N²(μ,σ²,ω) with:

[4.36]

We shall use the data to construct an ML estimate for the unknown mean vector

[4.37]

The PDF ϕ(x|μ) of the mixed normal distribution is

[4.38]

which has gradient

[4.39]

Based upon our data, the log-likelihood function log(L(μ)) is

[4.40]

Its gradient is

[4.41]

To maximize the log-likelihood function, we set [4.41] equal to 0 and solve for μ using Newton’s method. There is a saddle point at μ = (0.4212, 0.4212). The global maximum we seek occurs at μ = (–1.3616, 1.8513).