4.4.4 Example: Mixed Normal Distribution
Consider the data of Exhibit 4.4.
We treat the data as a realization of a sample for a mixed normal distribution N2(μ,σ2,ω) with:
We shall use the data to construct an ML estimate for the unknown mean vector
The PDF ϕ(x|μ) of the mixed normal distribution is
which has gradient
Based upon our data, the log-likelihood function log(L(μ)) is
Its gradient is
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).
Reproduce the results of the example of this section. Use different seed values with Newton’s method to locate both the saddle point and global maximum.