Question

Provide two examples of the invariance property of the maximum likelihood estimator.

Answer 1

Suppose the weights of randomly selected American female college students are normally distributed with unknown mean μ and standard deviation σ. A random sample of 10 American female college students yielded the following weights (in pounds):

115 122 130 127 149 160 152 138 149 180   

Based on the definitions given above, identify the likelihood function and the maximum likelihood estimator of μ, the mean weight of all American female college students. Using the given sample, find a maximum likelihood estimate of μ as well.

Solution :- The probability density function of Xi is:

f(xi;μ,σ2)=1σ2π−−√exp[−(xi−μ)22σ2]

for −∞ < x < ∞. The parameter space is Ω = {(μ, σ): −∞ < μ < ∞ and 0 < σ < ∞}. Therefore, (you might want to convince yourself that) the likelihood function is:

L(μ,σ)=σ−n(2π)−n/2exp[−12σ2∑i=1n(xi−μ)2]

for −∞ < μ < ∞ and 0 < σ < ∞. It can be shown (we'll do so in the next example!), upon maximizing the likelihood function with respect to μ, that the maximum likelihood estimator of μ is:

μ^=1n∑i=1nXi=X¯

Based on the given sample, a maximum likelihood estimate of μ is:

μ^=1n∑i=1nxi=110(115+⋯+180)=142.2

pounds. Note that the only difference between the formulas for the maximum likelihood estimator and the maximum likelihood estimate is that:

the estimator is defined using capital letters (to denote that its value is random), and

the estimate is defined using lowercase letters (to denote that its value is fixed and based on an obtained sample)