4.3.1 MLE for Transformed Parameters
The MLE of the transformed parameter, α = g(θ) is given by:
where 
is the MLE of θ. If g is not one-to-one function (i.e., not invertible) then 
is obtained as the MLE of transformed likelihood function, pT (x; α), which is defined as:
4.3.2 Example
![[ N∑ -1 ] p(x;A ) = ----1--N-exp - -1-- (x[n ] - A )2 - ∞ < A < ∞ (2πσ2 )2- 2σ2 n=0](images/3.jpg)
![[ ] 1 1 N∑ -1 pT(x; α) = -------N-exp - ---- (x[n ] - lnα )2 α > 0 (2π σ2)2 2σ2 n=0](images/4.jpg)
![x [n ] = ln α + w[n] n = 0,1,...,N - 1](images/5.jpg)
![N∑- 1 1 (x[n] - ln ˆα )--= 0 n=0 ˆα](images/6.jpg)
being the MLE of A, so we have 
, the α is not one-to-one transformation of A. If we take 
only then some possible PDFs will be missing. To characterize all possible PDFs, we need to consider two sets of PDFs ![[ N∑- 1 -- ] pT1(x;α) = ----1----exp - -1-- (x [n ] - √ α)2 α ≥ 0 (2πσ2 )N2- 2σ2 n=0 [ N- 1 ] ----1---- -1--∑ √ --2 pT2(x;α) = 2 N2-exp - 2σ2 (x [n ] + α) α > 0 (2πσ ) n=0](images/11.jpg)
![αˆ = argmaxα [pT1(x;α),pT 2(x; α)]](images/12.jpg)
. Repeat for all α > 0 to form 
. Note that 
. ![√ -- √ -- ˆα = arg max [pT1(x; α ),pT 2(x; - α)] [ α ] √ -- √ -- 2 = arg m√ax {pT 1(x; α),pT2(x;- α )} [ α≥0 ] 2 = arg -∞ma<xA<∞ p (x; A) = Aˆ2 = ¯x2](images/16.jpg)
4.3.3 MLE for General Linear Model
Consider the general linear model of the form:
x = Hθ + w
where H is a known N × p matrix, x is an N × 1 observation vector with N samples, and w is N × 1 noise vector with PDF 
(0,C). The PDF of the observed data is:
![1 [ 1 ] p(x;θ) = ----N----1----exp - -(x - H θ)TC -1(x - H θ) (2 π)2 det2(C ) 2](images/17.jpg)
and the MLE of θ is found by differentiating the log-likelihood which can be shown to yield:
which upon simplification and setting to zero becomes:
and this yields the MLE of θ as:
which turns out to be same as the MVU estimator.