Essay Example on Deriving the Likelihood Function: Assumptions & Independently Distributed Data

During the derivation of the likelihood function, some assumptions will be made:

Data has to be independently distributed.

The Data used is identically distributed (Ohad 2009). This means that the answer of one respondent, on whether they will be willing to pay a larger amount of given tax A or not, does not depend on the answer of another respondent and that each question that the researcher asks must be identical and at no point should the researcher change the form of the question.

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The parameters μ and σ will be represented as a tuple of the form θ.

θ=μ,σ

Considering there are n respondents, Mathematically it will look like this

fx1,X2,…xn|θ

Since these results are a series of events, using the data set, we can calculate joint probability density as follows:

fx1,X2,…xn|θ=fx1θ⋅fx2θ…⋅f(xn|θ) =Πfxi1θ

To denote this mathematically, we can use the argmax with respect to θ in that

θMLE=argmax1iN1fxi1θUsing the above equation derivatives with respect to θ is to calculated and then set this derivative to term so that it equals zero, and the result will be used to find the approximate location of the peak(in a normal distribution graph) along the θ-axis. With respect to that, the argmax of the joint probability term is then equated to the derivative of the joint probability density term considering θ equals zero. The equation below is a demonstration

θMLE=argmax1iN1fxi1θ →∂∂θinfxiθ = 0

Since the equation above isn't exactly an easy calculation to do, it can be simplified by changing the equations derivative term using the concept of a monotonic function. This would simplify the derivative calculation without necessarily changing the end result. The monotonic function that will be used here will take the form of a natural logarithm.

So we can write about our problem as follows:

∂∂θinfxiθ≈∂∂θln4Πfxiθ = ∂20Σn<fxiθ> = ln∂∂θnLn<fxiθ = 0 (Towards Data Science group 2017)

References

Ohad, Barak. 2009. "Function and Error Function." By Tele Aviv University, 25.

Towards Data Science group. 2017. "Likelihood estimation explained." Normal Distribution.