Question

MLE for Pareto distribution Let X₁​,…,Xn​ be an iid random sample from a Pareto(γ) distribution with pdf

f(x∣γ)= γ/(1+x)ʸ⁺¹ I₍₀,[infinity]₎ )​(x) where γ>0.Find the MLE for γ.

Answer 1

In this case, the MLE for γ in a Pareto(γ) distribution is given by: Y= n / ∑[i=1 to n] log(1+X`ᵢ).

Step-by-step explanation:

To find the maximum likelihood estimator (MLE) for the parameter γ in a Pareto distribution, we need to maximize the likelihood function L(γ), which is the product of the probability density functions (pdfs) of the observed random sample.

Given that the pdf of a Pareto(γ) distribution is f(x∣γ) = γ/(1+x)⁽γ⁺¹⁾), where γ > 0, we can express the likelihood function as follows:

L(γ) = ∏[i=1 to n] (γ/(1+X`ᵢ)⁽γ⁺¹⁾)

To simplify the calculation, it is often easier to work with the log-likelihood function, which is the natural logarithm of the likelihood function:

log(L(γ)) = ∑[i=1 to n] log(γ) - (γ+1)log(1+X`ᵢ)

To find the MLE for γ, we need to find the value of γ that maximizes the log-likelihood function.

Differentiating the log-likelihood function with respect to γ, we get:

d[log(L(γ))]/dγ = ∑[i=1 to n] [1/γ - log(1+X`ᵢ)]

Setting this derivative equal to zero and solving for γ, we can find the critical point:

∑[i=1 to n] [1/γ - log(1+X`ᵢ)] = 0

Rearranging the terms, we get:

∑[i=1 to n] 1/γ = ∑[i=1 to n] log(1+X`ᵢ)

Since the summation on the right side of the equation does not depend on γ, we can simplify further:

n/γ = ∑[i=1 to n] log(1+X`ᵢ)

Solving for γ, we find:

γ = n / ∑[i=1 to n] log(1+X`ᵢ)

Therefore, the MLE for γ in a Pareto(γ) distribution is given by:

Y= n / ∑[i=1 to n] log(1+X`ᵢ)