Maximum Likelihood Estimation of Cauchy Distribution Parameters
This assignment delves into the Cauchy distribution, focusing on its probability density function and the derivation of the log-likelihood function. The task includes graphing the likelihood function, implementing maximum likelihood estimation (MLE) for the parameter Theta using various numerical methods: Newton's method, the Bisection method, Fixed-point iteration, and the Secant method. Additionally, a comparison will be made with the Normal distribution (Theta, 1) to highlight key differences in estimations. The results will demonstrate the advantages and limitations of each method.
Maximum Likelihood Estimation of Cauchy Distribution Parameters
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Presentation Transcript
Applied Statistics Assignment Question One Student :Majid Instructor: Gao Xin
Cauchy Distribution • The formula for the probability density function of the Cauchy distribution with (Theta,1) is: f(x)=1/[pi(1+(x-Theta)^2]
Log Likelihood function Log Likelihood function is : SUM(log(1/(pi*(1+(x-t)^2)))) First derivative: Sum(pi * (2*(x-t))/(pi*(1+(x-t)^2))^2/(1/(pi*(1+(x-t)^2)))) Second derivative is : 2*sum(((x-t)^2-1)/(1+(x-t)^2)^2)
a) Continue MLE for Theta(Newton method) Newton.htm
b)Bisection Method Bisection Method.htm c) Fixed-Point_Method d) Secant Method
d) Normal (Theta,1) Normal Distribution
