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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.

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Maximum Likelihood Estimation of Cauchy Distribution Parameters

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  1. Applied Statistics Assignment Question One Student :Majid Instructor: Gao Xin

  2. 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]

  3. 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)

  4. a)Graph the likelihood function

  5. a) Continue MLE for Theta(Newton method) Newton.htm

  6. b)Bisection Method Bisection Method.htm c) Fixed-Point_Method d) Secant Method

  7. Conclusion

  8. d) Normal (Theta,1) Normal Distribution

  9. Conclusion for N(Theta,1)

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