Statistical Modelling and Inference - Cauchy Distribution - Median - Statistics Assessment Answer

Assessment Task:

Statistical Modelling and Inference Statistics Assessment Answer

Task

Statistical Modelling and Inference

Example: The Cauchy distribution has p.d.f. f(y) = π−1(1 + (x − θ)2)−1, where parameter θ equals the median of the distribution. A sample of data from such a distribution can be simulated using the R command rcauchy(n, theta). A possible estimator for θ is QP − tan{π(p − 0.5)}, where QP is the sample p-quantile and p is any value in the interval (0, 1). Different choices for p define different estimators. Consider the three estimators defined by p = 1/2, 2/3 and 3/4. Use simulation to estimate the biases and standard errors of these estimators for each of the sample sizes n = 10, 20. ., 90 when the true value of θ is θ = 0. Describe using suitable plots or tables how the biases and standard errors vary with n and explain which of the three estimators you prefer. 

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