Question

Multi-part question:

Let X1, .... ,Xn be random variables that describe the monthly salary (in thousands of dollars) that
people in San Francisco receive. By law, there is a minimum monthly salary that people should receive
and it is unknown. Denote this minimum salary θ.

A) What is the statistical model?

B) Assume that the random variables that describe the monthly income have the following p.d.f.

fx(X|θ) = 2*((θ)^2)*((x)^-3)     θ > 0, and θ ≤ x < ∞

Show that E(X) = 2θ (do this using the definition of expectation) and find the moment estimator
of the minimum salary, θ.

C) Under the assumptions in (B), plot the likelihood function and find the maximum likelihood
estimator of the minimum salary, θ.

D) Ten random citizens from San Francisco were chosen and its monthly salary was recorded. The
monthly salaries (in thousand of dollars) of these citizens are 2.3, 4.1, 2.2, 3.8, 5.3,
6.8, 2.9, 4.8, 8.0. Compute the moment estimate and the maximum likelihood estimate.

Answer 1

$A.~Here~we~use~statistical~ model~ based ~on ~Pareto~distribution.\\ B.~f_X(x|\theta)=\frac{2\theta^2}{x^3},~x\geq \theta\\ =0~otherwise\\ E(X)=\int_\theta^\infty xf_X(x|\theta)dx=2\theta^2\int_\theta^\infty \frac{1}{x^2}dx=2\theta^2\left[-\frac{1}{x} \right ]^\infty_\theta=2\theta.\\ E(\bar{X})=\frac{1}{n}\sum_{i=1}^nE(X_i)=2\theta.~Hence~E(\bar{X}/2)=\theta.\\ Hence~MME~of~\theta=\frac{\bar{X}}{2}.\\$

$(C)~The~likelihood~function~of~\theta\\=L(\theta)=\frac{2^n\theta^{2n}}{\prod_{i=1}^nx_i^3},~\theta\leq x_1,x_2,...,x_n<\infty\\ =\frac{2^n\theta^{2n}}{\prod_{i=1}^nx_i^3}~if~x_{(1)}\geq \theta;~x_{(1)}=\min\{x_1,x_2,...,x_n\}\\=\frac{2^n\theta^{2n}}{\prod_{i=1}^nx_i^3}C(x_{(1)},\theta)\\ where,~C(x_{(1)},\theta)=1~if~x_{(1)}\geq \theta \\ =0~if~x_{(1)}<\theta.$

$MLE~of~\theta=X_{(1)}=\min\{X_1,X_2,...,X_n\}\\$

$(D)~Moment~estimate~of~\theta\\ =(2.3&plus; 4.1&plus; 2.2&plus; 3.8&plus; 5.3&plus; 6.8&plus; 2.9&plus; 4.8&plus; 8.0)/18\\=2.233\times 1000=2233 dollars\\ MLE~of~\theta=2.2\times 1000=2200~dollars.$