Question

use R software

Let X be a non-negative random variable with μ = E[X] < ∞. For a random sample x₁, …, x_n from the distribution of X, the Gini ratio is defined by

G=12n2μn∑j=1n∑i=1|xi−xj|.G=12n2μ∑j=1n∑i=1n|xi−xj|.

The Gini ratio is applied in economics to measure inequality in income distribution (see, e.g., [168]). Note that G can be written in terms of the order statistics x_(i) as

G=1n2μn∑i=1(2i−n−1)x(i).G=1n2μ∑i=1n(2i−n−1)x(i).

If the mean is unknown, let ˆGG^ be the statistic G with μ replaced by ¯xx¯. Estimate by simulation the mean, median and deciles of ˆGG^ if X is standard lognormal. Repeat the procedure for the uniform distribution and Bernoulli(0.1). Also construct density histograms of the replicates in each case.

Answer 1

This is the code using standard lognormal distribution. Mean and median in this case is 0.514 and 0.511

For unifrom distribution, rlnorm( n ) is replaced by runif( n ) and mean and median in this case is 0.331 and 0.330

For bernoulli distribution, rlnorm( n ) is replaced by rbinom( n, size = 1, prob = 0.1 ) and mean and median in this case is 0.899 and 0.9.