To illustrate, suppose that

```{code-cell} ipython3

rng = np.random.default_rng()

n = 10_000 # size of sample

w = np.exp(np.random.randn(n)) # lognormal draws

w = np.exp(rng.standard_normal(n)) # lognormal draws

```

is data representing the wealth of 10,000 households.

n = 10_000 # size of each sample

fig, ax = plt.subplots()

for a in a_vals:

u = np.random.uniform(size=n)

u = rng.uniform(size=n)

y = u**(-1/a) # distributed as Pareto with tail index a

f_vals, l_vals = qe.lorenz_curve(y)

ax.plot(f_vals, l_vals, label=f'$a = {a}$')

fig, ax = plt.subplots()

for a in a_vals:

y = np.random.weibull(a, size=n)

y = rng.weibull(a, size=n)

ginis.append(qe.gini_coefficient(y))

ginis_theoretical.append(1 - 2**(-1/a))

ax.plot(a_vals, ginis, label='estimated gini coefficient')

simulation.

```{code-cell} ipython3

rng = np.random.default_rng()

a_vals = np.linspace(1, 10, 25) # Pareto tail index

ginis = np.empty_like(a_vals)

n = 1000 # size of each sample

fig, ax = plt.subplots()

for i, a in enumerate(a_vals):

y = np.random.uniform(size=n)**(-1/a)

y = rng.uniform(size=n)**(-1/a)

ginis[i] = qe.gini_coefficient(y)

ax.plot(a_vals, ginis, label='sampled')

ax.plot(a_vals, 1/(2*a_vals - 1), label='theoretical')