MLE with exponential distribution



In [1]:

    
import numpy as np
import matplotlib.pylab as plt
import seaborn as sns

np.set_printoptions(precision=4, suppress=True)
sns.set_context('notebook')

%matplotlib inline

Draw exponential density

$$f\left(y_{i},\theta\right)=\theta\exp\left(-\theta y_{i}\right),\quad y_{i}>0,\quad\theta>0$$



In [2]:

    
theta = 1
y = np.linspace(0, 10, 100)
f = theta * np.exp(-theta * y)

# plot function
plt.plot(y, f)
plt.xlabel('y')
plt.ylabel('f')
plt.show()

Draw several densities



In [3]:

    
# try several parameters
theta = [1, .5, .25]
y = np.linspace(0, 10, 100)

# for each parameter value
for t in theta:
    f = t * np.exp( - t * y)
    plt.plot(y, f)

plt.xlabel('y')
plt.ylabel('f')
plt.legend(theta)
plt.show()

Simulate data and draw histogram



In [4]:

    
n = 100
theta = 1
# simulate data
y = np.random.exponential(1 / theta, n)

# plot data
plt.hist(y, bins=10, normed=True)
plt.xlabel(r'$y_i$')
plt.ylabel(r'$\hat{f}$')
plt.show()

Simulate data and estimate model parameter by MLE

MLE estimator is

$$\hat{\theta}=\frac{n}{\sum_{i=1}^{n}y_{i}}=\overline{y}^{-1}$$



In [5]:

    
# sample size
n = int(1e2)
# true parameter value
theta = 1
# simulate data
y = np.sort(np.random.exponential(1 / theta, n))
# MLE estimator
theta_hat = n / np.sum(y)
print('Estimate is: theta = ', theta_hat)
# function of exponential density
f = lambda theta: theta * np.exp( - theta * y)

# plot results
plt.hist(y, bins = 10, normed = True, alpha = .2, lw = 0)
plt.plot(y, f(theta), c = 'black')
plt.plot(y, f(theta_hat), c = 'red')
plt.xlabel(r'$y_i$')
plt.ylabel(r'$\hat{f}$')
plt.legend(('True', 'Fitted','Histogram'))
plt.show()









    



Estimate is: theta =  0.900626527495