Study the properties of the bivariate Student's t distribution through simulation
python xbivariate_student_t.py gives
Parameters used:
nu = 5.0
sigma_x = 1.0
sigma_y = 2.0
rho = 0.5
n_samples = 300000
band h = 0.05
x0_grid = [-2. -1. 0. 1. 2.]
Sample covariance matrix of (X, Y) for Student t:
[[0.99631851 1.00310255]
[1.00310255 4.00489814]]
Student t: comparison of empirical and theoretical Var(Y | X = x):
x0 n_points empirical_var theoretical_var ratio_empirical/theoretical
-2.0 1219 5.147096 5.25 0.980399
-1.0 6249 3.051619 3.00 1.017206
0.0 14674 2.213105 2.25 0.983602
1.0 6104 2.941086 3.00 0.980362
2.0 1113 5.225092 5.25 0.995256
Sample covariance matrix of (X, Y) for normal:
[[1.00009646 0.99924968]
[0.99924968 3.99636017]]
Normal: comparison of empirical and theoretical Var(Y | X = x):
x0 n_points empirical_var theoretical_var ratio_empirical/theoretical
-2.0 1623 2.934992 3.0 0.978331
-1.0 7404 2.973429 3.0 0.991143
0.0 12099 3.027863 3.0 1.009288
1.0 7179 3.010133 3.0 1.003378
2.0 1576 3.000733 3.0 1.000244
If (x,y) data has the bivariate Student's t distribution, the dependence of y on x is linear, as
with the normal distribution, but the conditional variance of y given x rises with the deviation of x from its mean,
as shown below. For the normal distribution the conditional variance is constant.
Here is a dot plot:
