Hello!
I am trying to understand why I am getting the following inconsistency with a particular result. Let's say we want to generate 5000 RVs for the Generalized Pareto Distribution. We could use the following snippet:
for j=1:5000
V(j) = random('GeneralizedPareto',1.5,2,0);
end
Using the following, snippet we can them estimate its parameters:
pd = fitdist(transpose(V),'GeneralizedPareto');
In this case, we get the results:
Generalized Pareto distribution
k = 1.52002 [1.45041, 1.58962]
sigma = 1.97559 [1.85692, 2.10184]
theta = 0
All seems right with the world so far. However, let's change the location parameter to theta = 5. Now we obtain:
Generalized Pareto distribution
k = 0.635262 [0.601225, 0.669299]
sigma = 10.9004 [10.4708, 11.3476]
theta = 0
From the looks of it, it seems as if fitdist ignores the location parameter theta. Since the Matlab documentation includes all three parameters (k, sigma, and theta) in the GeneralizedPareto distribution, then I would expect that it would also address theta in the fitdist function.
You may say that simply shifting the data "pre fitdist" should be sufficient to address this limitation. However, this is not always sufficient. The next item is a little heavy on mathematical statistics so bear with me.
Let X and A be a random variable that is independent and identically distributed (iid) such that:
X ~ exponential(A) A ~ exponential(B) B ~ Pareto(1,1)
Then
p(X=x) = (-x + (1+x) ln(1+x))/(x^2 (1+x)) , x > 0
If we try to estimate this with a Pareto distribution, then we get
Generalized Pareto distribution
k = 1.18503 [1.13854, 1.23153]
sigma = 0.82043 [0.784495, 0.858012]
theta = 0
which is in agreement with the analytical results, however if we try the same for a slightly different hierarchical model such as:
X ~ InvGaussian(A,B) A ~ Pareto(1,1) B ~ Pareto(1,1)
Then the results no longer agree. We obtain:
Generalized Pareto distribution
k = 0.480286 [0.456964, 0.503608]
sigma = 1.86124 [1.80809, 1.91595]
theta = 0
Using another program to validate my concerns, I used a program called EasyFit for the same data. I obtained: k = 0.93914, sigma = 0.85407, and theta = 0.45094.
I think the problem is with how Matlab does not calculate theta but I could be wrong.
I would greatly appreciate any insight you can provide and if you need any further information, please let me know.
