Calculate Parameters of Shifted Exponential Density from Measurements

[hkBattousai]

I have a density function:

f(t) = L * e^-L(t-t₀) * u(t-t₀)

where u(t) is the unit step function.


And I have a column vector X, which is randomly chosen samples from f(t):

X = [x₁ x₂ ... x_n]^T


How can I estimate the unknown values t₀ and L from this X vector?

 

[winterfors]

You can use Bayes' theorem to calculate a (posterior) probability density function f(L, t0 | X) for the parameters L and t0 given the vector of sample times X:

If we call
f(t | L, t₀) = L * e^-L(t-t₀) * u(t-t₀)

and
f(X | L, t₀) = f(x₁ | L, t₀)*f(x₂ | L, t0)*...*f(x_n | L, t₀)

We can calculate the posterior PDF

f(L, t₀ | X) = f(X | L, t₀) / K

where K is a nornlization constant
$$K = \int_{L=0}^{\infty} \int_{t_0=0}^{\infty} f(X | L, t0) dt_0 dL$$

(above I have assumed a homogeneous prior distributions for L and t₀)

You can then e..g. calculate expectations of L and t₀ from f(L, t₀ | X)

$$\hat{L} = \int_{L=0}^{\infty} \int_{t_0=0}^{\infty} L* f(L, t0 | X) dt_0 dL$$
$$\hat{t_0} = \int_{L=0}^{\infty} \int_{t_0=0}^{\infty} t_0* f(L, t0 | X) dt_0 dL$$


The integrals are probably difficult to solve analytically, but you can solve them numarically given your X-vector.