# How to calculate parameters of shifted exponential density from a set of measurements

1. Apr 19, 2010

### 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 ... xn]T

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

2. Apr 19, 2010

### winterfors

Re: How to calculate parameters of shifted exponential density from a set of measurem

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(xn | 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.