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

  1. Apr 19, 2010 #1
    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. jcsd
  3. Apr 19, 2010 #2
    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
    [tex]K = \int_{L=0}^{\infty} \int_{t_0=0}^{\infty} f(X | L, t0) dt_0 dL[/tex]

    (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)

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


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