Log Likelihood using R

I have a probability density function (PDF)

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(1-cos(x-theta))/(2*pi)

theta is the unknown parameter. How do I write a log likelihood function for this PDF? I am confused; the x will come from my data, but how do I handle the theta in the equation. Thanks

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You need to use an optimisation or maximisation function in R to compute the value of theta that maximises the log-likelihood. See help(nlmin) for starters.

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The function you wrote is a likelihood function of theta given the known x:

ll(theta|x) = log((1-cos(x-theta))/(2*pi))

if you have many iid observations from this distribution, x1,x2,...xn just take the sum of the above:

ll(theta|x1,x2,...) = Sum[log((1-cos(xi-theta))/(2*pi))]

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If f(x_i) = (1-cos(x_i-theta))/(2*pi) for observation i, then likelihood function L(Theta)=product(f(x_i)) and logL(theta)=sum(f(x_i)), of course assuming that x_i are independent.

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I think log-likelihood only works for normal-distributions. The special property of the log-function is, that it cancels out the exp-function, but here's no exp-function.

Btw., your PDF is periodic and theta just manipulates the phase of that function. Where does this PDF come from? What should it describe?