We shall determine the Fisher information I θ in X The point mass function of X

# We shall determine the fisher information i θ in x

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We shall determine the Fisher information I ( θ ) in X . The point mass function of X is f ( x | θ ) = θ x (1 - θ ) 1 - x for x = 1 or x = 0 . Therefore l ( x | θ ) = log f ( x | θ ) = x log θ + (1 - x ) log(1 - θ )

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X is a random variable and f ( x ; θ ) is a statistical model for X . Here θ is a parameter. X 1 , . . . , X n is a random sample from X . We want to construct good estimators for θ using the random sample. Here is an example from Protheroe, et al. “Interpretation of cosmic ray composition - The path length distribution,” ApJ., 247 1981. X is length of paths. X is modeled as an exponential variable with density, f ( x ; θ ) = θ - 1 exp( - x ) , x > 0 . Under this model, E ( X ) = Z 0 x f ( x ; θ ) dx = θ.
Maximizing ln L using calculus, we get ˆ μ = ¯ X . Continuing with LF for globular clusters, suppose now that both μ and σ 2 are unknown. We have now likelihood function of two variables, L ( μ, σ 2 ) = f ( x 1 ; μ, σ 2 ) · · · f ( x n ; μ, σ 2 ) = 1 (2 πσ 2 ) n / 2 exp " - 1 2 σ 2 n X i =1 ( x i - μ ) 2 # ln L = - n 2 ln(2 π ) - n 2 ln( σ 2 ) - 1 2 σ 2 n X i =1 ( x i - μ ) 2

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Let x (1) be the smallest observation in the sample. “All x i θ ” is equivalent to “ x (1) θ ”. Thus L ( θ ) = exp( - n x - θ )) , if θ x (1) 0 , if x (1) < θ Conclusion: ˆ θ = X (1) .
General Properties of the MLE ˆ θ : (a) ˆ θ may not be unbiased. We often can remove this bias by multiplying ˆ θ by a constant.

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