2.Let X_1…X_n be i.i.d random variables with pdf f_θ(x)= θx^(θ-1)1(0<x<1) where θ is some positive number
a) is the parameter θ identificable
Y
N
b)
Compute the maximum likelihood estimator Ӫ of θ.
(Enter Sigma_i(g(X_i)) for the sum ∑ni=1g(Xi), e.g. enter Sigma_i(X_i^2) for ∑ni=1X2i, enter Sigma_i(ln(X_i)) for ∑ni=1ln(Xi). Do not forget any necessary n in your answer, e.g. ˉXn will need to be entered as Sigma_i(X_i)/n . Do not worry about the parser not rendering correctly, as the grader will still work independently. If you would like proper rendering, enclose Σi(g(Xi)) in parentheses i.e. use (Σi(g(Xi))).)
Maximum likelihood estimator Ӫ
To determine if the parameter θ is identifiable, we need to check if different values of θ result in different probability density functions (pdfs).
The pdf f_θ(x) = θx^(θ-1)1(0<x<1) is defined for 0 < x < 1 with θ being a positive number.
To check if the parameter θ is identifiable, we can compare the pdfs for two different values of θ, say θ_1 and θ_2.
For θ_1 ≠ θ_2, the pdf f_θ_1(x) = θ_1x^(θ_1-1)1(0<x<1) and f_θ_2(x) = θ_2x^(θ_2-1)1(0<x<1).
By comparing the two pdfs, we can see that if θ_1 ≠ θ_2, then f_θ_1(x) ≠ f_θ_2(x) for at least one value of x in the range (0, 1).
Therefore, the parameter θ is identifiable because different values of θ result in different pdfs.
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To compute the maximum likelihood estimator (MLE) Ӫ of θ, we need to find the value of θ that maximizes the likelihood function.
The likelihood function L(θ) is given by the product of the pdfs f_θ(x) for the given data X_1, X_2, ..., X_n:
L(θ) = ∏(f_θ(X_i)), where i = 1 to n.
Taking the natural logarithm of the likelihood function helps simplify the calculations, so we define the log-likelihood function ln(L(θ)).
ln(L(θ)) = Σ[(ln(θ) + (θ-1)ln(X_i))], where i = 1 to n.
To maximize the log-likelihood function, we take the derivative with respect to θ, set it equal to zero, and solve for θ.
d/dθ ln(L(θ)) = Σ[(1/θ + ln(X_i))] = 0.
Simplifying and solving for θ, we get:
Σ[(1/θ + ln(X_i))] = 0.
Σ[1/θ] + Σ[ln(X_i)] = 0.
(Σ[1])/θ + Σ[ln(X_i)] = 0.
Σ[ln(X_i)] = (-Σ[1])/θ.
-Σ[1] = n, since we are summing 1 over n terms.
Σ[ln(X_i)] = -n/θ.
Now, solving for θ, we have:
θ = -n/Σ[ln(X_i)].
Therefore, the maximum likelihood estimator (MLE) Ӫ of θ is -n/Σ[ln(X_i)].
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Note: In the expression Σ[ln(X_i)], the notation Σ represents the sum of ln(X_i) over i = 1 to n.