Question 1198696: Let x1Let x1, x2,......., xn be a random sample from the distribution having a pdf F(x/Ɵ1Ɵ2)
F(x/Ɵ1Ɵ2) = [1/Ɵ2 ℮^ -(x-Ɵ1/Ɵ2); x≥0, -ᵆ<Ɵ1<ᵆ
Find the maximum likelihood estimator of Ɵ1 and Ɵ2
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! **1. Likelihood Function**
* The likelihood function, L(θ1, θ2 | x1, ..., xn), is the joint probability density function of the observed data (x1, ..., xn) given the parameters θ1 and θ2. Since the observations are independent, the likelihood function is the product of the individual probability density functions:
L(θ1, θ2 | x1, ..., xn) = ∏(1/θ2) * exp[-(xi - θ1)/θ2]
= (1/θ2^n) * exp[-∑(xi - θ1)/θ2]
* where ∏ denotes the product over all observations (i = 1 to n) and ∑ denotes the sum over all observations.
**2. Log-Likelihood Function**
* For easier differentiation, we often work with the log-likelihood function:
log L(θ1, θ2 | x1, ..., xn) = -n*log(θ2) - ∑(xi - θ1)/θ2
**3. Partial Derivatives**
* To find the maximum likelihood estimates (MLEs) of θ1 and θ2, we need to find the values that maximize the log-likelihood function. We do this by taking partial derivatives with respect to θ1 and θ2 and setting them to zero.
* ∂/∂θ1 [log L(θ1, θ2 | x1, ..., xn)] = ∑(1/θ2) = n/θ2 = 0
* This equation has no solution for θ2 ≠ 0.
* ∂/∂θ2 [log L(θ1, θ2 | x1, ..., xn)] = -n/θ2 + ∑(xi - θ1)/θ2^2 = 0
**4. Solve the Equations**
* From the first equation, there is no solution for θ2 ≠ 0.
* However, we can use the second equation to solve for θ1:
-n/θ2 + ∑(xi - θ1)/θ2^2 = 0
-nθ2 + ∑(xi - θ1) = 0
nθ2 = ∑xi - nθ1
θ1 = (∑xi)/n - θ2
* The MLE for θ1 is the sample mean: **θ̂1 = (1/n) * ∑xi**
* Substitute θ1 = (∑xi)/n into the second equation:
-n/θ2 + ∑(xi - (∑xi)/n)/θ2^2 = 0
-nθ2 + ∑(xi) - n*(∑xi)/n = 0
-nθ2 + ∑(xi) - ∑(xi) = 0
-nθ2 = 0
θ2 = 0
* However, θ2 cannot be zero in the given probability density function. This indicates that there might be an issue with the provided PDF or the constraints on the parameters.
**Conclusion**
* Based on the calculations, finding the maximum likelihood estimators for θ1 and θ2 in this specific case presents a challenge.
* The derived equations do not yield a valid solution for θ2.
* It's possible that there might be constraints or assumptions about the parameters that have not been explicitly stated.
**Disclaimer:** This analysis assumes that the provided PDF is correct and that the necessary conditions for finding the MLEs are met. If there are any constraints or additional information about the parameters, those should be taken into account in the derivation.
|
|
|
