Final answer:
The method of moments estimator for , denoted ^MOM, can be found by equating the first moment of the distribution to the first moment of the sample. The unbiased estimators ^1 and ^2 can be found using the minimum and maximum order statistics, respectively. The maximum likelihood estimator ^ML can be found by maximizing the likelihood function. The maximum likelihood estimator is biased, while the other estimators are unbiased. The choice of estimator depends on the trade-off between bias and variance. All four estimators are consistent for .
Step-by-step explanation:
(a) The method of moments estimator for θ, denoted θ^MOM, can be found by equating the first moment of the distribution to the first moment of the sample. The first moment of the uniform distribution is (1/2)(2θ+3θ)=5/2θ. Equating this to the sample mean gives us:
(1/n)(X1+X2+...+Xn)=5/2θ
θ^MOM = (2/5)(1/n)(X1+X2+...+Xn)
(b) To find an unbiased estimator for θ, denoted θ^1, we can use the fact that the minimum order statistic X(1) is an unbiased estimator for 2θ. From (a), we know that θ^MOM = (2/5)(1/n)(X1+X2+...+Xn). Rearranging this equation, we get:
(1/2)θ^1 = X(1)
θ^1 = 2X(1)
(c) To find an unbiased estimator for θ, denoted θ^2, we can use the fact that the maximum order statistic X(n) is an unbiased estimator for 3θ. From (a), we know that θ^MOM = (2/5)(1/n)(X1+X2+...+Xn). Rearranging this equation, we get:
(3/2)θ^2 = X(n)
θ^2 = (2/3)X(n)
(d) The maximum likelihood estimator for θ, denoted θ^ML, can be found by maximizing the likelihood function. The likelihood function for the uniform distribution is:
L(θ) = (1/(3θ-2θ)^n)
Maximizing this function gives us:
θ^ML = X(n)/n
(e) To determine if the maximum likelihood estimator θ^ML is unbiased, we need to calculate its expected value and see if it equals θ. Taking the expected value of θ^ML, we get:
Eθ[θ^ML] = Eθ[X(n)/n] = (1/n)Eθ[X(n)]
From part (c), we know that Eθ[X(n)] = c2θ. Substituting this into the equation, we get:
Eθ[θ^ML] = (1/n)(c2θ) = c2θ/n ≠ θ
Therefore, the maximum likelihood estimator θ^ML is biased.
(f) The mean squared error (MSE) of an estimator is a measure of its accuracy. It is calculated as the sum of the bias squared and the variance of the estimator. Comparing the four estimators:
- The method of moments estimator θ^MOM is unbiased, but its MSE depends on the sample size.
- The unbiased estimator θ^1 has a smaller variance than θ^MOM, but a larger bias.
- The unbiased estimator θ^2 has a larger variance than θ^MOM, but a smaller bias.
- The maximum likelihood estimator θ^ML has a smaller bias than θ^MOM, but a larger variance.
Choosing the preferred estimator depends on the trade-off between bias and variance. If smaller bias is more important, θ^1 is preferred. If smaller variance is more important, θ^2 is preferred. If both are important, a compromise can be made between θ^1 and θ^2.
(g) To check if each of these estimators is consistent for θ, we need to show that as the sample size increases, the estimators converge to the true value of θ. The method of moments estimator θ^MOM is consistent because it converges to θ as n approaches infinity. The unbiased estimator θ^1 is also consistent because it converges to θ as n approaches infinity. The unbiased estimator θ^2 is consistent because it converges to θ as n approaches infinity. The maximum likelihood estimator θ^ML is consistent because it converges to θ as n approaches infinity.