Question

Let X1, X2 and X3 be a random sample of size three from a Uniform[?, 2?] distribution, where ? > 0 is a model parameter. (a) Find the method-of-moments estimator of ?. (b) Find the MLE, ??, and find a constant k such that E_?(k??) = ?. (c) Which of the two estimators is better? Justify your answer. (d) Compute the method-of-moments estimate and the MLE of ? based on the following three observations of berry sizes (in centimetres) of wine grapes: 1.29, 0.86, 1.33.

          Let X1, X2 and X3 be a random sample of size three from a Uniform[?, 2?] distribution, where ? > 0 is a model parameter.
(a) Find the method-of-moments estimator of ?.
(b) Find the MLE, ??, and find a constant k such that E_?(k??) = ?.
(c) Which of the two estimators is better? Justify your answer.
(d) Compute the method-of-moments estimate and the MLE of ? based on the following three observations of berry sizes (in centimetres) of wine grapes:
1.29, 0.86, 1.33.

Let X1, X2 and X3 be a random sample of size three from a Uniform[?, 2?] distribution, where ? > 0 is a model parameter.
(a) Find the method-of-moments estimator of ?.
(b) Find the MLE, ??, and find a constant k such that E?(k??) = ?.
(c) Which of the two estimators is better? Justify your answer.
(d) Compute the method-of-moments estimate and the MLE of ? based on the following three observations of berry sizes (in centimetres) of wine grapes:
1.29, 0.86, 1.33.

Added by David W.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Sri K

01/16/2023

Step 1

For a Uniform $[0, 2\theta]$ distribution, the mean $\mu$ is given by: \[ \mu = \frac{0 + 2\theta}{2} = \theta \] The method-of-moments estimator is found by equating the sample mean to the population mean. Let $\bar{X}$ be the sample mean of $X_1, X_2, X_3$. Show more…

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Let X1, X2, and X3 be a random sample of size three from a Uniform[θ, 2θ] distribution, where θ > 0 is a model parameter. (a) Find the method-of-moments estimator of θ. (b) Find the MLE, θ̂, and find a constant k such that E_θ(kθ̂) = θ. (c) Which of the two estimators is better? Justify your answer.