Question
Let $Y_{1}, Y_{2}, \ldots, Y_{n}$ denote independent and identically distributed uniform random variables on the interval $(0,3 \theta)$. Derive the method-of-moments estimator for $\theta$.
Step 1
Step 1: The first moment (or the expected value) of a uniform random variable on the interval $(0,3\theta)$ is given by $\frac{1}{2}(0+3\theta) = \frac{3\theta}{2}$. Show more…
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Key Concepts
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Let $Y_{1}, Y_{2}, \ldots, Y_{n}$ be a random sample from the probability density function given by $$f(y | \theta)=\left\{\begin{array}{ll} \frac{\Gamma(2 \theta)}{[\Gamma(\theta)]^{2}}\left(y^{\theta-1}\right)(1-y)^{\theta-1}, & 0 \leq y \leq 1 \\ 0, & \text { elsewhere } \end{array}\right.$$Find the method-of-moments estimator for $\theta$
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The Method of Moments
Use the method described in Exercise 9.26 to show that, if $Y_{(1)}=\min \left(Y_{1}, Y_{2}, \ldots, Y_{n}\right)$ when $Y_{1}, Y_{2}, \ldots, Y_{n}$ are independent uniform random variables on the interval $(0, \theta),$ then $Y_{(1)}$ is not a consistent estimator for $\theta$. [Hint: Based on the methods of Section $6.7, Y_{(1)}$ has the distribution function
Consistency
Y1, Y2,..., Yn are independently and identically distributed with theta degrees of freedom. Find the method of moments estimator for theta.
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