Math Problem Statement
Let X1, . . . , Xn be a random sample of size n, from a distribution with probability density
f(x) = ( 1 θ , if 2θ ≤ x ≤ 3θ, 0, elsewhere, where θ > 0. What is the maximum likelihood estimator of θ ?
Solution
To find the maximum likelihood estimator (MLE) of , let’s go through the steps systematically.
Step 1: Write the Likelihood Function
Given that is a random sample from the distribution with the probability density function:
\frac{1}{\theta} & \text{if } 2\theta \le x \le 3\theta, \\ 0 & \text{otherwise,} \end{cases}$$ the likelihood function $$L(\theta)$$ for a sample $$X_1, X_2, \dots, X_n$$ is the product of the individual densities: $$L(\theta) = \prod_{i=1}^n f(X_i; \theta).$$ Since $$f(X_i; \theta) = \frac{1}{\theta}$$ if $$2\theta \le X_i \le 3\theta$$, we have: $$L(\theta) = \frac{1}{\theta^n} \quad \text{if } 2\theta \le X_i \le 3\theta \text{ for all } i.$$ This constraint, $$2\theta \le X_i \le 3\theta$$, for all $$i = 1, 2, \dots, n$$, implies: $$2\theta \le \min(X_i) \quad \text{and} \quad \max(X_i) \le 3\theta.$$ Rearranging these inequalities gives: $$\frac{\min(X_i)}{2} \le \theta \le \frac{\max(X_i)}{3}.$$ Therefore, the likelihood function $$L(\theta)$$ is: $$L(\theta) = \frac{1}{\theta^n} \quad \text{for } \frac{\min(X_i)}{2} \le \theta \le \frac{\max(X_i)}{3}.$$ Outside this interval, $$L(\theta) = 0$$. ### Step 2: Find the Maximum of the Likelihood Function To maximize $$L(\theta)$$, observe that $$L(\theta) = \frac{1}{\theta^n}$$ is a decreasing function of $$\theta$$. Thus, $$L(\theta)$$ is maximized when $$\theta$$ is as small as possible, within the interval: $$\frac{\min(X_i)}{2} \le \theta \le \frac{\max(X_i)}{3}.$$ The smallest possible value of $$\theta$$ in this interval is $$\theta = \frac{\max(X_i)}{3}$$. ### Conclusion The maximum likelihood estimator (MLE) of $$\theta$$ is: $$\hat{\theta} = \frac{\max(X_i)}{3}.$$ Would you like further details or have any questions about this solution? Here are five related questions you might consider: 1. How does the range of $$X_i$$ values affect the likelihood function for other distributions? 2. What are the implications of the MLE's dependency on $$\max(X_i)$$ in this case? 3. How would the MLE change if the range of $$X_i$$ were different? 4. What is the role of order statistics in finding maximum likelihood estimators? 5. How does this MLE estimator behave for small sample sizes? **Tip:** When solving MLE problems, first express the likelihood function and constraints before seeking to maximize or minimize it.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Statistics
Maximum Likelihood Estimation
Probability Density Functions
Order Statistics
Formulas
Likelihood function L(θ) = Π f(Xi; θ) for all i
f(x; θ) = 1/θ if 2θ ≤ x ≤ 3θ
MLE estimator θ̂ = max(Xi) / 3
Theorems
Maximum Likelihood Estimation
Properties of Likelihood Functions
Order Statistics
Suitable Grade Level
Undergraduate Statistics
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