What is the Maximum Likelihood Estimator for Uniform Distribution Endpoints?

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SUMMARY

The maximum likelihood estimator (MLE) for the endpoints θ1 and θ2 of a uniform distribution over the interval [θ1, θ2] is given by the minimum and maximum of the observed data, respectively. Specifically, for n independent observations, the MLE for θ1 is the minimum value of the sample, and the MLE for θ2 is the maximum value of the sample. Consequently, the expected MLE for the mean of the uniform distribution can be derived as (θ1 + θ2) / 2, which reflects the average of the estimated endpoints.

PREREQUISITES
  • Understanding of maximum likelihood estimation (MLE)
  • Familiarity with uniform distribution properties
  • Knowledge of statistical inference techniques
  • Basic proficiency in probability theory
NEXT STEPS
  • Study the derivation of maximum likelihood estimators for different distributions
  • Explore the properties of uniform distributions in statistical analysis
  • Learn about likelihood functions and their applications in estimation
  • Investigate the implications of MLE in real-world data analysis scenarios
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Statisticians, data analysts, and students studying statistical inference who seek to understand maximum likelihood estimation in the context of uniform distributions.

das1
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I need help on this problem, anyone know how to do it?

Suppose you have n independent observations from a uniform distribution over the interval [𝜃1, 𝜃2].

a. Find the maximum likelihood estimator for each of the endpoints θ1 and θ2.
b. Based on your result in part (a), what would you expect the maximum likelihood estimator to be for the mean? Explain or prove your result.
 
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The answer will depend on what you know. For example, do you know an expression for the likelihood?
 

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