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Estimating Parameter of Uniform Distribution from Sample Mean, Median, MLE

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This document describes estimating an unknown parameter θ based on a random sample from a distribution with density f(x) = x^θ for 0 < x < 1 using three estimation methods: method of moments using the mean, method of quantiles using the median, and maximum likelihood estimation. It asks to calculate and compare the values of the three estimators given a sample with observations of 1/3, 1/2, and 5/6.

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Let X1, X2, ..., Xn be a random sample from a distribution with density: f(x) = x1 x (0, 1) 0 otherwise where > 0 is an unknown parameter. Find the method of moments and method of quantiles estimators for (based on the mean and median, respectively) and the m.l.e. of . Compare the values of the three estimators for a sample consisting of the following three observations: 1/3 , 1/2 , 5/6 .

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Estimating Parameter of Uniform Distribution from Sample Mean, Median, MLE

  • 1. Let X1, X2, ..., Xn be a random sample from a distribution with density: f(x) = x1 x (0, 1) 0 otherwise where > 0 is an unknown parameter. Find the method of moments and method of quantiles estimators for (based on the mean and median, respectively) and the m.l.e. of . Compare the values of the three estimators for a sample consisting of the following three observations: 1/3 , 1/2 , 5/6 .