Statistics MLE vs MME Shiu-Sheng Chen Department of Economics - PowerPoint PPT Presentation

Statistics MLE vs MME Shiu-Sheng Chen Department of Economics National Taiwan University Fall 2019 Shiu-Sheng Chen (NTU Econ) Statistics Fall 2019 1 / 4

A Simple Example i = 1 ∼ i . i . d . U [ 0, θ ] , θ > 0 . Let { X i } n Find the maximum likelihood estimator for θ , denoted by ˆ θ ML 1 Is ˆ θ ML unbiased? 2 Is ˆ θ ML consistent? 3 Use ˆ θ ML to construct a unbiased estimator, denoted by ˜ θ ML 4 Is ˜ θ ML consistent? 5 Find the method of moments estimator for θ , denoted by ˆ θ MM 6 Is ˆ θ MM unbiased? 7 Is ˆ θ MM consistent? 8 Compare Var ( ˆ θ ML ) and Var ( ˆ θ MM ) 9 Shiu-Sheng Chen (NTU Econ) Statistics Fall 2019 2 / 4

MLE vs MME Three observations were collected on a continuous uniform random variable X ∼ U [ 0, θ ] . The data recorded were x 1 = 3.2, x 2 = 2.9, x 3 = 13.1. Hence, X = 23.2 + 2.9 + 13.1 θ MM = 2 ¯ ˆ = 12.8 3 The trouble with this estimate is that it is so obviously wrong. A probability model defined to take values only over the range [ 0, 12.8 ] would not permit an observation as high as 13.1 , the third value obtained in the sample. Alternative, ˆ θ ML = X ( n ) = 13.1 , which is a valid estimate. Shiu-Sheng Chen (NTU Econ) Statistics Fall 2019 3 / 4

MLE vs MME MLE vs MME is indeed R.A. Fisher (1890–1962) vs. Karl Pearson (1857–1936) Shiu-Sheng Chen (NTU Econ) Statistics Fall 2019 4 / 4