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 Let {Xi}n

i=1 ∼i.i.d. U[0, θ], θ > 0.

1

Find the maximum likelihood estimator for θ, denoted by ˆ θML

2

Is ˆ θML unbiased?

3

Is ˆ θML consistent?

4

Use ˆ θML to construct a unbiased estimator, denoted by ˜ θML

5

Is ˜ θML consistent?

6

Find the method of moments estimator for θ, denoted by ˆ θMM

7

Is ˆ θMM unbiased?

8

Is ˆ θMM consistent?

9

Compare Var(ˆ θML) and Var(ˆ θMM)

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 x1 = 3.2, x2 = 2.9, x3 = 13.1. Hence, ˆ θMM = 2 ¯ X = 23.2 + 2.9 + 13.1 3 = 12.8 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 θML = 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