ECON 451 - Winter 2020: Homework 6 on Point and Interval Inference

This document contains the complete solution to Homework 6 for ECON 451: Intermediate Introduction to Statistics and Econometrics I. The assignment focuses on point estimation techniques, specifically the Method of Moments (MOM) and Maximum Likelihood (ML) estimators, as well as interval inference. The solution provides detailed steps and calculations for finding the MOM estimators for given probability density functions (PDFs) and the maximum likelihood estimators for unknown parameters, utilizing log-likelihood functions. The assignment covers different scenarios, including discrete and continuous distributions, requiring the application of statistical concepts to estimate parameters. Furthermore, the solution presents the estimation of parameters based on given data points and the determination of maximum likelihood estimates within specified intervals. The document references Myung's (2003) tutorial on maximum likelihood estimation, providing a comprehensive approach to solving the problems.

Running head ECON 451 1
ECON 451 Intermediate Introduction to Stats. and Econometrics I
Student’s Name
Institutional Affiliation

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ECON 451 INTERMEDIATE INTRODUCTION TO STATS. AND ECONOMETRICS I
Homework 6 covers material of point (MOM and ML) and interval inference.
1. Find the method of moment estimators for the unknown parameters in the pdf
below:
a.
Solution:
This is based on a random sample Y1, Y2,...., Yn
The first and second theoretical moments for this distribution;
Hence the solution is satisfied.
b.
Use E[Y] and E[Y2]
Solution
This is based on a random sample Y1, Y2,....., Yn
The first and second theoretical moments for the normal distribution are;
The first and second sample moments are;
Solving the equation results to
and
Therefore the method of moment’s estimate of the above is;

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Hence satisfied
2. Find the maximum likelihood estimator/estimate of the following unknown
parameters (use log likelihood for both):
a.
Solution
of the above function can be give as
b.
=Considering the points given values;
The average is 0.46 hence estimate is 0.46
3.

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According to the interval 0<y<1 for the size of 0.7,0.63, 0.92,0.86,0.43 and 0.21 the maximum
likelihood estimate of θ. The maximum value in this function is 1 and θ estimator Max(yi)
4.
Note that for every i then 0 < <min
Thus the MLE is the largest possible value for estimate is Min (Yi)
Reference
Myung, I. J. (2003). Tutorial on maximum likelihood estimation. Journal of mathematical
Psychology, 47(1), 90-100.

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