• Home
• About
• Academics
•     Courses
•     Undergraduate
•     Graduate
• Alumni
• Career Guidance
• Giving
• Research
• Resources
• Stat Consulting Center
• Directory
• Contact Us

Stat 4620: Introduction to Mathematical Statistics

Instructor:  Joshua D. Naranjo E-mail: joshua.naranjo@wmich.edu Phone: (269)387-4548 Office: 5507 Everett Tower

WEEK 13 Thursday April 18

• Practice final (from Spring 2016)

Tuesday April 16

• Notes (Likelihood estimation in logistic regression (GLM) )
• Journal article: Association between homocysteine and risk of stroke

WEEK 12 Thursday April 11

• Notes (Section 6.5: Multiparameter testing)
• Homework 21: Problem 6.5.2 (7th Ed) or 6.5.4 (8th Ed)

Tuesday April 9

• Notes (Section 6.4: Multiparameter estimation)
• Homework 20
  1. Find the information matrix I(mu, sigma^2) directly.
  2. Show: Var(S^2)= 2 sigma^4/(n-1) where S^2 is the unbiased sample variance. (Hint: (n-1)S^2/sigma^2 has a chi-square distribution with n-1 degrees of freedom.)

WEEK 11 Thursday April 4

• Notes (Section 6.3 con't: Asymptotic distribution of LRT, other MLE-based tests)
• Homework 19
  Given the following sample of size 20 from a Poisson distribution with mean Θ,
        9, 12, 9, 5, 7, 5, 7, 9, 7, 4, 7, 8, 8, 10, 6, 8, 5, 5, 8, 7
  test H0: Θ=8 against H1:Θ ≠ 8, using
  1. -2 ln(Λ)
  2. Wald test
  3. Scores test

Tuesday April 2

• Notes (Section 6.3: The Likelihood ratio test)
• Homework 18
  1. Problem 6.3.8 (7th Ed) or 6.3.9 (8th Ed)
  2. Problem 6.3.10 (7th Ed) or 6.3.11 (8th Ed)

WEEK 10 Thursday March 28

• Notes (Section 6.2: Likelihood-based confidence intervals, Delta Method)
• Homework 17: Given a random sample (x1,...,xn) from N(0, theta)
  1. Construct a likelihood-based confidence interval for the population variance theta.
  2. Construct a likelihood-based confidence interval for the population standard deviation sqrt(theta).
  3. Conduct a simulation in R by generating n=20 from N(0, 16). Calculate confidence intervals for theta and sqrt(theta) and confirm that coverage levels are approximately .95.

Tuesday March 26

• Notes (Section 6.2: Aymptotic normality and efficiency of mle)
• Homework 16
  1. Problem 6.2.10
  2. Simulation of ARE
     1. Generate 30 observations from Laplace distribution.
        • Calculate the mean and median
        • Repeat 1000 times, and confirm that ARE(median, mean)=2.0
     2. Generate 30 observations from N(0,1) distribution.
        • Calculate the mean and median
        • Repeat 1000 times, and confirm that ARE(median, mean)=0.64

WEEK 9 Thursday March 21

• Notes (Section 6.2: Information and Rao-Cramer Lower Bound)
• Homework 15
  1. Problem 6.2.7 (a), (b)
  2. Problem 6.2.8 (a), (b)

Tuesday March 19

• Notes (Section 6.1: Maximum likelihood theory)
• Homework 14 R solution
• Homework 14: Simulation
  1. Generate 100 observations from Normal distribution.
     • Calculate the mle under normal, Laplace, and logistic
     • Repeat 1000 times, and compare the simulated variance of the three estimators. Which one has smallest variance?
  2. Repeat (1) for 100 observations from Laplace. Which estimator has smallest variance?
  3. Repeat (1) for 100 observations from Logistic. Which estimator has smallest variance?

WEEK 8 Thursday March 14

• Notes (Section 4.9: Bootstrap)
• Homework 13
  1. Problem 4.9.6 (7th Ed) or 4.9.7 (8th Ed)

Tuesday March 12

• Notes (Section 4.8: Monte Carlo Method)
• Homework 12
  1. Problem 4.8.1
  2. Problem 4.8.6 (Generate and print a sample of n=15 observations from given pdf. Calculate the mean and sd of the sample, and compare to the true mean and sd.)

WEEK 7 Thursday Feb 21

• Notes (Section 4.7: Chi-square tests: Homogeneity and independence)

Tuesday Feb 19

• Notes (Section 4.7: Chi-square tests: Goodness-of-fit)
• Homework 11
  1. Problem 4.7.3 (7th Ed) or 4.7.4 (8th Ed)
  2. Problem 4.7.9
  WEEK 6 Thursday Feb 14
  • Notes (Section 4.6: Hypothesis test, con't.)
  • R code (Welch-t vs pooled-t simulation of size and power)
  • Homework 10
    1. Problem 4.6.4
    2. Problem 4.6.7
  Tuesday Feb 12
  • Notes (Section 4.5: Intro to hypothesis test)
  • Homework 9
  WEEK 5 Thursday Feb 7
  • Notes (Section 4.4.1: Quantiles)
  • Homework 8
    1. Problem 4.4.25
    2. Problem 4.4.28
  Tuesday Feb 5
  • Notes (Section 4.4: Order Statistics)
  • Homework 7
    1. Problem 4.4.6(a)
    2. Problem 4.4.13
  WEEK 4 Thurday Jan 31
  • No class (weather)
  Tuesday Jan 29
  • No class (weather)
  WEEK 3 Thursday Jan 24
  • Notes (Comparing dependent means and proportions, paired-t, McNemar's test)
  • BMR handout (illustrates independent and dependent data)
  • Homework 6: Answer Investigator's questions 1 and 2 in the BMR handout.

  Tuesday, Jan. 22
  

  • Notes (Welch-t vs pooled-t, two proportions, test of hypothesis, Fisher exact)
  • Homework 5
    1. Problem 4.2.21 (Compare pooled-t and Welch-t SE, CI, and p-value.)
    2. Problem 4.2.26
  • The following papers compare means and proportions
    • Effectiveness of Maternal Influenza Immunization in Mothers and Infants (see Table 1, Table 2)
    • Antimicrobial treatment of acute otitis media (see Figure 3, Table 2)
    • Naltrexone in the treatment of alcohol dependence (see Table 4)
  WEEK 2 Thursday, Jan. 17
  
  • Notes (CI for proportion, CI for difference between two means)
  • Homework 4
  Tuesday, Jan. 15
  
  • Notes (Confidence interval for mean, pivot method of constructing CI)
  • Homework 3: Problem 4.2.1, 4.2.2, 4.2.9 7th Ed (this is 4.2.8 in 8th Ed)
  • The following paper uses both SD and SE of the mean
    • Cognitive Function in postmenopausal women treated with Raloxifene
      
  • R session: Homework 2
  WEEK 1 Thursday, Jan. 10
  
  • Notes (MLE for multiple parameters, nonparametric density estimation)
  • Homework 2: Problem 4.1.8
  • The following papers use kernel density estimation
    • Variability of particulate emissions from woodstoves in New Zealand
      
    • Length selectivity of commercial fish traps
      
  Tuesday, Jan. 8
  
  • Notes (Point estimation, MLE)
  • Homework 1: Using the data in Exercise 4.1.3
    1. Find the MLE of theta
    2. Using R, numerically confirm that your answer in (1) maximizes the likelihood function
  • Syllabus (pdf)
    
  • Ex. 4.4.1: Exponential MLE example in R
  • The following paper uses both means and proportions
    • Effectiveness of Maternal Influenza Immunization in Mothers and Infants
  TOP