02 Statistical model and estimation Plan of the
- Slides: 38
02 Statistical model and estimation
Plan of the course 2
Learning goals • Concept of estimation • Construct estimators • Compare estimators • Exponential family of distribution
Recall •
Independent and identically distributed • iid
Example • iid
Point estimate and point estimator iid • iid
Nuisance parameter Parameter of interest
Constructing estimators
Moments of a random variable •
Example: 10 th raw moment of Bernoulli distribution •
Example: 2 nd moment of a uniform distribution U(0, 1) •
Method of moments •
Method of moments •
Example 2, waiting times •
Method of moments - summary • What is the relationship between the parameters of interest and moments? How many moments are needed? • Compute sample moments based on your data • Compute the estimate of the parameter of interest based on the sample moments • Homework: method of moments for finding parameters of Gamma distribution
Questions?
Exponential family of distributions
Exponential family of distributions •
Equivalent forms
Why exponential family of distributions? • Much of maximum likelihood estimation revolves around the derivative of the log-likelihood (called the score) • The score for a distribution in the exponential family straightforward to compute
Example: Bernoulli distribution
Example: Bernoulli distribution
Example: Poisson distribution •
Example: Poisson distribution •
Example: Normal distribution, known variance
Questions?
Least squares estimator •
Least square estimation •
Comparing estimators
Bias •
Example: Sample mean
Bias and mean square error
Example: mean square error
Questions? https: //forms. gle/VH 4 ebjyy. Ymqsj. K 4 BA