1. C19: Unbiased Estimators
MATH 3033 based on Dekking et al. A Modern Introduction to Probability and Statistics Slides by Nathan Weiser Format by Tim Birbeck Instructor Longin Jan Latecki
C19: Unbiased Estimators

2. 19.1 – Estimators
The parameters that determine the model distribution are called the model parameters
We focus on a situation where a parameter correspond to a feature of the model distribution that can be described by the model parameters themselves or by some function of the model parameters. This is known as the parameter of interest.

3. 19.1 – Estimators
ESTIMATE: an estimate is a value t that only depends on the dataset x1, x2,…,xn, i.e., t is some function that of the dataset only:
t = h(x1, x2,…,xn)
ESTIMATOR: Let t = h(x1, x2,…,xn) be an estimate based on the dataset x1, x2,…,xn. Then t is a realization of the random variable
T= h(X1, X2,…,Xn).
The random variable T is called an estimator.
Estimator refers to the method or device for estimation
Estimate refers to the actual value computed from a dataset

4. 19.2 Investigating the behavior of an estimator
Estimating the probability p0 of zero arrivals, which is an unknown number between 0 and 1. Possible estimators:

5. 19.3 The Sampling Distribution and Unbiasedness
Desireable:
E[S]=p0
The Sampling Distribution: Let T= h(X1, X2,…,Xn) be an estimator based on a random sample X1, X2,…,Xn. The probability distribution of T is called the sampling distribution of T.
Sampling Distribution of S:
Where Y is the number Xi equal to zero
Thus is follows that:

6. 19.3 The Sampling Distribution and Unbiasedness
Definition: An estimator T is called an unbiased estimator for the parameter Ө, if
E[T] = Ө
Irrespective of the value of Ө. The difference E[T] – Ө is called the bias of T; if this difference is nonzero, then T is called biased

7. 19.3 The Sampling Distribution and Unbiasedness
Definition: An estimator T is called an unbiased estimator for the parameter Ө, if
E[T] = Ө
Irrespective of the value of Ө. The difference E[T] – Ө is called the bias of T; if this difference is nonzero, then T is called biased

8. 19.4 Unbiased Estimators for Expectation and Variance
Suppose X1, X2,…,Xn is a random sample from a distribution with finite
expectation µ and finite variance σ2. Then:
Is an unbiased estimator for µ and
Is an unbiased estimator for σ2