Parameter estimation
-
Upload
ravi-prasad-kj -
Category
Documents
-
view
107 -
download
2
Transcript of Parameter estimation
![Page 1: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/1.jpg)
PARAMETER ESTIMATION Chapter 7
![Page 2: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/2.jpg)
EVALUATING A POINT ESTIMATOR
Let X = (X1, . . . , Xn) be a sample from a population whose distribution is specified up to an unknown parameter θ.
Let d = d(X) be an estimator of θ.
How are we to determine its worth as an estimator of θ ?
![Page 3: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/3.jpg)
EVALUATING A POINT ESTIMATOR
r(d, θ) : the mean square error of the estimator d
An indicator of the worth of d as an estimator of θ
No single estimator d that minimized r(d, θ) for all
possible values of θ.
![Page 4: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/4.jpg)
EVALUATING A POINT ESTIMATOR
Minimum mean square estimators rarely exist
Possible to find an estimator having the smallest mean square error among all estimators that satisfy a certain property.
One such property is that of unbiasedness.
An estimator is unbiased if its expected value always equals the value of the parameter it is attempting to estimate.
The bias of d as an estimator of θ is defined as below:
![Page 5: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/5.jpg)
EVALUATING A POINT ESTIMATOR
Example: Let X = (X1, . . . , Xn) be a sample from a population whose distribution is specified up to an unknown parameter θ.
Let d = d(X) be an estimator of θ.
Find the bias for the following estimators:
![Page 6: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/6.jpg)
EVALUATING A POINT ESTIMATOR
![Page 7: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/7.jpg)
EVALUATING A POINT ESTIMATOR
![Page 8: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/8.jpg)
EVALUATING A POINT ESTIMATOR
Combining Independent Unbiased Estimators:
Let d1 and d2 denote independent unbiased estimators of θ,
having known variances σ12 and σ2
2. For i = 1, 2,
New estimator
from the old
ones
![Page 9: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/9.jpg)
EVALUATING A POINT ESTIMATOR
It will be unbiased.
To determine the value of λ that results in d having the smallest possible mean square error:
![Page 10: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/10.jpg)
EVALUATING A POINT ESTIMATOR
The optimal weight to give an estimator is inversely proportional to its
variance (when all the estimators are unbiased and independent).
![Page 11: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/11.jpg)
EVALUATING A POINT ESTIMATOR
A generalization of the result that the mean square error of an unbiased estimator is equal to its variance
is that
the mean square error of any estimator is equal to its variance plus the square of its bias.
![Page 12: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/12.jpg)
EVALUATING A POINT ESTIMATOR
![Page 13: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/13.jpg)
EVALUATING A POINT ESTIMATOR
The mean square error of any estimator is equal to its variance plus the square of its bias
![Page 14: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/14.jpg)
EVALUATING A POINT ESTIMATOR
Let X = (X1, . . . , Xn) be a sample from
a uniform distribution (0, θ) with the unknown parameter θ.
Let us evaluate the following estimators:
![Page 15: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/15.jpg)
EVALUATING A POINT ESTIMATOR
Is d1 is unbiased?
![Page 16: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/16.jpg)
EVALUATING A POINT ESTIMATOR
![Page 17: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/17.jpg)
How to find the mean and variance of this estimator?
First we have to find the distribution :-
![Page 18: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/18.jpg)
![Page 19: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/19.jpg)
![Page 20: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/20.jpg)
![Page 21: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/21.jpg)
![Page 22: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/22.jpg)
IS THE FOLLOWING ESTIMATOR BIASED?
![Page 23: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/23.jpg)
![Page 24: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/24.jpg)
![Page 25: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/25.jpg)
The (biased) estimator (n + 2)/ (n + 1) maxi Xi has about
half the mean square error of the MLE maxi Xi.
![Page 26: Parameter estimation](https://reader030.fdocuments.net/reader030/viewer/2022012402/55a68b9f1a28ab895f8b46bb/html5/thumbnails/26.jpg)
Thank you for your attention