Transcript C19_CIS2033

CIS 2033 based on
Dekking et al. A Modern Introduction to Probability and Statistics. 2007
Slides by Nathan Weiser
Format by Tim Birbeck
Instructor Longin Jan Latecki
C19: Unbiased Estimators
19.1 – Estimators
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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.
19.1 – Estimators
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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
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:
number of X i equal to zero
S
n
T e
Xn
19.3 The Sampling Distribution and
Unbiasedness
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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:
Y
S
n
Where Y is the number Xi equal to zero

Thus is follows that:
E[Y ] np0
E[ S ] 

 p0
n
n
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
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
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:
X 1  X 2  ...  X n
Xn 
n
Is an unbiased estimator for µ and
1 n
2

(
X

X
)
n
i
S n n 1 
i 1
2
Is an unbiased estimator for σ2