November 2012
Intermediate to advanced
794 pages
22h 9m
English
book
Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
by John J. Shynk
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9.2 STATISTICS
Definition: Statistic The statistic T is a function of random variables X which itself is a random variable.
About the notation, the function of X is not random; we could write T = g(X), but this will not be necessary until later in the chapter. For specific outcomes X = x, the value of the statistic is T = t. The symbol T is commonly used in the statistics literature for estimators (so we initially use it in this chapter), and should not be confused with T used for a time duration.
Definition: Estimator An estimator of θ is a statistic T that is a function of the samples X and is chosen to have certain desirable properties.
Example 9.1. The sample mean is probably the most widely used statistic:
(9.1) 
It is an unbiased estimator of the actual mean 
. The following statistic
(9.2) 
contains essentially the same information about 
as 
, but it is a biased estimator. T1 and T2 are statistically equivalent because they are related by one-to-one function :
(9.3)
where is the ...
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