File Name: unbiased and biased estimators .zip
Theoretical Statistics pp Cite as.
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In many scientific research fields, statistical models are used to describe a system or a population, to interpret a phenomenon, or to investigate the [Page 85] relationship among various measurements. These statistical models often contain one or multiple components, called parameters , that are unknown and thus need to be estimated from the data sometimes also called the sample. An estimator, which is essentially a function of the observable data, is biased if its expectation does not equal the parameter to be estimated. Let be its estimator based on an observed sample. Then is a biased estimator if , where E denotes the expectation operator.
In statistics , the bias or bias function of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called unbiased. In statistics, "bias" is an objective property of an estimator. Bias can also be measured with respect to the median , rather than the mean expected value , in which case one distinguishes median -unbiased from the usual mean -unbiasedness property. Bias is a distinct concept from consistency. Consistent estimators converge in probability to the true value of the parameter, but may be biased or unbiased; see bias versus consistency for more. All else being equal, an unbiased estimator is preferable to a biased estimator, although in practice, biased estimators with generally small bias are frequently used.
An estimator of a given parameter is said to be unbiased if its expected value is equal to the true value of the parameter. In other words, an estimator is unbiased if it produces parameter estimates that are on average correct. The estimate is usually obtained by using a predefined rule a function that associates an estimate to each sample that could possibly be observed. The function is called an estimator. Definition An estimator is said to be unbiased if and only if where the expected value is calculated with respect to the probability distribution of the sample. The following table contains examples of unbiased estimators with links to lectures where unbiasedness is proved. The bias of an estimator is the expected difference between and the true parameter:.
Example 3. Definition 3. In order to compute its expectation, we need to obtain its p. We can derive it from Exercise 2. The c. Checking whether it is unbiased requires its p. Then, by the additive property of the gamma see Exercise 1.
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