statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population parameter(which is called the "estimand"); an "estimate" is the result from the actual application of the function to a particular sampleof data. Many different estimators are possible for any given parameter. Some criterion is used to choose between the estimators, although it is often the case that a criterion cannot be used to clearly pick one estimator over another.To estimate a parameter of interest (e.g., a population mean, a binomial proportion, a difference between two population means, or a ratio of two population standard deviation), the usual procedure is as follows:
#Select a random
samplefrom the population of interest.
#Calculate the point estimate of the parameter.
#Calculate a measure of its variability, often a
#Associate with this estimate a measure of variability.
There are two types of estimators:
point estimators and interval estimators.
Suppose we have a fixed "parameter" that we wish to estimate. Then an "estimator" is a function that maps a "sample design" to a set of "sample estimates". An estimator of is usually denoted by the symbol . A "sample design" can be thought of as an ordered pair where is a set of samples (or outcomes), and is the probability density function. The probability density function maps the set to the closed interval [0,1] , and has the property that the sum (or integral) of the values of , over all in , is equal to 1. For any given subset of , the sum or integral of over all in is .
For all the properties below, the value , the estimation formula, the set of samples, and the set probabilities of the collection of samples, can be considered "fixed". Yet since some of the definitions vary by sample (yet for the same set of samples and probabilities), we must use in the notation. Hence, the estimate for a given sample is denoted as .
We have the following definitions and attributes.
# For a given sample , the "error" of the estimator is defined as , where is the estimate for sample , and is the parameter being estimated. Note that the error depends not only on the estimator (the estimation formula or procedure), but on the sample.
# The "
mean squared error" of is defined as the expected value (probability-weighted average, over all samples) of the squared errors; that is, . It is used to indicate how far, on average, the collection of estimates are from the single parameter being estimated. Consider the following analogy. Suppose the parameter is the bull's-eye of a target, the estimator is the process of shooting arrows at the target, and the individual arrows are estimates. Then high MSE means the average distance of the arrows from the target is high, and low MSE means the average distance from the target is low. The arrows may or may not be clustered. For example, even if all arrows hit the same point, yet grossly miss the target, the MSE is still relatively large. Note, however, that if the MSE is relatively low, then the arrows are likely more highly clustered (than highly dispersed).
# For a given sample , the "
sampling deviation" of the estimator is defined as , where is the estimate for sample , and is the expected value of the estimator. Note that the sampling deviation depends not only on the estimator, but on the sample.
# The "
variance" of is simply the expected value of the squared sampling deviations; that is, . It is used to indicate how far, on average, the collection of estimates are from the "expected value" of the estimates. Note the difference between MSE and variance. If the parameter is the bull's-eye of a target, and the arrows are estimates, then a relatively high variance means the arrows are dispersed, and a relatively low variance means the arrows are clustered. Some things to note: even if the variance is low, the cluster of arrows may still be far off-target, and even if the variance is high, the diffuse collection of arrows may still be unbiased. Finally, note that even if all arrows grossly miss the target, if they nevertheless all hit the same point, the variance is zero.
# The "bias" of is defined as . It is the distance between the average of the collection of estimates, and the single parameter being estimated. It also is the expected value of the error, since . If the parameter is the bull's-eye of a target, and the arrows are estimates, then a relatively high absolute value for the bias means the average position of the arrows is off-target, and a relatively low absolute bias means the average position of the arrows is on target. They may be dispersed, or may be clustered. The relationship between bias and variance is analogous to the relationship between
accuracy and precision.
# is an "unbiased estimator" of
if and only if. Note that bias is a property of the estimator, not of the estimate. Often, people refer to a "biased estimate" or an "unbiased estimate," but they really are talking about an "estimate from a biased estimator," or an "estimate from an unbiased estimator." Also, people often confuse the "error" of a single estimate with the "bias" of an estimator. Just because the error for one estimate is large, does not mean the estimator is biased. In fact, even if all estimates have astronomical absolute values for their errors, if the expected value of the error is zero, the estimator is unbiased. Also, just because an estimator is biased, does not preclude the error of an estimate from being zero (we may have gotten lucky). The ideal situation, of course, is to have an unbiased estimator with low variance, and also try to limit the number of samples where the error is extreme (that is, have few outliers). Yet unbiasedness is not essential. Often, if we permit just a little bias, then we can find an estimator with lower MSE and/or fewer outlier sample estimates.
# The MSE, variance, and bias, are related: :i.e. mean squared error = variance + square of bias.
standard deviationof an estimator of θ (the square rootof the variance), or an estimate of the standard deviation of an estimator of θ, is called the "standard error" of θ.
A consistent sequence of estimators is a sequence of estimators that converge in probability to the quantity being estimated as the index (usually the
sample size) grows without bound.
Mathematically, a sequence of estimators is a consistent estimator for
parameterif and only if, for all , no matter how small, we have
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