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rom Sample to Estimation: What is the Mysterious Formula Behind the Estimator
In statistics, an estimator is a rule used to calculate unknown parameters based on observed data. In this process, we distinguish between the estimator (rule), the quantity of interest (estimand) and its result (estimate). For example, the sample mean is widely used as an estimator of the population mean. Such estimators can be divided into point estimators and interval estimators. The former provides an estimate of a single value, while the latter gives a range of feasible values.
"A single value does not necessarily mean a single number, but can also be an estimator of a vector value or a function value."
Estimation theory focuses on the properties of estimators, i.e., the defining properties used to compare the performance of different estimators on the same data. These properties can help us judge which estimation rules are better in a particular situation. However, in robust statistics, the study involves not only considering good properties under narrow assumptions, but also involving the balance between poor properties that may arise under broader conditions.
Background
The so-called "estimator" or "point estimate" is a term in statistics that refers to the statistics used to infer the values of unknown parameters in a statistical model. More specifically, "an estimator is a method chosen to obtain estimates of unknown parameters." The parameter being estimated is often called an estimand, which can be of finite dimension (in parametric and semiparametric models) or infinite dimension (in semiparametric and nonparametric models).
"As a function of the data, the estimator itself is also a random variable; a specific realization of the random variable is called an 'estimate'."
Although in practice, the definition of estimators places few restrictions on the functional form of the data used, their attractiveness often depends on their properties, such as unbiasedness, mean square error, consistency, and asymptotic distribution. . These properties provide a theoretical basis for the construction and comparison of estimators. In the context of decision theory, an estimator is viewed as a decision rule whose performance can be evaluated through a loss function.
Properties of the estimator
The following definitions and properties are related:
Error
For a given sample x, the "error" of the estimator is defined as:
e(x) = estimated value(x) - true value
Mean square error
Mean square error (MSE) is the expected value of the squared error between the estimated value and the true value. The formula is:
MSE = E[(estimated value(X) - true value)²]
"If you think of the parameters as the bullseye of the target, then the estimator is the shooting process, and each arrow is the estimate."
Deviation
The bias of the estimator (Bias) is defined as the distance between the expected value of the estimated value and the true value, expressed as:
Bias(estimator) = E(estimator) - true value
For situations where an unbiased estimator is desired, the unbiased estimator will not systematically produce estimates that are larger or smaller than the true value. In practical problems, if a small amount of bias is acceptable, it is possible to find an estimator with a smaller mean square error.
Unbiased
For an estimator, unbiasedness is an expectation property that indicates that the estimator does not deviate systematically from the true parameters in the long run. Ideally, an unbiased estimator should minimize the mean square error.
Finally, these statistical properties not only help us understand the performance of estimators, but also continue to promote the development of data analysis. How do different types of data and parameter requirements affect the selection and utilization of estimators?
