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Lecture

Basic Principles of Point Estimation

In course

MATH-442: Statistical theory

This course gives a mostly rigourous treatment of some statistical methods outside the context of standard likelihood theory. ...

Description

This lecture covers the basic principles of point estimation, focusing on the Method of Moments, Bias-Variance tradeoff, Consistency, Plug-In Principle, and Likelihood Principle. It explains how estimators are constructed, the Bias-Variance decomposition of Mean Squared Error, and the importance of consistency in estimators. The lecture also delves into the Likelihood Function, Maximum Likelihood Estimators, and the Moment Principle. Examples are provided to illustrate concepts such as Density Estimation, Bias, and Mean Squared Error. The instructor emphasizes the importance of understanding the quality of estimators and the tradeoffs involved in choosing different estimation methods.

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Instructor

Yoav Zemel

Official source

Ontological neighbourhood

Statistics

Statistical inference: Mathematical statistics

Data analysis: Regression analysis

Related concepts (32)

Maximum likelihood estimation

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference.

Bias of an estimator

In statistics, the bias of an estimator (or bias function) 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 property of an estimator. 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.

Estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, the sample mean is a commonly used estimator of the population mean. There are point and interval estimators. The point estimators yield single-valued results. This is in contrast to an interval estimator, where the result would be a range of plausible values.

Mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the actual value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive (and not zero) is because of randomness or because the estimator does not account for information that could produce a more accurate estimate.

Estimation theory

Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the measured data. An estimator attempts to approximate the unknown parameters using the measurements.

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