Parameter Estimation is a branch of statistics that involves using sample data to estimate the parameters of a distribution.

## Methods of Parameter Estimation

The techniques used for parameter estimation are called estimators.

Some estimators are:

- Probability Plotting: A method of finding parameter values where the data is plotted on special plotting paper and parameters are derived from the visual plot
- Rank Regression (Least Squares): A method of finding parameter values that minimizes the sum of the squares of the residuals.
- Maximum Likelihood Estimation: A method of finding parameter values that, given a set of observations, will maximize the likelihood function.
- Bayesian Estimation Methods: A family of estimation methods that tries to minimize the posterior expectation of what is called the utility function. In practice, what this means is that existing knowlege about a situation is formulated, data is gathered, and then posterior knowledge is used to update our beliefs.

## Qualities of Estimators

If the value an estimator estimates for the parameter, θ’, always converges to the actual parameter value θ as the quantity of data used for parameter estimation increases, we say an estimator is **consistent**.

The bias of an estimator is the deviation of the expectation from the actual true value. If, for a given estimator, the bias is zero, we say that that estimator is **unbiased**.

A third statistic that tells us about the reliability of an estimator is the variance. If an estimator has lower variance than another we say it is more **efficient**, and we can calculate the **efficiency** of estimator p relative to estimator q as (Var(θ’_{p}))/Var(θ’_{q}).

## References

Levy, Roger. Parameter Estimation. Chapter 4 in Probabilistic Models in the Study of Language, a work in progress. Retrieved from http://idiom.ucsd.edu/~rlevy/pmsl_textbook/chapters/pmsl_4.pdf on August 12, 2018.

Cadima, Emygdio. Estimation of Parameters. Chapter 7 in Fish Stock Assessment Manual. Fod and Agricultural Organization of the United Nations. Rome, 2003 Retrieved from http://www.fao.org/docrep/006/X8498E/x8498e0e.htm on August 12, 2018.

Eshky, Ashiel. Bayesian Methods of Parameter Estimation. Retrieved from http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/AV0809/eshky.pdf on August 16, 2018.

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