# Parameter Estimation Task

A parameter estimation task is a point estimation task that produces a parameter estimate (for a population parameter).

**AKA:**Parameter Fitting.**Context:**- It can (typically) require a Loss Function.
- It can be solved by a Parameter Estimation System (that applies a Parameter Estimation algorithm).

**Example(s):****Counter-Example(s):****See:**Sample Statistic, Function Selection Task, Function Optimization, Experiment Design, Estimation Theory.

## References

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/estimation_theory Retrieved:2015-6-12.
**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.For example, it is desired to estimate the proportion of a population of voters who will vote for a particular candidate. That proportion is the parameter sought; the estimate is based on a small random sample of voters.

Or, for example, in radar the goal is to estimate the range of objects (airplanes, boats, etc.) by analyzing the two-way transit timing of received echoes of transmitted pulses. Since the reflected pulses are unavoidably embedded in electrical noise, their measured values are randomly distributed, so that the transit time must be estimated.

In estimation theory, two approaches are generally considered.

- The probabilistic approach (described in this article) assumes that the measured data is random with probability distribution dependent on the parameters of interest
- The set-membership approach assumes that the measured data vector belongs to a set which depends on the parameter vector.

- For example, in electrical communication theory, the measurements which contain information regarding the parameters of interest are often associated with a noisy signal. Without randomness, or noise, the problem would be deterministic and estimation would not be needed.

### 2009

- http://research.microsoft.com/en-us/um/people/zhang/inria/publis/tutorial-estim/node3.html
- Parameter estimation is a discipline that provides tools for the efficient use of data for aiding in mathematically modeling of phenomena and the estimation of constants appearing in these models [2]. It can thus be visualized as a study of inverse problems. Much of parameter estimation can be related to four optimization problems:
- criterion: the choice of the best function to optimize (minimize or maximize)
- estimation: the optimization of the chosen function
- design: optimal design to obtain the best parameter estimates
- modeling: the determination of the mathematical model which best describes the system from which data are measured.

- Parameter estimation is a discipline that provides tools for the efficient use of data for aiding in mathematically modeling of phenomena and the estimation of constants appearing in these models [2]. It can thus be visualized as a study of inverse problems. Much of parameter estimation can be related to four optimization problems:

### 2003

- (Myung, 2003) ⇒ In Jae Myung. (2003). “Tutorial on Maximum Likelihood Estimation.” In: Journal of Mathematical Psychology, 47. doi:10.1016/S0022-2496(02)00028-7
- QUOTE: As these laws and principles are not directly observable, they are formulated in terms of hypotheses. In mathematical modeling, such hypotheses about the structure and inner working of the behavioral process of interest are stated in terms of parametric families of probability distributions called models. The goal of modeling is to deduce the form of the underlying process by testing the viability of such models. Once a model is specified with its parameters, and data have been collected, one is in a position to evaluate its goodness of fit, that is, how well it fits the observed data. Goodness of fit is assessed by finding parameter values of a model that best fits the data — a procedure called parameter estimation.

### 1977

- (Beck & Arnold, 1977) ⇒ J. V. Beck, and K. J. Arnold. (1977). “Parameter Estimation in Engineering and Science." Wiley series in probability and mathematical statistics.

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