Minimum Chi Squared Estimator

TL;DR

• Estimates population parameters from sample data by minimizing a chi-squared measure of discrepancy.
• Particularly useful for large datasets and complex models because it yields parameter estimates with relatively low computational cost.
• Commonly applied in contexts such as survey analysis and financial parameter estimation.

Definition

The minimum chi-squared estimator is a statistical method used to estimate the parameters of a population based on a sample by minimizing a chi-squared statistic that measures the difference between observed and expected values of model parameters.

Explanation

The method proceeds by specifying a parametric model and the parameters to estimate. From the collected sample, observed values for the model parameters are computed (for example, sample proportions, sample means, or sample standard deviations). Expected values for those parameters are derived from the model under candidate parameter values.

A chi-squared statistic is then calculated as the sum, over parameters or categories, of the squared differences between observed and expected values divided by the expected values. Minimizing this chi-squared statistic with respect to the model parameters yields the minimum chi-squared estimates. Numerical optimization algorithms, such as the Nelder-Mead simplex method or the Levenberg-Marquardt algorithm, are typically used to find these minimizing parameter values.

The parameter estimates obtained by this procedure are the minimum chi-squared estimator of the population parameters and can be used for predictions and inference about the population based on the sample.

$$\chi^2 \;=\; \sum_i \frac{(O_i - E_i)^2}{E_i}$$

(where O_i are observed values and E_i are expected values for each parameter or category, as described in the source text).

Examples

Survey data

A researcher wants to estimate the proportion of individuals in a population who have a certain trait (for example, a disease or risk factor). The researcher collects a sample and records trait status. The observed proportion in the sample is compared to the expected proportion under candidate parameter values for the population, the chi-squared statistic is computed, and the parameter value that minimizes this statistic is taken as the minimum chi-squared estimate of the population proportion.

Financial data

A financial analyst aims to estimate the expected return and volatility of a portfolio. The analyst collects data on returns and volatilities of individual assets, computes observed sample means and standard deviations, and compares these to expected values implied by candidate parameter settings for a multivariate model. The chi-squared statistic formed from these discrepancies is minimized (commonly via numerical optimization) to produce minimum chi-squared estimates of the portfolio’s expected return and volatility.

Use cases

• Survey analysis (estimating population proportions)
• Finance (estimating expected returns and volatilities of portfolios)
• More generally, parameter estimation for complex models and large datasets, as noted in the source

Related terms

• Chi-squared statistic
• Binomial distribution
• Multivariate normal distribution
• Nelder-Mead simplex method
• Levenberg-Marquardt algorithm
• Sample mean and sample standard deviation