Minimum chi-square estimation
In statistics, minimum chi-square estimation is a method of estimation of unobserved quantities based on observed data.[1]
In certain chi-square tests, one rejects a null hypothesis about a population distribution if a specified test statistic is too large, when that statistic would have approximately a chi-square distribution if the null hypothesis is true. In minimum chi-square estimation, one finds the values of parameters that make that test statistic as small as possible.
Among the consequences of its use is that the test statistic actually does have approximately a
Illustration via an example
Suppose a certain
The
Finding the minimum chi-square estimate
The minimum chi-square estimate of the population mean λ is the number that minimizes the chi-square statistic
where a is the estimated expected number in the "> 8" cell, and "20" appears because it is the sample size. The value of a is 20 times the probability that a Poisson-distributed random variable exceeds 8, and it is easily calculated as 1 minus the sum of the probabilities corresponding to 0 through 8. By trivial algebra, the last term reduces simply to a. Numerical computation shows that the value of λ that minimizes the chi-square statistic is about 3.5242. That is the minimum chi-square estimate of λ. For that value of λ, the chi-square statistic is about 3.062764. There are 10 cells. If the null hypothesis had specified a single distribution, rather than requiring λ to be estimated, then the null distribution of the test statistic would be a chi-square distribution with 10 − 1 = 9 degrees of freedom. Since λ had to be estimated, one additional degree of freedom is lost. The expected value of a chi-square random variable with 8 degrees of freedom is 8. Thus the observed value, 3.062764, is quite modest, and the null hypothesis is not rejected.
Notes and references
- JSTOR 2240587.