A statistical test for the mean squared error

Abstract

The likelihood ratio test for a specified value of the mean squared error is derived assuming observation from a normal distribution. The mean squared error is a well established tool for assessing closeness to a target value when bias as well as sampling error (or measurement error) is taken into account. While the distributional properties of the mean squared error are well known, so far no simple tests for hypotheses concerning the MSE have been established. An explicit expression for the likelihood ratio test statistic is provided together with a simple intuitive interpretation. The distribution of the test statistic under the null-hypothesis is discussed and it is shown that the specified value of the MSE enters as a scale-parameter. This means that it is only necessary to determine the distribution of the test-statistic when the hypothesized MSE equals one. In principle, the null distribution of the test-statistic depends on the partition of the MSE into the bias component and the variance. However, our results show that for moderate to large sample sizes, the effect of this partition is negligible. For practical application it is therefore recommended that the test is performed as a conservative test. It is shown that the test also applies to situations where the relative mean squared error is of interest. The theory is illustrated with data from a chemical analysis on blood levels of lead.

Keywords:

• Mean squared error
• relative mean squared error
• likelihood ratio test
• normal distribution