A modified chi-square statistics of the linear estimator for inter-laboratory comparison

Chi-square statistics of the uncertainty weighted mean, which is a linear estimator, is widely used in data analysis of inter-laboratory comparison. However, the chi-square statistics of other linear estimators is not investigated. In this study, a modified chi-square statistics, which comprises the linear estimator, is proposed under the condition that comparison results are Gaussian distributed with a common mean. The proposed statistics is analyzed through Monte Carlo simulation by combining the weights of linear estimator into a multi-dimension vector. Simulation results show that the proposed statistics is (n−1)th-order chi-square distributed when the weights vector of linear estimator is located in a particular subspace, which is influenced by the uncertainties of participants. Furthermore, this chi-square statistics of arithmetic mean is applied to the common mean and random effects models as examples. For the common mean model, the statistics can be applied to the hypothesis testing of arithmetic mean; for the random effects model, the statistics can be applied to the variance estimation of random effects.