t ‐statistics for weighted means in credit risk modeling

Purpose – The purpose of this paper is to describe a generalization of the familiar two‐sample t ‐test for equality of means to the case where the sample values are to be given unequal weights. This is a natural situation in financial risk modeling when some samples are considered more reliable than others in predicting a common mean. We also describe an example with real credit data showing that ignoring this modification of the two‐sample test can lead to the wrong statistical conclusion. Design/methodology/approach – We follow the analysis of the classical two‐sample tests in the more general situation of weighted means. We also test our methods against some market data to assess the importance of the findings. Findings – We formulate some explicit test statistics that should be used when the sample values are to be assigned differing known weights. Different cases are presented depending on how much is known about the variances. In the most typical case (the unpooled two‐sample test), we approximate the test statistic with a t ‐distribution. Proofs are given where possible. Research limitations/implications – In the unpooled case, we still only have an approximate t ‐distribution. This is related to the classical Behrens‐Fisher problem, which is still not fully solved. We also focus on the case where the sample values are normally distributed. It would be valuable to see how far the discussion can be extended to non‐normal distributions. Practical implications – Researchers should use the two‐sample test statistics given in this paper instead of the standard ones when testing for equality of weighted means. Originality/value – Weighted means occur frequently in situations when the credibility or reliability of data vary. However, standard tests for equality of means do not take weights into account. These results will be of value to any researchers studying statistical means of data of varying reliability, such as corporate bond spreads.