a1 Mathematical Institute, University of Edinburgh
The problem with which this paper is concerned arose in the discussion of a series of chronometric observations, but it is of more general application, and is capable of wide extension. Pairs of readings (xi yi) were taken at times ti, i = 1, 2, …, n. These readings were known to be affected by respective errors (ξi ηi) from sources different but possessing some common part. It was important to have an estimate of the consequent correlation and to assess its precision. The assumptions made in the particular experiment were that x and y were both linear in t, representable by x = a0 + a1t, y = b0 + b1t, and that the distributions of error in x and y were normal. The parameters a0 and a1, b0 and b1 were therefore obtained from two separate sets of normal equations, and the unknown correlation was then estimated from the sum of products of corresponding residuals ui, vi, one from each set. In the corresponding situation in n samples (xi,yi) from a bivariate normal distribution the mean value of is (n − 1) ρσ1σ2, where σ12, σ22 are the variances of x and y and ρσ1σ2 is their product moment. One might therefore anticipate, by analogy, that in the present case the mean value of Σuivi would be (n − 2)ρσ1σ2. So indeed it proves to be, and the sampling variance of Σuivi conforms likewise with standard results; but it is desirable, by an extension of the problem, both to see why this is so and to take notice of cases where the analogy fails to hold.
(Received January 06 1946)