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It is desired to find an approximate distribution of simple form for the statistic r = x₁x₂ + + xTx₁ x₁² + + xT² (r is an estimate of the serial correlation coefficient in a circular universe) in the case that O in the universe. Such a distribution is obtained by smoothing the joint characteristic function of the numerator and denominator of the expression for r. The first two moments are calculated; from these r is seen to be a consistent estimate of. A graph of this distribution for sample size T = 20 and various values of is given. In addition, an approximate distribution for p = x²₁ + + x²T is derived which reduces to the exact (²-) distribution if = 0. From a formula which yields all moments, it is concluded that, at least up to the degree of approximation attained, p/T is an unbiased and consistent extimate of ².
Roy B. Leipnik (Sat,) studied this question.