Brief Discussion: Asymptotic Value of the Probability that the First Order Statistic is from Null Hypothesis

For an independent random vector X = (X1;X2; ;Xn), assume that the cumulative distribution function (cdf) of an element of X is F0 and F1 with probability p and 1 p, respectively. We have shown that the probability S0 that the rst order statistic of X is originally under F0 can be expressed as S0 = np ∫∞ −∞ f1 (1 p)F1(x) pF0(x)gn−1 dF0(x). Next, denoting by [xi;1] the support of Fi(x) for i = 0; 1, we have shown that lim n→∞ S0 = p p+(1−p) , where = lim x→x F1(x) F0(x) and x = max (x0; x1), and provided some examples. Applications and implications of the results are brie y discussed in the performance of wideband spectrum sensing schemes. Keywords: Energy; order statistic; probability; spectrum sensing.