Asymptotic Value of the Probability That the First Order Statistic Is from Null Hypothesis

When every element of a random vector   1 2 , , , n X X X X   assumes the cumulative distribution function 0 F and 1 F with probability and p p  1 , respectively, we have shown that the probability that the first order statistic of 0  X is originally under 0 F can be expressed as. We have also shown that        0 1 0 1 1 d np p F x pF x F x

, respectively, we have shown that the probability that the first order statistic of 0  X is originally under 0 F can be expressed as .We have also shown that , where

Introduction
In wideband spectrum sensing (WSS) of wireless communications with a multiple of receive antennas, the primary goal is to find vacant subbands in a wideband channel composed of a multitude of frequency bands [1][2][3][4].It is beneficial in the WSS to have an accurate estimate of the noise variance.For example, in the detection scheme proposed in [5], estimation of the noise variance in a subband is performed based on the observations in all the subbands.The accuracy of the estimate of the noise variance can be shown to depend on the distributions of the observations under the null and alternative hypotheses.The key parameter in the estimation is the probability that the subband with the lowest energy is under the null hypothesis.
In this paper, we focus on the asymptotic value of the probability and discuss its implications in the performance of WSS schemes.

The Probability
the probability that the first order statistic   1 X of X is one of the random variables [6,7] having the cdf i F by for and 1. 0 Lemma 1.The probability can be expressed as and where
We can assume that j X F  for and for under without loss of generality.Then we easily get Denoting the pdf of i X and the joint pdf of X under by Thus, from (4) and ( 5), we have Finally, recollecting that ] and combining (3) and ( 6), we get (1) as Following similar steps, we can show (2).
It is straightforward to see that irrespective of the values of and , by letting , and therefore, .1) and (9) since and .

Asymptotic Value of 0 
Let us now obtain the value more specifically.
Proof 2. Recollecting that the pdf   0 f x is nonnegative and (9) holds at any point x , it is clear that from (10), we immediately have (13) from ( 14).Theorem 1.For 0,1 i  , let the support of the cdf where assumes a non-negative value with   , and F .Based on this observation, (16) can be expressed as Proof 3. (Proof of Theorem ) Assume where T x is a number in the interval   Here, recollect that 0 Next, when 0 for a sufficiently small interval   0 0 , x x of x , where 0 0 x x  .Then, we can rewrite as using (10) since , where is a number larger than .Now, choosing the number p 0 x in the open interval , where 0 x is now a sufficiently small negative, yet finite, number satisfying Finally, following steps similar to those leading to (22) obtained when 0 and  is finite, we can show that quite immediately by symmetry when 0 and  is infinite: Combining this result with the relation shown in (8), we have , where   1 u x  for and 0 for 0 Thus we have , which can also be obtained directly from (16) as x , and


We have derived the probability that the first order statistic of a number of independent random variables is originally under the null hypothesis.We have also obtained the asymptotic value of the probability as the sample size tends to infinity, and then we discuss an application and implications of the results in the performance of wideband spectrum sensing schemes.(25) This value, which can also be obtained directly from ( 22) and ( 23 2) we can estimate the noise variance correctly by increasing both the sample size and SNR in the doubleexponential and logistic cases, but  3) we cannot estimate the noise variance correctly with probability higher than by increasing the sample size or SNR in the Cauchy case.


 and    , respectively.On the other hand,when 0 1 x x , the value of the non-negative parameter  depends on 0 F and 1

1 :
This result and (8) will after some steps provide us with 0 n , and consequently, 0 n and  is finite, we can similarly show that (22) holds by employing the approximation supported by the National Research Foundation (NRF) of Korea, under Grant 2010-0015175, for which the authors would like to express their thanks.