Research Article
The Relations Among the Order Statistics of Uniform Distribution
Department of Statistics, Payame Nour University of Amol, Iran
H. Pazira
Department of Statistics, Payame Nour University of Tehran, Iran
Let X1,X2,....,Xn be a random sample from U(,) distribution. We show the order statistic with X(i) for this sample. We know:
where, the parameter and satisfy in
According to the order statistics transformation theorem (Appendix):
thus:
where, according to identically distributed theorem (Appendix):
(1) |
Now, since X(i) and (-) Vi+ are identically distributed, they have a same density and moments and have a same moment generating function generally (if exist) (Behbodian, 2003b).
Therefore, when we intend to compute, for example, variance of X(i), we can use the variance of (-) Vi+ , because they have a equal result.
MOMENTS OF ORDER STATISTICS
Here, we present and prove some important moments of the order statistics of the random variable X (XU(-)). For universalization this moment to all of the order statistics, this moments be computed generally. Also these moments are useful to proof of the theorem 1:
(a) |
(b) |
where, 1i<jn, i, jεN:
(c) |
(d) |
(e) |
where, 1i<jn, i, jεN:
(f) |
where, 1i<jn, I, jεN.
According to the above equations, these equations are easily obtainable; as Eq. a, b and c are given by using the expectation definition, Eq. 1 and identically distributed theorem. Equation e and d are given by definition of variance and covariance and using a, b and c and last equation is given by using Eq. e and d. As an example, we obtain equation c.
We know, by using Eq. 1 and identically distributed theorem, it is sufficient to use the expectation of [ (-)Vi+]2 in stead of the expectation of X(i)2, where Vi Bet(i,n+1i). Thus we must compute E(Vi) and E(Vi2):
also:
thus:
thus part (c) is true, (Behbodian, 2003a; Mood et al., 1998).
Notice, since 0<Corr(X(i)X(j))≤1, order statistics of uniform distribution have a positive correlation.
We express the important result of above equations in the follow theorem.
Theorem 1: Let X1,X2,....,Xn be a random sample from U(,) distribution with order statistics X(1),X(2),....,X(n), then the follow equation always is true:
(2) |
where, 1i<jn, i, jεN.
Proof: We use the above moment equations for prove this theory. We use the part d for prove first equal in left hand:
Thus the first equal is true. We use the part e for prove second equal.
First we consider. Since that is i<n+1i, we have:
thus:
Now, if that is i<n+1i, we can write:
hence:
and proof is complete.
We know that, if we sort X1,X2,....,Xn as ascendant (so that obtained the order statistics), X(i) and X(n+1i) are the order statistics, so that, with respect to the medianorX:
are concurrent. Indeed, those have equal distance to median and theorem 1 use for such statistics.
Theorem 2: Let X1,X2,....,Xn be a random sample from U(,)istribution with order statistics X(1),X(2),....,X(n). Then both following formula are true if 1i<jn:
(3) |
(4) |
Proof: For prove, we define the order statistics X(i) and X(j) with attention to Eq. 1 and condition 1i<jn:
According to the Identically distributed and order statistics transformation theorem (Appendix) we have:
and since know (Behbodian, 2003b):
then by using Eq. 1 we can write:
so Eq. 3 is proved. With a little change in this proof, we can prove Eq. 4. According to identically distributed theorem we have:
and since we know (Behbodian, 2003b):
thus:
by using Eq. 1 we can write:
thus Eq. 4 is proved and the proof is complete.
APPENDIX
Identically distributed theorem: Let X and Y be random variables. If (or P(X≤z) = P(Y≤z) = F(z)) and g(t) be a arbitrary function, then (Behbodian, 2003b, Chapter 1, Theorem 2).
Order statistics transformation theorem: Let X1,X2,....,Xn be a random sample so that has a probability distribution function (p.d.f.) of the form f and a cumulative distribution function (p.d.f.) of the form F. We show the order statistics with X(1),X(2),....,X(n). Let U1,U2,....,Un be a random sample from U(0,1) distribution, we show the order statistics it with V1, V2,....,Vn. Then we have (Behbodian, 2003b):
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