Chapter 5: Order StatisticsGiven a random sample, we are interested in the smallest, largest, or middleobservations. Examples:• the highest flood waters (useful when planning for future emergencies)• the lowest winter temperature recorded in the last 50 years• the median price of houses sold in last month• the median salary of NBA players (better measure than sample mean)Definition:Given a random sample, X1, · · · , Xn, the sample order statistics are thesample values placed in ascending order,X(1)= min1≤i≤nXi,X(2)= second smallest Xi,... = ...X(n)= max1≤i≤nXi.Example: Suppose four numbers are observed as a sample of size 4. Thesample values are x1= 6, x2= 9, x3= 3, x4= 8.. What are the orderstatistics?Remarks:• Order statistics are random variables themselves (as functions of arandom sample).• Order statistics satisfyX(1)≤ · · · ≤ X(n).• Though the samples X1, · · · , Xnare independently and identically dis-tributed, the order statistics X(1), · · · , X(n)are never independent be-cause of the order restriction.• We will study their marginal distributions and joint distributions1Distributions of Order Statistics - Continuous CaseMarginal distributionsAssume X1, · · · , Xnare from a continuous population with cdf F (x) andpdf f(x). Then(1) The nth order statistic, or the sample maximum, X(n)had the pdffX(n)(x)= n[F (x)]n−1f(x)(2) The first order statistic, or the sample minimum, X(1)had the pdffX(1)(x)= n[1 − F (x)]n−1f(x)(3) More generally, the jth order statistic X(j)has the bvgpdffX(j)(x) =n!(j − 1)!(n − j)!f(x)[F (x)]j−1[1 − F (x)]n−j.2Joint distributions(4) For 1 ≤ i < j ≤ n, the joint pdf of X(i)and X(j)isfX(i),X(j)(u, v) =n!(i − 1)!(j − i − 1)!(n − j)!f(u)f(v)[F (u)]i−1[F (v)−F (u)]j−i−1[1−F (v)]n−j.if −∞ < u < v < ∞; = 0 otherwise.(5) The joint pdf X(1), · · · , X(n)isfX(1),··· ,X(n)(x1, · · · , xn) =(n!f(x1) · · · f (xn) if − ∞ < x1< · · · < xn< ∞,0 otherwise.3Example: X1, · · · , Xnare iid from unif [0, 1].(1) Show that X(j)∼ Beta(j, n + 1 − j)(2) The joint pdf of X(1)and X(n).(3) The conditional pdf of X(1)given X(n)X(1)|X(n)∼ X(n)Beta(1, n − 1)(4) The conditional pdf of X(i)given X(j)(for any i < j)X(i)|X(j)∼ X(j)Beta(i, j − i)(5) Let n = 5. Derive the joint pdf of X(1), · · · , X(5).(6) Let n = 5. Derive the joint pdf of X(2)and X(4).4Example: compute P (X(1)> 1, X(n)≤ 2).P (X(1)> x, X(n)≤ y) =nYi=1P (x < Xi≤ y) = [F (y) − F (x)]n.5Discrete CaseAssume X1, · · · , Xnare a random sample with a discrete pmfP (X = xi) = pi, x1< x2< · · · (countable).Define the cumulative sumr0= 0, r1= p1, r2= p1+ p2, · · · , ri=iXk=1pk, · · · .ThenP (X(j)≤ xi) =nXk=jnkrki(1 − ri)n−k,andP (X(j)= xi) =nXk=jnkhrki(1 − ri)n−k− rki−1(1 − ri−1)n−ki.6Common statistics based on order statistics:sample range:R = X(n)− X(1)sample midrange:V =X(n)+ X(1)/2sample median:M =(X((n+1)/2)if n is oddX(n/2)+ X(n+1)/2)/2 if n is even.sample percentile: For any 0 < p < 1, the (100p)th sample percentileis the observation such that about np of the observations are less than thisobservation and n(1 − p)th of the observations are larger.• sample median M is 50th sample quantile (the second sample quartile)• denote Q1as 25th sample quantile (the first sample quartile)• denote Q3as 75th sample quantile (the third sample quartile)• interquartile range IQR=Q3− Q1(describing the spread about themedian)Remark: Sample Mean vs Sample MedianRemark: Sample Median vs Population