1 اعداد : د. زكية الصيعري كلية العلوم للبنات Dr. Zakeia A.Alsaiary

Refrences: المراجع 1- First course in order statistics Arnold, Balakrishnan and Nagaraja 2- Order statistics H. A. David 3- الإحصاءات الترتيبية للدكتورة ثروت محمد عبدالمنعم- مكتبة المتنبي 2

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Definition The order statistics of a random sample X1,...,Xn are the sample values placed in ascending order. They are denoted by X(1),...,X(n). The smallest of the Xi,s is denoted by X1:n, the second smallest is denoted by X2:n,…, and finally the largest is denoted by Xn:n thus order statistics are random variables that satisfy X(1) < X(2) <· · · · < X(n). The 4Dr. Zakeia A.Alsaiary

5 الإحصاءات المرتبة : هي عناصر عينة عشوائية مرتبة من الأصغر إلى الأكبر. وفي أغلب مناقشاتنا للإحصاءات المرتبة سوف نعتبر العينة العشوائية تتبع توزيعات متصلة. Example: Let x1 = 0.62, x2 = 0.98, x3= 0.31, x4 = 0.81, x5 = 0.53 are observation for independent expeirement, the order statistics of it are: Y1 = 0.31, y2 = 0.53, y3 = 0.62, y4 = 0.81, y5= 0.98 Y3 = 0.62 is the median and the range = y5 - y1= 0.98 – 0.31 = 0.67 ; Dr. Zakeia A.Alsaiary

6 ORDER STATISTICS If X 1, X 2,…,X n be a r.s. of size n from a population with continuous pdf f(x), then the joint pdf of the order statistics X (1), X (2),…,X (n) is = 0 (elsewhere) Dr. Zakeia A.Alsaiary

7 ORDER STATISTICS Example 1:كتاب الإحصاءات الترتيبية صفحة 4 Find the joint pdf of the order statistics for the uniform distribution, the standard exponential distribution and normal distribution? Solution: p.d.f for the uniform is: Dr. Zakeia A.Alsaiary

8 ORDER STATISTICS Solution: p.d.f for the standard Exponential distribution is: Dr. Zakeia A.Alsaiary

9 ORDER STATISTICS Solution: p.d.f for the standard normal distribution is: Dr. Zakeia A.Alsaiary

10 The marginal distributions for the Oder Statistics p.d.f of the rth order statistics: Theorem: If X 1, X 2,…,X n be a r.s. of size n from a population with continuous pdf f(x), then the p.d.f. of the rth order statistics X (r) is given as: PROOF: Dr. Zakeia A.Alsaiary

11 ORDER STATISTICS r -th Order Statistic y y1y1 y2y2 y r-1 yryr y r+1 ynyn … … P(X<y r )P(X>y r ) f(y r ) # of possible orderings n!/{(r  1)!1!(n  r)!} Dr. Zakeia A.Alsaiary

12 The marginal distributions for the oder statistics p.d.f of the largest order statistics: Theorem: If X 1, X 2,…,X n be a r.s. of size n from a population with continuous pdf f(x), then the p.d.f. of the Largest order statistics Y (n) is given as: PROOF: Dr. Zakeia A.Alsaiary

13 The marginal distributions for the oder statistics p.d.f of the smallest order statistics: Theorem: If X 1, X 2,…,X n be a r.s. of size n from a population with continuous pdf f(x), then the p.d.f. of the smallest order statistics X (1) is given as: PROOF: Dr. Zakeia A.Alsaiary

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15 Example Let are an O. S. of sample size n = 6 and the p.d.f. of this sample is Find: Solution: Dr. Zakeia A.Alsaiary

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17 Example Let is a random sample from beta population with parameters. Let the order statistics of the sample, Find: Dr. Zakeia A.Alsaiary

18Dr. Zakeia A.Alsaiary Solution:

Solution Dr. Zakeia A.Alsaiary19

20 Example Let is a random sample from standard uniform distribution. Find: p.d.f. for the median? ________________________________________ m m Dr. Zakeia A.Alsaiary

21 Joint p.d.f. of i -th and j-th Order Statistic (for i<j) Theorem: If X 1, X 2,…,X n be a r.s. of size n from a population with continuous pdf f(x), and Y 1 < Y 2 <…<Y n are the order statistics of that sample, then the p.d.f. of the two order statistics Y i < Y j, i<j and i,j = 1,2, …,n is given as Dr. Zakeia A.Alsaiary

22 ORDER STATISTICS Joint p.d.f. of i -th and j-th Order Statistic (for i<j) y y1y1 y2y2 y i-1 yiyi y i+1 ynyn … … P(X<y i )P(y i <X<y j ) f(y i ) # of possible orderings n!/{(i  1)!1!(j-i-1)!1!(n  j)!} y j-1 yjyj y j+1 i-1 itemsj-i-1 itemsn-j items 1 item P(X>y j ) f(y j ) Dr. Zakeia A.Alsaiary

23 Example Let is a random sample from beta population with parameters. Let the order statistics of the sample, Find: 1- Joint p.d.f. for ( ) 2- Let n = 4 then find Dr. Zakeia A.Alsaiary

24 Example Let the order statistics of random sample with size n = 5, from distribution with p.d.f. Prove that the two O.S. Are independent. Dr. Zakeia A.Alsaiary

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Solution Dr. Zakeia A.Alsaiary26

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29 c.d.f of the rth order statistics: Theorem: If X 1, X 2,…,X n be an independent and identical r.s. of size n from a population with pdf f(x) and cdf F(x), then the c.d.f. of the rth order statistics X (r) is given as: PROOF: Dr. Zakeia A.Alsaiary

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Special cases: Dr. Zakeia A.Alsaiary31 p.d.f of the rth order statistics:

32 c.d.f of the rth order statistics: Theorem: If X 1, X 2,…,X n be an independent and identical r.s. of size n from a population with pdf f(x) and cdf F(x), then the c.d.f. of the rth order statistics X (r) is given as: PROOF: Dr. Zakeia A.Alsaiary

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Special case: Dr. Zakeia A.Alsaiary34 p.d.f of the rth order statistics:

صور أخرى لدالة التوزيع التراكمية للإحصاء المرتب الرائي مثبتة في الكتب: Dr. Zakeia A.Alsaiary35

36 Example Let the order statistics of the sample with n = 5, from the distribution Find: Dr. Zakeia A.Alsaiary

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38 Example Let the order statistics of the sample with n = 5, from the distribution Find: Dr. Zakeia A.Alsaiary