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Statistical Theory II

by: Jordane Kemmer

Statistical Theory II ST 522

Jordane Kemmer
GPA 3.79

Hao Zhang

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Hao Zhang
Class Notes
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This 7 page Class Notes was uploaded by Jordane Kemmer on Thursday October 15, 2015. The Class Notes belongs to ST 522 at North Carolina State University taught by Hao Zhang in Fall. Since its upload, it has received 20 views. For similar materials see /class/223947/st-522-north-carolina-state-university in Statistics at North Carolina State University.

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Date Created: 10/15/15
Chapter 5 Order Statistics Given a random sample we are interested in the smallest largest or middle observations Examples 0 the highest flood waters useful when planning for future emergencies o the lowest Winter temperature recorded in the last 50 years 0 the median price of houses sold in last month 0 the median salary of NBA players better measure than sample mean De nition Given a random sample X1 a Xn the sample order statistics are the sample values placed in ascending order Xu minigtgn Xi X9 second smallest Xi Xm maxggn X22 Example Suppose four numbers are observed as a sample of size 4 The sample values are an 6 avg 9 avg 3 4 8 What are the order statistics Remarks 0 Order statistics are random variables themselves as functions of a random sample Order statistics satisfy Xlt1gt S S Xltngtgt Though the samples X1 a Xn are independently and identically dis tributed the order statistics X0 a X are never independent be cause of the order restriction We will study their marginal distributions and joint distributions Distributions of Order Statistics Continuous Case Marginal distributions Assume X1 Xn are from a continuous population with cdf and pdffw Then 1 The nth order statistic or the sample maximum Xm had the pdf fXltmltzgt an wgtl 1fwgt 2 The rst order statistic or the sample minimum X0 had the pdf fXlt1I nll 7 3 More generally the jth order statistic X has the bvgpdf 3 7 fwFwgtl7 11 Fwgtl 3 W 7 Joint distributions 4 For 1 S 2 lt j S n the joint pdf of X9 and X0 is n fXltinltjgt 7W W v u F7JgtFugtj i 11Fvgt 7gt if 700 lt u lt 1 lt 00 0 otherwise 5 The joint pde1 XW is nlfw1 fwn if fooltw1lt ltwnltoo 0 otherwise fXlt1Xltnw17 7 Example X1 7Xn are iid from unif 01 1 Show that X0 N Betajn 179 2 The joint pdf of XO and X01 3 The conditional pdf of XO given Xm N XnBeta1n 7 4 The conditional pdf of X9 given X0 for any 2 lt j N XjBetaij 7 5 Let n 5 Derive the joint pdf of Xu X5 6 Let n 5 Derive the joint pdf of X9 and X4 Example compute PX1 gt 1Xn S 2 PX1 gt wXn S 3 HPW lt Xi S 3 W31 7 NM 2391 Discrete Case Assume X1 Xn are a random sample with a discrete pmf PX pi 31 lt 32 lt countable De ne the cumulative sum 239 T007T1P17 T2P1P27 w T Zpk k1 Then n PXjgt S 062 2 ml MM k V Common statistics based on order statistics sample range R Xltngt Xlt1gt sample mldranye V Xw Xlt1gt 2 sample median M Xn12 if n is odd Xn2 Xn12 2 if n is even sample percentile For any 0 lt p lt 17 the 100pth sample percentile is the observation such that about np of the observations are less than this observation and n1 7 pth of the observations are larger 0 sample median M is 50th sample quantile the second sample quartile 0 denote Q1 as 25th sample quantile the rst sample quartile 0 denote Q3 as 75th sample quantile the third sample quartile o interquartile range IQRQ3 7 Q1 describing the spread about the median Remark Sample Mean vs Sample Median Remark Sample Median vs Population Median


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