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Our treatment of systematic errors



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Our treatment of systematic errors Sept 2003 PHYSTAT 21 What is a systematic error This is why people are so frightened of systematic errors and most other textbooks avoid the subject altogether You never know whether you have got them and can never be sure that you have not like an insidious disease The good news however is that despite popular prejudices and superstitions once you know what your systematic errors are they can be handled with standard statistical methods R J Barlow Statistics Sept 2003 PHYSTAT 22 Imagine that two experimental groups have measured a quantity with the results shown OK what is the value of This is very analogous to what happens in global analysis of PDF s But in the case of PDF s the systematic differences are only visible through the PDF s Sept 2003 PHYSTAT 23 We use 2 minimization with fitting of systematic errors For statistical errors define N Di Ti i 1 i2 2 2 Di data value Ti theoretical value statistical error S D i Ti Ti a1 a2 ad a function of d theory parameters Minimize 2 w r t a optimal parameter values a0 All this would be based on the assumption that Di Ti a0 i ri 2 e r 2 dP dr 2 Sept 2003 PHYSTAT 24 Treatment of the normalization error In scattering experiments there is an overall normalization uncertainty from uncertainty of the luminosity We define 1 fN f N Di Ti a f N 2 norm i 1 i 2 2 N 2 where fN overall normalization factor Minimize 2 w r t both a and fN Sept 2003 PHYSTAT 25 A method for general systematic errors K i statistical error of Di ij set of systematic errors j 1 K of Di Di Ti a0 i ri ij r j j 1 Define N 2 i 1 D s i j ij j i2 Ti 2 K 2 s j j 1 quadratic penalty term Minimize 2 with respect to both shape parameters a and optimized systematic shifts sj Sept 2003 PHYSTAT 26 Because 2 depends quadratically on sj we can solve for the systematic shifts analytically K 2 0 0 implies s j s j a A 1 s j m 1 jm Bm where N ij ik i 1 i2 A jk jk and K x K matrix Sept 2003 N Di Ti ij i 1 i2 Bj K x 1 matrix PHYSTAT 27 Now let



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