Feldman-Cousins Limit Setting

Contents

Construction of Confidence Bands

  1. Measured Top Width Distributions
  2. Likelihood Ratio Distributions - Discrete
  3. Fits to the Width Distributions
  4. Likelihood Ratio Function - Continuous
  5. Confidence Interval in the width function
  6. 95% CL Confidence Band

Pseudo Experiments on Limit Setting

  1. Top Width Limits
  2. Top Width Limits with systematics
    • Convoluted Width Distributions
    • Likelihood Functions
    • Confidence Band
    • PEs
  3. Comparison of 90%CL vs 95%CL

  1. Measured Top Width Distributions
  2. This is the output of the Likelihood fit with quadratic dependency on the top width.
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  3. Likelihood Ratio (lambda) distribution
  4. Likelihood ratio = p(# PE | width) / p(# PE | best)
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  5. parameterization fit to the width distribution
  6. We characterize the shapes of the top width distributions to use the Feldman-Cousins construction in a continuous manner.
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    difference between fit and the width distribution

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  7. Likelihood ratio function
  8. likelihood ratio function. The red line is the 95% CL region in width distribution.
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  9. 95% CL interval in the width function
  10. Shaded region is the 95%CL region in the top width function (and not in the top width distribution)
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  11. Confidence Band!!
  12. 95% CL band.
    The left plot is the confidence bands drawn only for the true width that we have MC samples. The right plot is the confidence bands drawn for every 1 GeV true top width. Red bands corresponds to the bands on the left plots (i.e. for our MC samples' top width).
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  13. PE for the top width limits without systematics
  14. This plot is PE from using lots of CL bands. The x-axis is the measured top width (output of the likelihood) and the y-axis is the upper and lower limit given that measured top width.
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    Lower and Upper limits for the input top width of 1.5 GeV
    Projection on Y axis to see the number of entries in each limit.
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    For the input top width of 30 GeV
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    Coverage vs input top width from 3000 PEs (at 95% CL)

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  15. Top Width limits with systematics included
  16. Measured Top width distribution with systematics (after convolution). On the right is the fit to this distribution.

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    Likelihood Ratio Function

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    Confidence Band with Systematics

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    PE results for 1.5 GeV with systematics at &Gammatopinput = 1.5 GeV and Mtop = 175 GeV

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    PE results for 1.5 GeV with x10 statistics and systematics at &Gammatopinput = 1.5 GeV and Mtop = 175 GeV

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    PE results for 30 GeV with systematics at &Gammatopinput = 1.5 GeV and Mtop = 175 GeV

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    Comparison of 90%CL and 95% CL bands

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