Unnecessary

Transcript

1. Unnecessary Bias
   The Consequences of Averaging Simulated
   Quantities of Interest
   Carlisle Rainey
   Assistant Professor
   Texas A&M University
   crainey@tamu.edu
   Holger Kern
   Assistant Professor
   Florida State University
   hkern@fsu.edu
   paper, data, and code at
   carlislerainey.com/talk
   

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2. Unnecessary Bias
   The Consequences of Averaging Simulated
   Quantities of Interest
   Carlisle Rainey
   Assistant Professor
   Texas A&M University
   crainey@tamu.edu
   Holger Kern
   Assistant Professor
   Florida State University
   hkern@fsu.edu
   paper, data, and code at
   carlislerainey.com/talk
   

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3. Should you average simulated
   quantities of interest?
   

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4. Should you average simulated
   quantities of interest?
   

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5. …the way King, Tomz, and Wittenberg (2000) say to do.
   Should you average simulated
   quantities of interest?
   

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6. …the way King, Tomz, and Wittenberg (2000) say to do.
   …the way Clarify and Zelig do.
   Should you average simulated
   quantities of interest?
   

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7. …the way King, Tomz, and Wittenberg (2000) say to do.
   …the way Clarify and Zelig do.
   …the way that Matt Golder’s marginal effects plot code does.
   Should you average simulated
   quantities of interest?
   

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8. …the way King, Tomz, and Wittenberg (2000) say to do.
   …the way Clarify and Zelig do.
   …the way that Matt Golder’s marginal effects plot code does.
   …the way that Hanmer and Kalkan’s (2013) code does.
   Should you average simulated
   quantities of interest?
   

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9. …the way King, Tomz, and Wittenberg (2000) say to do.
   …the way Clarify and Zelig do.
   …the way that Matt Golder’s marginal effects plot code does.
   …the way that Hanmer and Kalkan’s (2013) code does.
   …the way that my own code does.
   Should you average simulated
   quantities of interest?
   

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10. …the way King, Tomz, and Wittenberg (2000) say to do.
    …the way Clarify and Zelig do.
    …the way that Matt Golder’s marginal effects plot code does.
    …the way that Hanmer and Kalkan’s (2013) code does.
    …the way that my own code does.
    …the way that Holger’s code does.
    Should you average simulated
    quantities of interest?
    

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11. …the way King, Tomz, and Wittenberg (2000) say to do.
    …the way Clarify and Zelig do.
    …the way that Matt Golder’s marginal effects plot code does.
    …the way that Hanmer and Kalkan’s (2013) code does.
    …the way that my own code does.
    …the way that Holger’s code does.
    …the way that your own code might.
    Should you average simulated
    quantities of interest?
    

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12. No.
    

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13. View Slide

14. TABLE 2. Models of Involvement in Militarized Disputes, 1950–1985:
    Assessing the Liberal Peace
    Variable Eqn 1 Eqn 2 Eqn 3 Eqn 4
    Democracy scoreL
    ß -0.0497 -0.0554 -0.0413 -0.0457
    SEß
    0.0074 0.0077 0.0083 0.0076
    p < .001 < .001 < .001 < .001
    Economic growth rateL
    -0.0223 -0.0318 -0.0297 -0.0220
    0.0085 0.0093 0.0094 0.0090
    .009 < .001 < .001 .02
    Allies -0.821 -0.868 -0.864 -0.815
    0.080 0.092 0.092 0.083
    < .001 < .001 < .001 < .001
    Contiguity 1.31 1.26 1.32 1.39
    0.08 0.09 0.09 0.08
    < .001 < .001 < .001 < .001
    Capability ratio -0.00307 -0.00345 -0.00356 -0.00279
    0.00042 0.00050 0.00051 0.00041
    < .001 < .001 < .001 < .001
    Dyadic trade-to-GDP ratioL
    -66.1 -43.8 -81.0
    13.4 12.5 15.2
    < .001 < .001 < .001
    Total trade-to-GDP ratioL
    -0.706 -0.512
    0.192 0.193
    < .001 .008
    Trend, dyadic trade-to-GDP -8.89
    ratioH
    2.93
    < .001
    Constant -3.29 -3.28 -3.20 -3.32
    0.08 0.10 0.10 0.08
    < .001 < .001 < .001 < .001
    Chi2 764.04 593.65 611.74 728.56
    P of chi2 < .0001 < .0001 < .0001 < .0001
    Log likelihood -3477.57 -2795.89 -2786.85 -3231.42
    N 20,990 17,709 17,709 19,772
    

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15. 0.08
    < .001
    Capability ratio -0.00307
    0.00042
    < .001
    Dyadic trade-to-GDP ratioL
    -66.1
    13.4
    < .001
    Total trade-to-GDP ratioL
    Trend, dyadic trade-to-GDP
    

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16. TABLE 3. Annual Probabilities of Involvement in a Militarized Dispute, 1950–1985,
    Based on the Estimated Coefficients for Equations 1 and 2 in Table 2
    Equation 1 Equation 2
    1. Democracy scoreL
    , economic growth rateL
    , dyadic .078 .070
    trade-to-GDP ratioL
    , and capability ratio at mean
    values; allies equals 0; contiguity equals 1
    2. Increase in democracyL
    of 1 std. dev.; other variables .057 .049
    at baseline values
    3. Increase in dyadic trade-to-GDP ratioL
    of 1 std. dev.; .047 —
    other variables at baseline values
    4. Increase in total trade-to-GDP ratioL
    of 1 std. dev.; — .061
    other variables at baseline values
    5. Increase in economic growth rateL
    of 1 std. dev.; other .073 .064
    variables at baseline values
    6. Allies equals 1; other variables at baseline values .036 .031
    7. Contiguity equals 0; other variables at baseline values .022 .021
    8. Increase in capability ratio of 1 std. dev.; other variables .021 .016
    at baseline values
    9. Capability ratio equals 1; other variables at baseline values .122 .117
    10. Capability ratio equals 10; other variables at baseline values .119 .114
    

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17. TABLE 3. Annual Probabilities of Involvement in a Militarized Dispute, 1950–1985,
    Based on the Estimated Coefficients for Equations 1 and 2 in Table 2
    Equation 1 Equation 2
    1. Democracy scoreL
    , economic growth rateL
    , dyadic .078 .070
    trade-to-GDP ratioL
    , and capability ratio at mean
    values; allies equals 0; contiguity equals 1
    2. Increase in democracyL
    of 1 std. dev.; other variables .057 .049
    at baseline values
    3. Increase in dyadic trade-to-GDP ratioL
    of 1 std. dev.; .047 —
    other variables at baseline values
    4. Increase in total trade-to-GDP ratioL
    of 1 std. dev.; — .061
    other variables at baseline values
    5. Increase in economic growth rateL
    of 1 std. dev.; other .073 .064
    variables at baseline values
    6. Allies equals 1; other variables at baseline values .036 .031
    7. Contiguity equals 0; other variables at baseline values .022 .021
    8. Increase in capability ratio of 1 std. dev.; other variables .021 .016
    at baseline values
    9. Capability ratio equals 1; other variables at baseline values .122 .117
    10. Capability ratio equals 10; other variables at baseline values .119 .114
    

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18. TABLE 3. Annual Probabilities of Involvement in a Militarized Dispute, 1950–1985,
    Based on the Estimated Coefficients for Equations 1 and 2 in Table 2
    Equation 1 Equation 2
    1. Democracy scoreL
    , economic growth rateL
    , dyadic .078 .070
    trade-to-GDP ratioL
    , and capability ratio at mean
    values; allies equals 0; contiguity equals 1
    2. Increase in democracyL
    of 1 std. dev.; other variables .057 .049
    at baseline values
    3. Increase in dyadic trade-to-GDP ratioL
    of 1 std. dev.; .047 —
    other variables at baseline values
    4. Increase in total trade-to-GDP ratioL
    of 1 std. dev.; — .061
    other variables at baseline values
    5. Increase in economic growth rateL
    of 1 std. dev.; other .073 .064
    variables at baseline values
    6. Allies equals 1; other variables at baseline values .036 .031
    7. Contiguity equals 0; other variables at baseline values .022 .021
    8. Increase in capability ratio of 1 std. dev.; other variables .021 .016
    at baseline values
    9. Capability ratio equals 1; other variables at baseline values .122 .117
    10. Capability ratio equals 10; other variables at baseline values .119 .114
    

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19. TABLE 3. Annual Probabilities of Involvement in a Militarized Dispute, 1950–1985,
    Based on the Estimated Coefficients for Equations 1 and 2 in Table 2
    Equation 1 Equation 2
    1. Democracy scoreL
    , economic growth rateL
    , dyadic .078 .070
    trade-to-GDP ratioL
    , and capability ratio at mean
    values; allies equals 0; contiguity equals 1
    2. Increase in democracyL
    of 1 std. dev.; other variables .057 .049
    at baseline values
    3. Increase in dyadic trade-to-GDP ratioL
    of 1 std. dev.; .047 —
    other variables at baseline values
    4. Increase in total trade-to-GDP ratioL
    of 1 std. dev.; — .061
    other variables at baseline values
    5. Increase in economic growth rateL
    of 1 std. dev.; other .073 .064
    variables at baseline values
    6. Allies equals 1; other variables at baseline values .036 .031
    7. Contiguity equals 0; other variables at baseline values .022 .021
    8. Increase in capability ratio of 1 std. dev.; other variables .021 .016
    at baseline values
    9. Capability ratio equals 1; other variables at baseline values .122 .117
    10. Capability ratio equals 10; other variables at baseline values .119 .114
    

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20. TABLE 3. Annual Probabilities of Involvement in a Militarized Dispute, 1950–1985,
    Based on the Estimated Coefficients for Equations 1 and 2 in Table 2
    Equation 1 Equation 2
    1. Democracy scoreL
    , economic growth rateL
    , dyadic .078 .070
    trade-to-GDP ratioL
    , and capability ratio at mean
    values; allies equals 0; contiguity equals 1
    2. Increase in democracyL
    of 1 std. dev.; other variables .057 .049
    at baseline values
    3. Increase in dyadic trade-to-GDP ratioL
    of 1 std. dev.; .047 —
    other variables at baseline values
    4. Increase in total trade-to-GDP ratioL
    of 1 std. dev.; — .061
    other variables at baseline values
    5. Increase in economic growth rateL
    of 1 std. dev.; other .073 .064
    variables at baseline values
    6. Allies equals 1; other variables at baseline values .036 .031
    7. Contiguity equals 0; other variables at baseline values .022 .021
    8. Increase in capability ratio of 1 std. dev.; other variables .021 .016
    at baseline values
    9. Capability ratio equals 1; other variables at baseline values .122 .117
    10. Capability ratio equals 10; other variables at baseline values .119 .114
    difference of -0.031
    

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21. two options
    

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22. two options
    use the invariance property
    

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23. two options
    use the invariance property
    Long (1997)
    

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24. two options
    use the invariance property
    average simulated QIs
    

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25. two options
    use the invariance property
    average simulated QIs
    Clarify in Stata
    

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26. two options
    use the invariance property
    average simulated QIs
    

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27. two options
    use the invariance property
    average simulated QIs
    t.i. bias
    

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28. two options
    use the invariance property
    average simulated QIs
    t.i. bias
    2x t.i. bias
    

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29. two options
    use the invariance property
    average simulated QIs
    t.i. bias
    2x t.i. bias
    2017 PA
    (1)
    

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30. two options
    use the invariance property
    average simulated QIs
    t.i. bias
    2x t.i. bias
    2017 PA
    What happens when you
    average simulated QIs?
    (2)
    (1)
    

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31. two options
    use the invariance property
    average simulated QIs
    t.i. bias
    2x t.i. bias
    2017 PA
    What happens when you
    average simulated QIs?
    Does any of this matter?
    (2)
    (3)
    (1)
    

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32. Rainey (2017)
    

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33. Long (1997)
    

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34. King (1998)
    

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35. margins
    

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36. predict()
    

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37. bias
    

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38. bias
    expected estimate - true value
    

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39. bias
    expected estimate - true value
    under repeated sampling
    

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40. bias
    expected estimate - true value
    under repeated sampling
    *IMPORTANT*
    property of procedure,
    not estimate
    

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41. View Slide

42. View Slide

43. View Slide

44. e.g., small-sample bias in logit
    

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45. Jensen’s inequality: E[g(X)] > g(E[X]) for convex g.
    e.g., small-sample bias in logit
    

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46. A Stark Illustration
    

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47. yi
    ⇠ N(µ, 1), for i = 1, 2, ..., 100
    ⌧(µ) = µ2
    µ = 0
    

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48. yi
    ⇠ N(µ, 1), for i = 1, 2, ..., 100
    ⌧(µ) = µ2
    µ = 0
    DGP or model
    

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49. yi
    ⇠ N(µ, 1), for i = 1, 2, ..., 100
    ⌧(µ) = µ2
    µ = 0
    DGP or model
    QI
    

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50. yi
    ⇠ N(µ, 1), for i = 1, 2, ..., 100
    ⌧(µ) = µ2
    µ = 0
    DGP or model
    QI
    truth
    

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51. sample ave. is a BUE of u.
    (…and MLE!)
    yi
    ⇠ N(µ, 1), for i = 1, 2, ..., 100
    ⌧(µ) = µ2
    µ = 0
    DGP or model
    QI
    truth
    

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52. 0.00
    0.05
    0.10
    0.15
    −0.4 −0.2 0.0 0.2 0.4
    µ
    ^
    τ
    ^
    Iteration 1
    

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53. ●
    τ
    ^mle
    µ
    ^mle
    0.00
    0.05
    0.10
    0.15
    −0.4 −0.2 0.0 0.2 0.4
    µ
    ^
    τ
    ^
    Iteration 1
    

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54. ●
    ●
    0.00
    0.05
    0.10
    0.15
    −0.4 −0.2 0.0 0.2 0.4
    µ
    ^
    τ
    ^
    Iteration 2
    

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55. ●
    ● ●
    0.00
    0.05
    0.10
    0.15
    −0.4 −0.2 0.0 0.2 0.4
    µ
    ^
    τ
    ^
    Iteration 3
    

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56. ●
    ● ●
    ●
    0.00
    0.05
    0.10
    0.15
    −0.4 −0.2 0.0 0.2 0.4
    µ
    ^
    τ
    ^
    Iteration 4
    

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57. ●
    ● ●
    ●
    ●
    0.00
    0.05
    0.10
    0.15
    −0.4 −0.2 0.0 0.2 0.4
    µ
    ^
    τ
    ^
    Iteration 5
    

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58. ●
    ● ●
    ●
    ●
    ●
    0.00
    0.05
    0.10
    0.15
    −0.4 −0.2 0.0 0.2 0.4
    µ
    ^
    τ
    ^
    Iteration 6
    

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59. ●
    ● ●
    ●
    ●
    ●
    ●
    0.00
    0.05
    0.10
    0.15
    −0.4 −0.2 0.0 0.2 0.4
    µ
    ^
    τ
    ^
    Iteration 7
    

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60. ●
    ● ●
    ●
    ●
    ●
    ●
    ●
    0.00
    0.05
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    0.15
    −0.4 −0.2 0.0 0.2 0.4
    µ
    ^
    τ
    ^
    Iteration 8
    

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61. ●
    ● ●
    ●
    ●
    ●
    ●
    ●
    ●
    0.00
    0.05
    0.10
    0.15
    −0.4 −0.2 0.0 0.2 0.4
    µ
    ^
    τ
    ^
    Iteration 9
    

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62. ●
    ● ●
    ●
    ●
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    ●
    ●
    ●
    ●
    0.00
    0.05
    0.10
    0.15
    −0.4 −0.2 0.0 0.2 0.4
    µ
    ^
    τ
    ^
    Iteration 10
    

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63. 0.00
    0.05
    0.10
    0.15
    −0.4 −0.2 0.0 0.2 0.4
    µ
    ^
    τ
    ^
    Iteration 10000
    

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64. µ
    ^
    τ
    ^
    −0.4 −0.2 0.0 0.2 0.4 0.00 0.05 0.10 0.15
    0
    30
    60
    90
    0
    1
    2
    3
    4
    Estimate
    Density
    Method Maximum Likelihood Average of Simulations
    

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65. ●
    µ
    ^
    τ
    ^
    −0.4 −0.2 0.0 0.2 0.4 0.00 0.05 0.10 0.15
    0
    30
    60
    90
    0
    1
    2
    3
    4
    Estimate
    Density
    Method Maximum Likelihood Average of Simulations
    

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66. ● ●
    µ
    ^
    τ
    ^
    −0.4 −0.2 0.0 0.2 0.4 0.00 0.05 0.10 0.15
    0
    30
    60
    90
    0
    1
    2
    3
    4
    Estimate
    Density
    Method Maximum Likelihood Average of Simulations
    

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67. One Theoretical Result
    

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68. One Theoretical Result
    Convex transformations lead to positive
    transformation-induced bias.
    

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69. two options
    use the invariance property
    average simulated QIs
    t.i. bias
    2x t.i. bias
    2017 PA
    What happens when you
    average simulated QIs?
    Does any of this matter?
    (2)
    (3)
    (1)
    

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70. Simulating Quantities of Interest
    King, Tomz, Wittenberg (2000)
    

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71. Simulating Quantities of Interest
    King, Tomz, Wittenberg (2000)
    1. Fit the model.
    

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72. Simulating Quantities of Interest
    King, Tomz, Wittenberg (2000)
    1. Fit the model.
    2. Simulate the coefficients.
    

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73. Simulating Quantities of Interest
    King, Tomz, Wittenberg (2000)
    1. Fit the model.
    2. Simulate the coefficients.
    3. Transform simulated coefficients into the quantities of
    interest.
    

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74. Simulating Quantities of Interest
    King, Tomz, Wittenberg (2000)
    1. Fit the model.
    2. Simulate the coefficients.
    3. Transform simulated coefficients into the quantities of
    interest.
    4. Average the simulations of the quantity of interest to
    obtain a point estimate.
    

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75. King, Tomz, Wittenberg (2000)
    

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76. Gelman and Hill (2006)
    

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77. mattgolder.com/interactions
    

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78. A Stark Illustration
    continued…
    

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79. sample ave. is a BUE of u.
    (…and MLE!)
    yi
    ⇠ N(µ, 1), for i = 1, 2, ..., 100
    ⌧(µ) = µ2
    µ = 0
    DGP or model
    QI
    truth
    

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80. sample ave. is a BUE of u.
    (…and MLE!)
    yi
    ⇠ N(µ, 1), for i = 1, 2, ..., 100
    ⌧(µ) = µ2
    µ = 0
    DGP or model
    QI
    truth
    SE is 1/sqrt(100)
    

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81. τ
    ^mle
    µ
    ^mle
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    

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82. τ
    ^mle
    µ
    ^mle
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    normal w/ mean of 0.2
    and SD of 0.1
    

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83. τ
    ^mle
    µ
    ^mle
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    

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84. τ
    ^mle
    µ
    ^mle
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    sim. coef.
    

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85. τ
    ^mle
    µ
    ^mle
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    sim. coef.
    translated into sim. QI
    

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86. τ
    ^mle
    µ
    ^mle
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    sim. coef.
    translated into sim. QI
    

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87. τ
    ^mle
    µ
    ^mle
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    τ
    ^avg.
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    

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88. τ
    ^mle
    µ
    ^mle
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 2
    

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89. τ
    ^mle
    µ
    ^mle
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 2
    

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90. τ
    ^mle
    µ
    ^mle
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 2
    

    View Slide

91. 0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 2
    

    View Slide

92. τ
    ^avg.
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 2
    

    View Slide

93. τ
    ^avg.
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 2
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 3
    

    View Slide

94. τ
    ^avg.
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 2
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 3
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 4
    

    View Slide

95. τ
    ^avg.
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 2
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 3
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 4
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 5
    

    View Slide

96. τ
    ^avg.
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 2
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 3
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 4
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 5
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 6
    

    View Slide

97. τ
    ^avg.
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 2
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 3
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 4
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 5
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 6
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 7
    

    View Slide

98. τ
    ^avg.
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 2
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 3
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 4
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 5
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 6
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 7
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 8
    

    View Slide

99. τ
    ^avg.
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 1
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 2
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 3
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 4
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 5
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 6
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 7
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 8
    0.0
    0.1
    0.2
    0.3
    0.4
    −0.3 0.0 0.3 0.6
    µ
    τ
    Iteration 9
    

    View Slide

100. ● ●
     µ
     ^
     τ
     ^
     −0.4 −0.2 0.0 0.2 0.4 0.00 0.05 0.10 0.15
     0
     30
     60
     90
     0
     1
     2
     3
     4
     Estimate
     Density
     Method Maximum Likelihood Average of Simulations
     

     View Slide

101. ● ●
     µ
     ^
     τ
     ^
     −0.4 −0.2 0.0 0.2 0.4 0.00 0.05 0.10 0.15
     0
     30
     60
     90
     0
     1
     2
     3
     4
     Estimate
     Density
     Method Maximum Likelihood Average of Simulations
     

     View Slide

102. ● ●
     ●
     µ
     ^
     τ
     ^
     −0.4 −0.2 0.0 0.2 0.4 0.00 0.05 0.10 0.15
     0
     30
     60
     90
     0
     1
     2
     3
     4
     Estimate
     Density
     Method Maximum Likelihood Average of Simulations
     

     View Slide

103. τ
     ^mle
     µ
     ^mle
     0.0
     0.1
     0.2
     0.3
     0.4
     −0.3 0.0 0.3 0.6
     µ
     τ
     Iteration 1
     τ
     ^avg.
     0.0
     0.1
     0.2
     0.3
     0.4
     −0.3 0.0 0.3 0.6
     µ
     τ
     Iteration 1
     

     View Slide

104. View Slide

105. For a convex transformation, the average
     of simulations is always greater than the
     (upwardly biased) ML estimate.
     1
     2
     3
     

     View Slide

106. For a convex transformation, the average
     of simulations is always greater than the
     (upwardly biased) ML estimate.
     So, of course the average of simulations is
     more biased than the ML estimate.
     1
     2
     3
     

     View Slide

107. For a convex transformation, the average
     of simulations is always greater than the
     (upwardly biased) ML estimate.
     So, of course the average of simulations is
     more biased than the ML estimate.
     This additional bias is roughly equal to the
     transformation-induced bias.
     1
     2
     3
     

     View Slide

108. two options
     use the invariance property
     average simulated QIs
     t.i. bias
     2x t.i. bias
     2017 PA
     What happens when you
     average simulated QIs?
     Does any of this matter?
     (2)
     (3)
     (1)
     

     View Slide

109. additional bias from simulation
     is similar to
     small-sample bias in logit
     

     View Slide

110. The additional bias grows larger…
     

     View Slide

111. The additional bias grows larger…
     as the standard errors increase and
     

     View Slide

112. The additional bias grows larger…
     as the standard errors increase and
     as the transformations become more nonlinear.
     

     View Slide

113. Three Examples
     

     View Slide

114. study George and Epstein (1992)
     key claim Supreme Ct. decisions depend on legal and
     extralegal factors.
     data 64 observations, 11 explanatory variables
     quantities of
     interest
     first difference: effect of a Solicitor General brief
     on the probability of a conservative decision for all
     64 combinations of the other covariates
     

     View Slide

115. ●
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     Maximum Likelihood Estimate
     Simulation Average Estimate
     Effect of a Solicitor General Brief
     

     View Slide

116. ●
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     0.0 0.2 0.4 0.6
     Maximum Likelihood Estimate
     Simulation Average Estimate
     Effect of a Solicitor General Brief
     sim. avg. about 20%
     smaller than MLE
     

     View Slide

117. ●
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     0.0 0.2 0.4 0.6
     Maximum Likelihood Estimate
     Simulation Average Estimate
     Effect of a Solicitor General Brief
     sim. avg. about 20%
     smaller than MLE
     sim. avg. about
     2x the MLE
     

     View Slide

118. ●
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     0.0 0.2 0.4 0.6
     Maximum Likelihood Estimate
     Simulation Average Estimate
     Effect of a Solicitor General Brief
     sim. avg. about 20%
     smaller than MLE
     sim. avg. about
     2x the MLE
     7 cases where sim. avg. >10x MLE
     

     View Slide

119. study Fearon and Laitin (2003)
     key claim Lots of factors lead to civil wars,
     but ethnic and religious diversity do not.
     data 6,327 observations (106 events),
     11 explanatory variables
     quantities of
     interest
     risk ratio: the effect of a 10% decrease in GDP per
     capita on the probability of civil war for all 6,327
     combinations of explanatory variables
     

     View Slide

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     100
     101
     102
     103
     104
     105
     100 100.5 101 101.5 102 102.5 103
     Maximum Likelihood Estimate
     Simulation Average Estimate
     100 100.5 101 101.5 102
     Ratio: Avg./MLE
     The Effect of a 10% Decrease in GDP on the Probability of Civil War
     

     View Slide

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     100
     101
     102
     103
     104
     105
     100 100.5 101 101.5 102 102.5 103
     Maximum Likelihood Estimate
     Simulation Average Estimate
     100 100.5 101 101.5 102
     Ratio: Avg./MLE
     The Effect of a 10% Decrease in GDP on the Probability of Civil War
     Kuwait from 1968-72
     

     View Slide

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     100
     101
     102
     103
     104
     105
     100 100.5 101 101.5 102 102.5 103
     Maximum Likelihood Estimate
     Simulation Average Estimate
     100 100.5 101 101.5 102
     Ratio: Avg./MLE
     The Effect of a 10% Decrease in GDP on the Probability of Civil War
     Kuwait from 1968-72
     year MLE s.a. ratio
     1968 101 743 7
     

     View Slide

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     100
     101
     102
     103
     104
     105
     100 100.5 101 101.5 102 102.5 103
     Maximum Likelihood Estimate
     Simulation Average Estimate
     100 100.5 101 101.5 102
     Ratio: Avg./MLE
     The Effect of a 10% Decrease in GDP on the Probability of Civil War
     Kuwait from 1968-72
     year MLE s.a. ratio
     1968 101 743 7
     1969 99 716 7
     

     View Slide

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     100
     101
     102
     103
     104
     105
     100 100.5 101 101.5 102 102.5 103
     Maximum Likelihood Estimate
     Simulation Average Estimate
     100 100.5 101 101.5 102
     Ratio: Avg./MLE
     The Effect of a 10% Decrease in GDP on the Probability of Civil War
     Kuwait from 1968-72
     year MLE s.a. ratio
     1968 101 743 7
     1969 99 716 7
     1970 65 311 5
     

     View Slide

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     100
     101
     102
     103
     104
     105
     100 100.5 101 101.5 102 102.5 103
     Maximum Likelihood Estimate
     Simulation Average Estimate
     100 100.5 101 101.5 102
     Ratio: Avg./MLE
     The Effect of a 10% Decrease in GDP on the Probability of Civil War
     Kuwait from 1968-72
     year MLE s.a. ratio
     1968 101 743 7
     1969 99 716 7
     1970 65 311 5
     1971 48 177 4
     

     View Slide

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     100
     101
     102
     103
     104
     105
     100 100.5 101 101.5 102 102.5 103
     Maximum Likelihood Estimate
     Simulation Average Estimate
     100 100.5 101 101.5 102
     Ratio: Avg./MLE
     The Effect of a 10% Decrease in GDP on the Probability of Civil War
     Kuwait from 1968-72
     year MLE s.a. ratio
     1968 101 743 7
     1969 99 716 7
     1970 65 311 5
     1971 48 177 4
     1972 50 189 4
     

     View Slide

127. study Holland (2017)
     key claim Politicians enforce laws less diligently when it
     helps them keep office
     data 19, 36, and 34 observations
     3 explanatory variables
     quantities of
     interest
     expected count: the expected number of
     enforcement operations for each observation.
     

     View Slide

128. ●
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     Bogota Lima Santiago
     10 20 30 10 20 30 40 50 2 4 6 8
     5
     10
     20
     40
     60
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     20
     30
     Maximum Likelihood Estimate
     Simulation Average Estimate
     Expected Number of Enforcement Operations
     

     View Slide

129. ●
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     Bogota Lima Santiago
     10 20 30 10 20 30 40 50 2 4 6 8
     5
     10
     20
     40
     60
     10
     20
     30
     Maximum Likelihood Estimate
     Simulation Average Estimate
     Expected Number of Enforcement Operations
     MLE is 24
     sim. avg. is 46
     sim. avg. is 90% bigger
     

     View Slide

130. ●
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     Bogota Lima Santiago
     10 20 30 10 20 30 40 50 2 4 6 8
     5
     10
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     40
     60
     10
     20
     30
     Maximum Likelihood Estimate
     Simulation Average Estimate
     Expected Number of Enforcement Operations
     MLE is 24
     sim. avg. is 46
     sim. avg. is 90% bigger
     MLE is 4.6
     sim. avg. is 5.8
     sim. avg. is 25% bigger
     

     View Slide

131. two options
     use the invariance property
     average simulated QIs
     t.i. bias
     2x t.i. bias
     2017 PA
     What happens when you
     average simulated QIs?
     Does any of this matter?
     (2)
     (3)
     (1)
     

     View Slide

132. Questions?
     

     View Slide

133. Appendix
     Stark Illustration
     • Illustration of Transformation-
     Induced Bias.
     
     Iteration 1
     
     Iteration 10
     
     Iteration 10,000
     
     • Illustration of Additional Bias
     
     Iteration 1
     
     Iteration 2
     
     Iterations 1-9
     
     Sampling Distributions
     Examples
     George and Epstein (1992)
     
     Fearon and Laitin (2003)
     
     Holland (2017)
     
     Logit Quantities of Interest
     
     Technical Details
     Theorem 1 from Rainey 2017
     
     Lemma 1
     
     Theorem 1
     
     Law of Iterated Expectations
     
     Blank
     

     View Slide

134. ●
     τ
     ^mle
     µ
     ^mle
     0.00
     0.05
     0.10
     0.15
     −0.4 −0.2 0.0 0.2 0.4
     µ
     ^
     τ
     ^
     Iteration 1
     toc
     

     View Slide

135. ●
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     µ
     ^
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     ^
     Iteration 10
     toc
     

     View Slide

136. 0.00
     0.05
     0.10
     0.15
     −0.4 −0.2 0.0 0.2 0.4
     µ
     ^
     τ
     ^
     Iteration 10000
     toc
     

     View Slide

137. τ
     ^mle
     µ
     ^mle
     0.0
     0.1
     0.2
     0.3
     0.4
     −0.3 0.0 0.3 0.6
     µ
     τ
     Iteration 1
     τ
     ^avg.
     0.0
     0.1
     0.2
     0.3
     0.4
     −0.3 0.0 0.3 0.6
     µ
     τ
     Iteration 1
     toc
     

     View Slide

138. τ
     ^mle
     µ
     ^mle
     0.0
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     0.2
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     −0.3 0.0 0.3 0.6
     µ
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     Iteration 1
     0.0
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     −0.3 0.0 0.3 0.6
     µ
     τ
     Iteration 2
     toc
     

     View Slide

139. τ
     ^avg.
     0.0
     0.1
     0.2
     0.3
     0.4
     −0.3 0.0 0.3 0.6
     µ
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     Iteration 1
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     −0.3 0.0 0.3 0.6
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     Iteration 6
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     τ
     Iteration 9
     toc
     

     View Slide

140. ● ●
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     ^
     τ
     ^
     −0.4 −0.2 0.0 0.2 0.4 0.00 0.05 0.10 0.15
     0
     30
     60
     90
     0
     1
     2
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     4
     Estimate
     Density
     Method Maximum Likelihood Average of Simulations
     toc
     

     View Slide

141. ●
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     Maximum Likelihood Estimate
     Simulation Average Estimate
     Effect of a Solicitor General Brief
     toc
     

     View Slide

142. ●
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     Maximum Likelihood Estimate
     Simulation Average Estimate
     Effect of a Solicitor General Brief
     sim. avg. about 20%
     smaller than MLE
     toc
     

     View Slide

143. ●
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     0.0 0.2 0.4 0.6
     Maximum Likelihood Estimate
     Simulation Average Estimate
     Effect of a Solicitor General Brief
     sim. avg. about 20%
     smaller than MLE
     sim. avg. about
     2x the MLE
     toc
     

     View Slide

144. ●
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     Maximum Likelihood Estimate
     Simulation Average Estimate
     Effect of a Solicitor General Brief
     sim. avg. about 20%
     smaller than MLE
     sim. avg. about
     2x the MLE
     7 cases where sim. avg. >10x MLE
     toc
     

     View Slide

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     100
     101
     102
     103
     104
     105
     100 100.5 101 101.5 102 102.5 103
     Maximum Likelihood Estimate
     Simulation Average Estimate
     100 100.5 101 101.5 102
     Ratio: Avg./MLE
     The Effect of a 10% Decrease in GDP on the Probability of Civil War
     toc
     

     View Slide

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     100
     101
     102
     103
     104
     105
     100 100.5 101 101.5 102 102.5 103
     Maximum Likelihood Estimate
     Simulation Average Estimate
     100 100.5 101 101.5 102
     Ratio: Avg./MLE
     The Effect of a 10% Decrease in GDP on the Probability of Civil War
     Kuwait from 1968-72
     toc
     

     View Slide

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     100
     101
     102
     103
     104
     105
     100 100.5 101 101.5 102 102.5 103
     Maximum Likelihood Estimate
     Simulation Average Estimate
     100 100.5 101 101.5 102
     Ratio: Avg./MLE
     The Effect of a 10% Decrease in GDP on the Probability of Civil War
     Kuwait from 1968-72
     year MLE s.a. ratio
     1968 101 743 7
     toc
     

     View Slide

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     100
     101
     102
     103
     104
     105
     100 100.5 101 101.5 102 102.5 103
     Maximum Likelihood Estimate
     Simulation Average Estimate
     100 100.5 101 101.5 102
     Ratio: Avg./MLE
     The Effect of a 10% Decrease in GDP on the Probability of Civil War
     Kuwait from 1968-72
     year MLE s.a. ratio
     1968 101 743 7
     1969 99 716 7
     toc
     

     View Slide

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     100
     101
     102
     103
     104
     105
     100 100.5 101 101.5 102 102.5 103
     Maximum Likelihood Estimate
     Simulation Average Estimate
     100 100.5 101 101.5 102
     Ratio: Avg./MLE
     The Effect of a 10% Decrease in GDP on the Probability of Civil War
     Kuwait from 1968-72
     year MLE s.a. ratio
     1968 101 743 7
     1969 99 716 7
     1970 65 311 5
     toc
     

     View Slide

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     100
     101
     102
     103
     104
     105
     100 100.5 101 101.5 102 102.5 103
     Maximum Likelihood Estimate
     Simulation Average Estimate
     100 100.5 101 101.5 102
     Ratio: Avg./MLE
     The Effect of a 10% Decrease in GDP on the Probability of Civil War
     Kuwait from 1968-72
     year MLE s.a. ratio
     1968 101 743 7
     1969 99 716 7
     1970 65 311 5
     1971 48 177 4
     toc
     

     View Slide

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     100
     101
     102
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     104
     105
     100 100.5 101 101.5 102 102.5 103
     Maximum Likelihood Estimate
     Simulation Average Estimate
     100 100.5 101 101.5 102
     Ratio: Avg./MLE
     The Effect of a 10% Decrease in GDP on the Probability of Civil War
     Kuwait from 1968-72
     year MLE s.a. ratio
     1968 101 743 7
     1969 99 716 7
     1970 65 311 5
     1971 48 177 4
     1972 50 189 4
     toc
     

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152. ●
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     Bogota Lima Santiago
     10 20 30 10 20 30 40 50 2 4 6 8
     5
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     40
     60
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     30
     Maximum Likelihood Estimate
     Simulation Average Estimate
     Expected Number of Enforcement Operations
     toc
     

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153. ●
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     Bogota Lima Santiago
     10 20 30 10 20 30 40 50 2 4 6 8
     5
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     60
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     30
     Maximum Likelihood Estimate
     Simulation Average Estimate
     Expected Number of Enforcement Operations
     MLE is 24
     sim. avg. is 46
     sim. avg. is 90% bigger
     toc
     

     View Slide

154. ●
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     Bogota Lima Santiago
     10 20 30 10 20 30 40 50 2 4 6 8
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     60
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     Maximum Likelihood Estimate
     Simulation Average Estimate
     Expected Number of Enforcement Operations
     MLE is 24
     sim. avg. is 46
     sim. avg. is 90% bigger
     MLE is 4.6
     sim. avg. is 5.8
     sim. avg. is 25% bigger
     toc
     

     View Slide

155. yi
     ⇠ Bernoulli(⇡i), for i = 1, 2, ..., 100
     logit(⇡i) = 1.5 + xi + 0.2wi + 0.5zi
     toc
     

     View Slide

156. probability marginal effect
     first difference risk ratio
     coefficient−induced bias transformation−induced bias additional bias coefficient−induced bias transformation−induced bias additional bias
     coefficient−induced bias transformation−induced bias additional bias coefficient−induced bias transformation−induced bias additional bias
     0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.05 0.10 0.15 0.20 0.25
     0.05 0.10 0.15 0.20 0.25
     0.05 0.10 0.15 0.20 0.25
     0.050 0.075 0.100 0.125 0.050 0.075 0.100 0.125 0.050 0.075 0.100 0.125 1.3 1.4 1.5 1.6 1.3 1.4 1.5 1.6 1.3 1.4 1.5 1.6
     True Quantity of Interest True Quantity of Interest
     True Quantity of Interest True Quantity of Interest
     Bias
     Bias
     Bias
     Bias
     −0.02
     −0.01
     0.00
     0.01
     0.02
     −0.04
     −0.02
     0.00
     −0.02
     −0.01
     0.00
     −0.04
     0.00
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     toc
     

     View Slide

157. Theorem 1 (t.i. bias, Rainey 2017) Suppose a non-degenerate estimator ˆ.
     Then any strictly convex (concave) ⌧ creates upward (downward) transformation-
     induced ⌧-bias.
     Proof The proof follows directly from Jensen’s inequality. Suppose that the
     non-degenerate sampling distribution of ˆ is given by S (b) so that ˆ ⇠ S (b).
     Then E(ˆ) =
     R
     B
     bS (b)db and E[⌧(ˆ)] =
     R
     B
     ⌧(b)S (b)db. Suppose first that ⌧ is
     convex. By Jensen’s inequality,
     R
     B
     ⌧(b)S (b)db > ⌧
     ⇥R
     B
     bS (b)db
     ⇤
     , which implies
     that E[⌧(ˆ)] > ⌧[E(ˆ)]. Because E[⌧(ˆ)] ⌧[E(ˆ)] > 0, the transformation-
     induced ⌧-bias is upward. By similar argument, one can show that for any
     strictly concave ⌧, E[⌧(ˆ)] ⌧[E(ˆ)] > 0 and that the transformation-induced
     ⌧-bias is downward. ⌅
     toc
     

     View Slide

158. toc
     Lemma 1 Suppose a maximum likelihood estimator ˆmle. Then any strictly
     convex (concave) ⌧ guarantees that ˆ
     ⌧avg. is strictly greater [less] than ˆ
     ⌧mle.
     Proof By definition,
     ˆ
     ⌧avg. = E
     h
     ⌧
     ⇣
     ˜
     ⌘i
     .
     Using Jensen’s inequality, we know that E
     h
     ⌧
     ⇣
     ˜
     ⌘i
     > ⌧
     h
     E
     ⇣
     ˜
     ⌘i
     , so that
     ˆ
     ⌧avg. > ⌧
     h
     E
     ⇣
     ˜
     ⌘i
     .
     However, because ˜ ⇠ N
     h
     ˆmle, ˆ
     V
     ⇣
     ˆmle
     ⌘i
     , E
     ⇣
     ˜
     ⌘
     = ˆmle, so that
     ˆ
     ⌧avg. > ⌧
     ⇣
     ˆmle
     ⌘
     .
     Of course, ˆ
     ⌧mle = ⌧
     ⇣
     ˆmle
     ⌘
     by definition, so that
     ˆ
     ⌧avg. > ˆ
     ⌧mle.
     The proof for concave ⌧ follows similarly. ⌅
     

     View Slide

159. Theorem 1 Suppose a maximum likelihood estimator ˆmle. Then for any strictly
     convex or concave ⌧, the transformation-induced ⌧-bias for ˆ
     ⌧avg. is strictly
     greater in magnitude than the transformation-induced ⌧-bias for ˆ
     ⌧mle.
     Proof According to Theorem 1 of Rainey (2017), E ˆ
     ⌧mle ⌧
     h
     E
     ⇣
     ˆmle
     ⌘i
     >
     0. Lemma 1 shows that for any convex ⌧, ˆ
     ⌧avg. > ˆ
     ⌧mle. It follows that
     E (ˆ
     ⌧avg.) ⌧
     h
     E
     ⇣
     ˆmle
     ⌘i
     | {z }
     t.i. ⌧-bias in ˆ
     ⌧avg.
     > E ˆ
     ⌧mle ⌧
     h
     E
     ⇣
     ˆmle
     ⌘i
     | {z }
     t.i. ⌧-bias in ˆ
     ⌧mle
     > 0. For the concave case,
     it follows similarly that E (ˆ
     ⌧avg.) ⌧
     h
     E
     ⇣
     ˆmle
     ⌘i
     | {z }
     t.i. ⌧-bias in ˆ
     ⌧avg.
     < E ˆ
     ⌧mle ⌧
     h
     E
     ⇣
     ˆmle
     ⌘i
     | {z }
     t.i. ⌧-bias in ˆ
     ⌧mle
     < 0
     ⌅
     toc
     

     View Slide

160. toc
     

     View Slide

161. toc
     

     View Slide