Compare Jackknife/Bootstrap methods for the slope of a regression
Starting random-number seed: 123456

X-coordinate of regressions (n=10, xbar=108.400, xsse=79582.4):
  2  18  34  57  60  78  134  215  229  257

REGRESSION WITH NORMAL ERRORS (n=10, mu=3, beta=3, sigma=20)
Normal regression data (n=10, ybar=339.983, xbar=108.400):
   (2, -6.47)   (18, 76.93)   (34, 113.38)   (57, 192.46)
   (60, 184.28)   (78, 276.54)   (134, 434.16)   (215, 665.46)
   (229, 687.44)   (257, 775.66)

Least-squares estimates:  BetaHat=2.994   Muhat=15.390
  True values:            Beta=3.000      Mu=3.000 
  Residual MSE=298.23  (true sigma^2=400)
  Standard Est(Var(betahat)):  0.00374743  True Var(betahat): 0.00502624
  t(8,0.975)=2.306   (z(0.975)=1.960)   CIwid=0.141

Full-sample estimates for true beta:
  BetaHat=2.994   95% CI (2.853, 3.136)   True beta=3.000

Coverage probability of beta for 95% normal CIs (true beta=3):
Normal samples of size 10:
Estimated coverage = 9503/10,000 = 95.03%

JACKKNIFE ESTIMATES OF BETA FOR NORMAL DATA:
Jackknife record mean: 2.989   95% CI: (2.593, 3.385)

Coverage probability of beta for 95% jackknife CIs (true beta=3):
Normal samples of size 10:
Estimated coverage = 9993/10,000 = 99.93%

BOOKSTRAP RECORD ESTIMATES OF BETA FOR NORMAL DATA:
Filling the bootstrap ``beta(star)'' array:
Sorting the bootstrap ``beta(star)'' array:
Bootstrap median bias (Obs_Beta=2.9944):  4992/10000 = 0.499

Bootstrap median and bootstrap 95% CI for beta:
  2.9946   (2.8923, 3.1853)   Observed beta=2.9944

Coverage probability of beta for 95% bootstrap record CIs (true beta=3):
Normal samples of size 10:
Estimated coverage = 9023/10,000 = 90.23%

BOOKSTRAP RESIDUAL ESTIMATES OF BETA FOR NORMAL DATA:
Residuals of  Y = 2.994*X + 15.390:
  -27.853  7.637  -3.821  6.388  -10.771
  27.587  17.517  6.273  -13.665  -9.293
Filling the bootstrap ``beta(star)'' array:
Sorting the bootstrap ``beta(star)'' array:
Bootstrap median bias (Obs_Beta=2.9944):  4907/10000 = 0.491

Bootstrap median and bootstrap 95% CI for beta:
  2.9957   (2.8883, 3.1008)   Observed beta=2.9944

Coverage probability of beta for 95% bootstrap residual CIs (true beta=3):
Normal samples of size 10:
Estimated coverage = 8796/10,000 = 87.96%

REGRESSION WITH EXPONENTIAL ERRORS (n=10, mu=3, beta=3, sigma=20)
Exponential regression data (n=10, ybar=358.451, xbar=108.400):
   (2, 37.37)   (18, 75.42)   (34, 105.02)   (57, 187.13)
   (60, 183.84)   (78, 370.66)   (134, 422.92)   (215, 677.93)
   (229, 720.65)   (257, 803.56)
Least-squares estimates:  BetaHat=3.029   Muhat=30.151
  True values:            Beta=3.000      Mu=3.000 
  Residual MSE=1624.48  (true sigma^2=400)
  Standard Est(Var(betahat)):  0.0204125  True Var(betahat): 0.00502624
  t(8,0.975)=2.306   (z(0.975)=1.960)   CIwid=0.329

Full-sample estimates for true beta:
  BetaHat=3.029   95% CI (2.699, 3.358)   True beta=3.000

Coverage probability of beta for 95% normal CIs (true beta=3):
Exponential samples of size 10:
Estimated coverage = 9533/10,000 = 95.33%

JACKKNIFE ESTIMATES OF BETA FOR EXPONENTIAL DATA:
Jackknife record mean: 3.026   95% CI: (2.653, 3.399)

Coverage probability of beta for 95% jackknife CIs (true beta=3):
Exponential samples of size 10:
Estimated coverage = 9995/10,000 = 99.95%

BOOKSTRAP RECORD ESTIMATES OF BETA FOR EXPONENTIAL DATA:
Filling the bootstrap ``beta(star)'' array:
Sorting the bootstrap ``beta(star)'' array:
Bootstrap median bias (Obs_Beta=3.02859):  4003/10000 = 0.400

Bootstrap median and bootstrap 95% CI for beta:
  3.0420   (2.8377, 3.1580)   Observed beta=3.0286

Coverage probability of beta for 95% bootstrap record CIs (true beta=3):
Exponential samples of size 10:
Estimated coverage = 9064/10,000 = 90.64%

BOOKSTRAP RESIDUAL ESTIMATES OF BETA FOR EXPONENTIAL DATA:
Residuals of  Y = 3.029*X + 30.151:
  1.161  -9.245  -28.101  -15.655  -28.028
  104.275  -13.058  -3.368  -3.046  -4.936
Filling the bootstrap ``beta(star)'' array:
Sorting the bootstrap ``beta(star)'' array:
Bootstrap median bias (Obs_Beta=3.02859):  5296/10000 = 0.530

Bootstrap median and bootstrap 95% CI for beta:
  3.0224   (2.7841, 3.2968)   Observed beta=3.0286

Coverage probability of beta for 95% bootstrap residual CIs (true beta=3):
Exponential samples of size 10:
Estimated coverage = 8910/10,000 = 89.10%

Summary of estimates and 95% confidence intervals:
True values of beta:  Normal: 3   Exponential: 3
  Normal, Classical t(8)            2.9944   ( 2.8532,  3.1356)
  Normal, Jackknife                 2.9891   ( 2.5928,  3.3854)
  Normal, Bootstrap record          2.9946   ( 2.8923,  3.1853)
  Normal, Bootstrap residual        2.9957   ( 2.8883,  3.1008)
  Exponential, Classical t(8)       3.0286   ( 2.6991,  3.3581)
  Exponential, Jackknife            3.0261   ( 2.6530,  3.3992)
  Exponential, Bootstrap record     3.0420   ( 2.8377,  3.1580)
  Exponential, Bootstrap residual   3.0224   ( 2.7841,  3.2968)

Summary of coverage probabilities of 95% confidence intervals:
True values of beta:  Normal: 3   Exponential: 3
  Normal, Classical t(8)           9503/10,000 = 95.03%
  Normal, Jackknife                9993/10,000 = 99.93%
  Normal, Bootstrap record         9023/10,000 = 90.23%
  Normal, Bootstrap residual       8796/10,000 = 87.96%
  Exponential, Classical t(8)      9533/10,000 = 95.33%
  Exponential, Jackknife           9995/10,000 = 99.95%
  Exponential, Bootstrap record    9064/10,000 = 90.64%
  Exponential, Bootstrap residual  8910/10,000 = 89.10%

Random numbers used:  4,001,309,726
Initial random-number seed: 123456