
BIWEIGHT CONFIDENCE LIMITSName:
Many statistics have one of these properties. However, it can be difficult to find statistics that are both resistant and have robustness of efficiency. Standard confidence intervals are base in the mean and variance. These are the optimal estimators if the data are in fact from a Gaussian population. However, they lack both resistance and robustness of efficiency. The biweight confidence interval is based on estimates of of location and scale that are both resistant and have robustness of efficiency. Therefore it should provide a reasonable confidence interval when the normality assumption cannot be validated. Note that it is still a symmetric confidence interval. However, symmetry is a much looser assumption than normality. The biweight confidence interval for the population biweight location is defined by:
where the biweight location and biweight scale are location and scale estimators based on the biweight and ν = 0.7*(n1). The definitions for the biweight location and biweight scale estimators are given in:
HELP BIWEIGHT SCALE
where <y> is the response variable, and where the <SUBSET/EXCEPT/FOR qualification> is optional.
BIWEIGHT CONFIDENCE LIMITS Y1 SUBSET TAG > 2
LET Y1 = NORMAL RANDOM NUMBERS FOR I = 1 1 100 LET Y2 = LOGISTIC RANDOM NUMBERS FOR I = 1 1 100 LET Y3 = CAUCHY RANDOM NUMBERS FOR I = 1 1 100 LET Y4 = DOUBLE EXPONENTIAL RANDOM NUMBERS FOR I = 1 1 100 SET WRITE DECIMALS 4 BIWEIGHT CONFIDENCE LIMTIS Y1 TO Y4Dataplot generates the following output: Confidence Limits for Biweight Location (TwoSided) Response Variable: Y1 Summary Statistics: Number of Observations: 100 Sample Biweight Location: 0.01272 Sample Biweight Scale 0.78155 Standard Error: 0.08840 Degrees of Freedom: 69.00000  Confidence t tValue X Lower Upper Value (%) Value StdErr Limit Limit  50.000 0.678 0.05994 0.04722 0.07266 75.000 1.160 0.10256 0.08983 0.11528 90.000 1.667 0.14739 0.13466 0.16011 95.000 1.994 0.17636 0.16364 0.18908 99.000 2.648 0.23418 0.22146 0.24690 99.900 3.437 0.30386 0.29114 0.31659 99.990 4.130 0.36514 0.35242 0.37787 99.999 4.767 0.42151 0.40879 0.43424 Confidence Limits for Biweight Location (TwoSided) Response Variable: Y2 Summary Statistics: Number of Observations: 100 Sample Biweight Location: 0.09524 Sample Biweight Scale 3.52551 Standard Error: 0.18776 Degrees of Freedom: 69.00000  Confidence t tValue X Lower Upper Value (%) Value StdErr Limit Limit  50.000 0.678 0.12731 0.03207 0.22255 75.000 1.160 0.21782 0.12258 0.31307 90.000 1.667 0.31304 0.21780 0.40829 95.000 1.994 0.37457 0.27933 0.46982 99.000 2.648 0.49738 0.40213 0.59262 99.900 3.437 0.64537 0.55013 0.74062 99.990 4.130 0.77553 0.68028 0.87077 99.999 4.767 0.89525 0.80000 0.99049 Confidence Limits for Biweight Location (TwoSided) Response Variable: Y3 Summary Statistics: Number of Observations: 100 Sample Biweight Location: 0.18511 Sample Biweight Scale 2.86058 Standard Error: 0.16913 Degrees of Freedom: 69.00000  Confidence t tValue X Lower Upper Value (%) Value StdErr Limit Limit  50.000 0.678 0.11468 0.07043 0.29980 75.000 1.160 0.19621 0.01109 0.38133 90.000 1.667 0.28198 0.09686 0.46710 95.000 1.994 0.33741 0.15229 0.52252 99.000 2.648 0.44802 0.26291 0.63314 99.900 3.437 0.58134 0.39622 0.76645 99.990 4.130 0.69858 0.51346 0.88369 99.999 4.767 0.80641 0.62130 0.99153 Confidence Limits for Biweight Location (TwoSided) Response Variable: Y4 Summary Statistics: Number of Observations: 100 Sample Biweight Location: 0.00512 Sample Biweight Scale 0.93957 Standard Error: 0.09693 Degrees of Freedom: 69.00000  Confidence t tValue X Lower Upper Value (%) Value StdErr Limit Limit  50.000 0.678 0.06572 0.07085 0.06060 75.000 1.160 0.11245 0.11757 0.10732 90.000 1.667 0.16160 0.16673 0.15648 95.000 1.994 0.19337 0.19849 0.18824 99.000 2.648 0.25676 0.26189 0.25164 99.900 3.437 0.33317 0.33829 0.32804 99.990 4.130 0.40036 0.40548 0.39523 99.999 4.767 0.46216 0.46729 0.45704  
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Date created: 11/21/2001 