BIWEIGHT CONFIDENCE LIMITS

BIWEIGHT CONFIDENCE LIMITS

Analysis Command

Generates a biweight based confidence interval for the location of a variable.

Mosteller and Tukey (see Reference section below) define two types of robustness:
1. resistance means that changing a small part, even by a large amount, of the data does not cause a large change in the estimate

2. robustness of efficiency means that the statistic has high efficiency in a variety of situations rather than in any one situation. Efficiency means that the estimate is close to optimal estimate given that we distribution that the data comes from. A useful measure of efficiency is:
   Efficiency = (lowest variance feasible)/(actual variance)

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:

\( \mbox{biweight confidence limits } = \mbox{biweight location } \pm t_{\nu}\sqrt{\frac{\mbox{biweight scale}}{n}} \)

where the biweight location and biweight scale are location and scale estimators based on the biweight and ν = 0.7*(n-1). The definitions for the biweight location and biweight scale estimators are given in:

HELP BIWEIGHT LOCATION
HELP BIWEIGHT SCALE

BIWEIGHT CONFIDENCE LIMITS <y>           <SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable, and where the <SUBSET/EXCEPT/FOR qualification> is optional.

BIWEIGHT CONFIDENCE LIMITS Y1
BIWEIGHT CONFIDENCE LIMITS Y1 SUBSET TAG > 2

A table of confidence intervals is printed for alpha levels of 50.0, 75.0, 90.0, 95.0, 99.0, 99.9, 99.99, and 99.999. The sample biweight location estiamte and sample biweight scale estimate are also printed. The t-value and t-value * \( \sqrt{s_{bi}^2} \) are printed in the table.

None

None

CONFIDENCE LIMITS	= Compute a Gaussian based confidence limit.
T-TEST	= Perform a t-test.
BIWEIGHT LOCATION	= Compute a biweight location estimate.
BIWEIGHT SCALE	= Compute a biweight scale estimate.

Mosteller and Tukey (1977), "Data Analysis and Regression: A Second Course in Statistics," Addison-Wesley, pp. 203-209.

Robust Data Analysis

2001/11

 
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 Y4
    

Dataplot generates the following output:

             Confidence Limits for Biweight Location
                           (Two-Sided)
  
 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      t-Value 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
                           (Two-Sided)
  
 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      t-Value 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
                           (Two-Sided)
  
 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      t-Value 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
                           (Two-Sided)
  
 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      t-Value 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