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Dataplot Vol 2 Vol 1

ORDER STATISTICS

Name:
    ORDER STATISTICS (LET)
Type:
    Let Subcommand
Purpose:
    Generate the N order statistics for a given statistic from a given distribution.
Description:
    Given a sample of size N from a distribution, the observations can be ordered from smallest to largest. The smallest value is called the first order statistic and the largest value is called the n-th order statistic. There is an order statistic corresponding to each observation (so there are N order statistics in all). Each of these order statistics has its own sampling distribution with its own statistics (e.g., mean, standard deviation).

    Dataplot supports analytic order statistics for a few specific statistics (median, mean, standard deviation) for a limited number of distributons. Enter HELP ORDER STATISTIC MEDIANS or HELP ORDER STATISTIC MEANS for details.

    This command estimates the order statistics for an arbitrary statistic and an arbitrary distribution. Specifically, N random numbers are generated from the specified distribution NSIM times. Then for each order statistic there will be NSIM values. The specified statistic is computed for these NSIM values for each order statistic.

Syntax:
    LET <resp> = <dist> ORDER STATISTICS <stat>
                            FOR I = <start> <inc> <stop>
    where <dist> specifies the distribution for which the order statistics are computed;
                <stat> specifies the desired statistic;
                <start> is a number or parameter that identifies the first row of <resp> in which the order statistics are saved (typically it has a value of 1);
                <inc> is a number or parameter that identifies the row increment of <resp> in which the order statistics are saved (typically it has a value of 1);
                <stop> is a number or parameter that identifies the last row of <resp> in which the order statistics are saved;
    and where <resp> is a variable where the order statistics are saved.
Examples:
    LET Y1 = NORMAL ORDER STATISTICS INTERQUARTILE RANGE FOR I = 1 1 100
    LET Y2 = EXPONENTIAL ORDER STATISTICS BIWEIGHT LOCATION FOR I = 1 1 1000
    LET Y1 = LOGISTIC ORDER STATISTICS MAD FOR I = 1 1 100
    LET Y1 = UNIFORM ORDER STATISTICS MIDRANGE FOR I = 1 1 100
    LET Y1 = EV1 ORDER STATISTICS MIDMEAN FOR I = 1 1 100

    LET GAMMA = 2.7
    LET Y1 = WEIBULL ORDER STATISTICS SKEWNESS FOR I = 1 1 100

    LET GAMMA = 2 LET Y1 = EV2 ORDER STATISTICS KURTOSIS FOR I = 1 1 100

Note:
    By default, 10,000 simulations are performed. To specify a different number of simulations, enter the command

      SET ORDER STATISTICS SAMPLE SIZE <value>

    where <value> is an integer between 100 and 1,000,000. Values less than 100 will be set to 100 and values greater than 1,000,000 will be set to 1,000,000.

Note:
    For cases where an analytic approach is supported, you can specify whether to use the analytic method (i.e., the ORDER STATISTIC MEDIAN or the ORDER STATISTIC MEAN commands) or the simulation method described here. By default, the analytic method will be used. To specify the simulation method, enter the command

      SET ORDER STATISTICS METHOD SIMULATION

    To reset the default, enter

      SET ORDER STATISTICS METHOD ANALYTIC
Note:
    To see a list of supported statistics, enter HELP STATISTICS. Note that only statistics based on a single response variable are supported by this command.

    To see a list of supported probability distributions, enter HELP PROBABILITY DISTRIBUTIONS.

Default:
    None
Synonyms:
    None
Related Commands: Applications:
    Extreme value analysis
References:
    Filliben (1975), "The Probability Plot Correlation Coefficient Test for Normality," Technometrics, pp. 111-117.
Implementation Date:
    2024/06
Program: 
     
. Step 1:   Set some defaults
.
let n = 100
set write decimals 5
.
set order statistic method simulation
set order statistic sample size 10000
.
. Step 2:   Weibull, Gamma, Frechet SD
.
let gamma = 2.9
let ysd1   = weibull order statistic sd      for i = 1 1 n
let ysd2   = gamma   order statistic sd      for i = 1 1 n
let ysd3   = frechet order statistic sd      for i = 1 1 n
.
print ysd1 ysd2 ysd1b ysd2b
.
. Step 3:   Weibull mean, median, midmean, biweight location
.
let yloc1   = weibull order statistic mean              for i = 1 1 n
let yloc2   = weibull order statistic median            for i = 1 1 n
let yloc3   = weibull order statistic midmean           for i = 1 1 n
let yloc4   = weibull order statistic biweight location for i = 1 1 n
.
print yloc1 yloc2 yloc3 yloc4
    The following output is generated 
     ------------------------------
            YSD1           YSD2
 ------------------------------
         0.06834        0.14600
         0.06268        0.14291
         0.05933        0.14160
         0.05677        0.14064
         0.05511        0.14083
         0.05383        0.14096
         0.05263        0.14093
         0.05141        0.14179
         0.05049        0.14234
         0.04958        0.14277
         0.04907        0.14274
         0.04874        0.14393
         0.04844        0.14524
         0.04810        0.14646
         0.04763        0.14729
         0.04699        0.14853
         0.04662        0.14947
         0.04656        0.15012
         0.04627        0.15084
         0.04585        0.15226
         0.04568        0.15290
         0.04546        0.15470
         0.04518        0.15642
         0.04504        0.15731
         0.04497        0.15872
         0.04474        0.15970
         0.04453        0.16048
         0.04445        0.16200
         0.04426        0.16315
         0.04413        0.16415
         0.04412        0.16517
         0.04393        0.16721
         0.04393        0.16823
         0.04388        0.16965
         0.04379        0.17103
         0.04366        0.17244
         0.04365        0.17393
         0.04361        0.17531
         0.04358        0.17749
         0.04351        0.17883
         0.04353        0.18056
         0.04347        0.18230
         0.04359        0.18370
         0.04344        0.18465
         0.04342        0.18605
         0.04341        0.18850
         0.04347        0.19014
         0.04328        0.19204
         0.04336        0.19446
         0.04350        0.19669
         0.04351        0.19916
         0.04355        0.20144
         0.04352        0.20289
         0.04364        0.20535
         0.04371        0.20770
         0.04383        0.21019
         0.04394        0.21293
         0.04411        0.21628
         0.04425        0.21930
         0.04439        0.22183
         0.04453        0.22437
         0.04467        0.22662
         0.04487        0.22911
         0.04514        0.23237
         0.04541        0.23531
         0.04553        0.23957
         0.04579        0.24231
         0.04619        0.24602
         0.04638        0.24906
         0.04657        0.25326
         0.04671        0.25835
         0.04691        0.26305
         0.04722        0.26708
         0.04734        0.27111
         0.04761        0.27695
         0.04789        0.28166
         0.04836        0.28584
         0.04874        0.29162
         0.04923        0.29980
         0.04971        0.30611
         0.05022        0.31234
         0.05085        0.31935
         0.05147        0.32724
         0.05230        0.33611
         0.05282        0.34340
         0.05389        0.35574
         0.05480        0.36704
         0.05598        0.38306
         0.05717        0.39759
         0.05840        0.41394
         0.06009        0.43488
         0.06197        0.46250
         0.06404        0.49152
         0.06658        0.52557
         0.07038        0.56692
         0.07439        0.62003
         0.08018        0.69657
         0.08893        0.81458
         0.10478        1.00407
         0.14329        1.54811
  
  
 ------------------------------------------------------------
           YLOC1          YLOC2          YLOC3          YLOC4
 ------------------------------------------------------------
         0.18201        0.18029        0.17932        0.18040
         0.24548        0.24635        0.24476        0.24398
         0.28814        0.28850        0.28726        0.28710
         0.32239        0.32237        0.32185        0.32110
         0.35088        0.35002        0.35034        0.34952
         0.37589        0.37471        0.37486        0.37409
         0.39788        0.39708        0.39757        0.39664
         0.41834        0.41716        0.41767        0.41679
         0.43710        0.43677        0.43623        0.43576
         0.45448        0.45392        0.45397        0.45339
         0.47094        0.47078        0.47073        0.47028
         0.48660        0.48708        0.48660        0.48607
         0.50163        0.50180        0.50134        0.50101
         0.51588        0.51553        0.51558        0.51543
         0.52975        0.52860        0.52949        0.52911
         0.54297        0.54237        0.54290        0.54239
         0.55588        0.55572        0.55569        0.55527
         0.56830        0.56803        0.56811        0.56764
         0.58075        0.57993        0.58029        0.57962
         0.59238        0.59215        0.59212        0.59142
         0.60392        0.60314        0.60380        0.60288
         0.61516        0.61410        0.61503        0.61404
         0.62600        0.62501        0.62611        0.62507
         0.63671        0.63611        0.63670        0.63592
         0.64711        0.64667        0.64724        0.64637
         0.65745        0.65695        0.65758        0.65670
         0.66756        0.66767        0.66767        0.66698
         0.67757        0.67783        0.67747        0.67703
         0.68736        0.68741        0.68727        0.68690
         0.69716        0.69688        0.69705        0.69670
         0.70677        0.70705        0.70663        0.70626
         0.71628        0.71664        0.71633        0.71576
         0.72568        0.72610        0.72571        0.72504
         0.73506        0.73535        0.73506        0.73431
         0.74424        0.74491        0.74419        0.74385
         0.75343        0.75389        0.75346        0.75304
         0.76262        0.76300        0.76262        0.76204
         0.77171        0.77235        0.77163        0.77105
         0.78081        0.78132        0.78062        0.78020
         0.78978        0.78997        0.78933        0.78919
         0.79858        0.79861        0.79813        0.79809
         0.80740        0.80756        0.80701        0.80693
         0.81610        0.81624        0.81608        0.81569
         0.82487        0.82520        0.82486        0.82452
         0.83370        0.83323        0.83380        0.83329
         0.84244        0.84240        0.84256        0.84185
         0.85124        0.85092        0.85127        0.85076
         0.85991        0.85964        0.86000        0.85957
         0.86872        0.86838        0.86882        0.86830
         0.87737        0.87713        0.87752        0.87703
         0.88604        0.88583        0.88628        0.88575
         0.89481        0.89489        0.89504        0.89445
         0.90357        0.90360        0.90392        0.90325
         0.91227        0.91239        0.91269        0.91193
         0.92100        0.92115        0.92130        0.92078
         0.92980        0.93035        0.93027        0.92951
         0.93864        0.93907        0.93903        0.93835
         0.94746        0.94763        0.94801        0.94743
         0.95644        0.95692        0.95722        0.95651
         0.96547        0.96579        0.96624        0.96552
         0.97448        0.97481        0.97534        0.97464
         0.98366        0.98390        0.98455        0.98395
         0.99293        0.99340        0.99359        0.99322
         1.00224        1.00280        1.00298        1.00243
         1.01164        1.01262        1.01247        1.01176
         1.02125        1.02225        1.02201        1.02141
         1.03089        1.03109        1.03171        1.03111
         1.04061        1.04118        1.04118        1.04091
         1.05048        1.05095        1.05110        1.05095
         1.06051        1.06079        1.06099        1.06097
         1.07077        1.07072        1.07114        1.07091
         1.08106        1.08087        1.08162        1.08123
         1.09168        1.09146        1.09203        1.09161
         1.10243        1.10221        1.10262        1.10229
         1.11342        1.11283        1.11333        1.11314
         1.12449        1.12390        1.12470        1.12407
         1.13566        1.13512        1.13587        1.13556
         1.14721        1.14656        1.14747        1.14706
         1.15900        1.15883        1.15910        1.15899
         1.17114        1.17112        1.17116        1.17115
         1.18368        1.18327        1.18363        1.18369
         1.19671        1.19642        1.19666        1.19637
         1.20985        1.21000        1.20988        1.20928
         1.22358        1.22398        1.22372        1.22306
         1.23815        1.23830        1.23810        1.23730
         1.25327        1.25333        1.25291        1.25231
         1.26897        1.26900        1.26876        1.26811
         1.28556        1.28530        1.28509        1.28458
         1.30282        1.30276        1.30263        1.30213
         1.32109        1.32143        1.32090        1.32056
         1.34117        1.34095        1.34070        1.34005
         1.36243        1.36320        1.36179        1.36166
         1.38575        1.38549        1.38529        1.38477
         1.41154        1.41130        1.41065        1.41005
         1.44054        1.43982        1.43957        1.43873
         1.47396        1.47274        1.47211        1.47172
         1.51391        1.51164        1.51176        1.51085
         1.56346        1.55900        1.55976        1.55948
         1.63294        1.62620        1.62722        1.62627
         1.75229        1.73998        1.74048        1.74120
Date created: 07/31/2024
Last updated: 07/31/2024
Please email comments on this WWW page to alan.heckert@nist.gov.