SED navigation bar go to SED home page go to Dataplot home page go to NIST home page SED Home Page SED Contacts SED Projects SED Products and Publications Search SED Pages
Dataplot Vol 1 Auxiliary Chapter

MAXIMUM LIKELIHOOD

Name:
    ... MAXIMUM LIKELIHOOD
Type:
    Analysis Command
Purpose:
    Compute the maximum likelihood estimates for the parameters of a statistical distribution.
Description:
    There are a number of approaches to estimating the parameters of a statistical distribution from a set of data.

    Maximum likelihood estimates are popular because they have good statistical properties. The primary drawback is that likelihood equations have to be derived for each specific distributions (other approaches, such as least squares or PPCC plots, allow a more general approach). In some cases, the maximum likelihood estimates are trivial while in other cases they are quite complex and may require specialized methods to solve.

    Dataplot currently supports maximum likelihood estimates for the following continuous distributions:

    1. normal
    2. 2-parameter lognormal
    3. exponential
    4. 2-parameter Weibull
    5. 2-parameter inverted Weibull
    6. 2-parameter gamma
    7. Gumbel (extreme value type 1)
    8. Frechet (extreme value type 2, maximum case only)
    9. beta
    10. Pareto
    11. Rayleigh
    12. logistic
    13. Cauchy
    14. double exponential
    15. inverse gaussian
    16. power
    17. uniform
    18. Johnson SB (method of moments, percentile)
    19. Johnson SU (method of moments, percentile)
    20. fatigue life
    21. geometric extreme exponential
    22. folded normal
    23. Generalized Pareto
    24. Asymmetric Double Exponential
    25. Maxwell
    26. mixture or normal distributions (number of components assummed known)

    Dataplot currently supports maximum likelihood estimates for the following discrete distributions:

    1. binomial
    2. Poisson
    3. Logarithmic series
    4. geometric
    5. beta binomial
    6. negative binomial
    7. hypergeometric
    8. Hermite
    9. Yule

    Additional distributions may be added in the future.

    We do not give the likelihood equations for the various distributions here. Most of them can be found in the sources listed in the Reference section.

    For a given distribution, the maximum likelihood command will generate one or more of the following outputs:

    1. Some summary statistics for the data and point estimates for the parameters of the distribution.

      This is the minimum output and is supported for all distributions.

    2. Confidence intervals for the parameters of the distribution. This is supported for the following 16 distributions:

        normal, 2-parameter lognormal, exponential, 2-parameter Weibull, 2-parameter gamma, Gumbel (extreme value type 1), beta, Pareto, Rayleigh, logistic, Cauchy, 2-parameter Frechet, 2-parameter Weibull, binomial, geometric, Poisson

      Confidence intervals are obtained in the following ways:

      1. In some cases, the sampling distribution, or an an approximation to the sampling distribution, may be known for the given parameter. In these cases, an explicit formula for the confidence interval or a numerically tabulated value can be used.

      2. If the standard error for the parameter can be determined, the large sample asymptotic normal approximation can be obtained as

          point estimate +/- NORPPF(alpha/2)*STDERR

        with STDERR, NORPPF, and alpha denoting the standard error of the parameter estimate, the percent point function of the standard normal distribution, and the desired significance level, respectively.

      3. For a few distributions, likelihood ratio methods are used to determine a confidence interval. These can be more accurate than the normal approximations, particularly for small samples where the asymptotic normality may not be as accurate.

    3. Confidence intervals for selected percentiles of distribution. The command

        SET MAXIMUM LIKELIHOOD PERCENTILE <NONE/DEFAULT/VARNAME>

      where NONE means no percentile confidence limits will be generated, DEFAULT generates percentile confidence limits for a default set of percentiles, and VARNAME specifies the name of a variable that contains the percentile values where the confidence limits will be generated.

      This option is supported for the following seven distributions:

        normal, 2-parameter lognormal, exponential, 2-parameter Weibull, gamma, beta, gumbel (maximum case only)

      Note that point estimates for selected percentiles can be generated by simply using the point estimates for the distribution parameters in the percent point function of the distribution.

    4. Data is sometimes censored. With censored data, we are typically interested in modeling failure or lifetime data. In censored data, we typically have r failure times and n-r censoring (or survival) times (a censoring time means the unit had not failed at the time the test was terminated).

      There are several types of censoring:

      1. A test is terminated at a given time. This is referred to as time censored data. Singly censored data means all censoring times are equal (i.e., all units were started at the same time). Multiply censored data means that censoring times are not necessarily the same (i.e., units may have different start times, this is common with data collected from the field rather than in a lab).

        Time censored data is also called type 1 censored data.

        This is the most common type of censoring.

      2. Alternatively, a test can be run until a pre-selected number of failures have occurred. Again, you can have singly or multiply censored data.

        Number of failures censored data is also called type 2 censored data.

      3. In some cases, the number of units, n, may not be known in advance (and may in fact be the quantity that we are trying to estimate). In this case, we observe the number of failures in a given time (i.e., we know r but not n). We typically need to estimate the parameters of the distribution and the value of n.

        At this time, Dataplot does not support maximum likelihood estimation for truncated data. However, we anticipate adding this support for a few select distributions in a future release.

      Censored data is supported for the following five distributions:

      • normal - multiply time censored data, estimates for selected percentiles supported

      • 2-parameter lognormal - singly time censored data, estimates for selected percentiles supported

      • exponential - both multiply time censored data and singly number of failures censored data, estimates for selected percentiles supported

      • 2-parameter Weibull - multiply time censored data, estimates for selected percentiles supported

      • 2-parameter gamma - multiply time censored data, estimates for selected percentiles supported
      • 2-parameter inverted Weibull - multiply time censored data, point estimates only

      For the exponential distribution, you can enter the following command to specify what type of censoring was used:

        SET CENSORING TYPE <1/2>

      The other distributions assume type 1 censoring.

    5. Data is sometimes only available in binned format. Maximum likelihood estimates for grouped data are supported for the following distributions:

        exponential
        normal mixture

    A number of these commands generate method of moment estimates or other quantitative parameter estimates in addition to the maximum likelihood estimates. The Johnson SB and Johnson SU case only generates method of moment or percentile estimates.

Syntax 1:
    <DIST> MAXIMUM LIKELIHOOD <y>
                            <SUBSET/EXCEPT/FOR qualification>
    where <DIST> is one of:
        NORMAL, LOGNORMAL, WEIBULL, EXPONENTIAL, DOUBLE EXPONENTIAL, EV1, GUMBEL, PARETO, GAMMA, INVERSE GAUSSIAN, POWER, LOGISTIC, UNIFORM, BETA, FATIGUE LIFE, GEOMETRIC EXTREME EXPONENTIAL, FOLDED NORMAL, CAUCHY, GENERALIZED PARETO, ASYMMETRIC DOUBLE EXPONENTIAL, RAYLEIGH, MAXWELL, NORMAL MIXTURE, JOHNSON SB, JOHNSON SU, JOHNSON, INVERTED WEIBULL, FRECHET, BINOMIAL, POISSON, LOGARITHMIC SERIES, GEOMETRIC, BETA BINOMIAL, NEGATIVE BINOMIAL, HYPERGEOMETRIC, HERMITE, or YULE;
                <y> is the response variable;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This generates maximum likelihood estimates for the full sample case.

Syntax 2:
    <DIST> MOMENTS <y>             <SUBSET/EXCEPT/FOR qualification>
    where <DIST> is JOHNSON SB, JOHNSON SU, or UNIFORM;
                <y> is the response variable;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Syntax 3:
    <DIST> PERCENTLE <y>             <SUBSET/EXCEPT/FOR qualification>
    where <DIST> is JOHNSON SB, JOHNSON SU, or JOHNSON;
                <y> is the response variable; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Syntax 4:
    <DIST> MAXIMUM LIKELIHOOD <y> <x>
                            <SUBSET/EXCEPT/FOR qualification>
    where <DIST> is one of:
        NORMAL, LOGNORMAL, EXPONENTIAL, WEIBULL, GAMMA, or INVERTED WEIBULL;
                   
                <y> is the response variable;
                <x> is the censoring variable;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    The censoring variable should contain 1's and 0's where 1 indicates a failure time and 0 indicates a censoring time.

Syntax 5:
    <DIST> GROUPED MAXIMUM LIKELIHOOD <y> <x>
                            <SUBSET/EXCEPT/FOR qualification>
    where <DIST> is EXPONENTIAL;
                <y> is the frequency variable;
                <x> is the bin mid-points variable;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax is used for grouped data. In this case, the keyword GROUPED is required to distinguish this from the censored data case.

Syntax 6:
    <DIST> MAXIMUM LIKELIHOOD <y> <x>
                            <SUBSET/EXCEPT/FOR qualification>
    where <DIST> is NORMAL MIXTURE;
                <y> is the frequency variable;
                <x> is the bin mid-points variable;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    In this case, the keyword GROUPED is omitted since censored data is not supported for these distributions.

Examples:
    NORMAL MAXIMUM LIKELIHOOD Y
    EXPONENTIAL MAXIMUM LIKELIHOOD Y
    WEIBULL MAXIMUM LIKELIHOOD Y X
    WEIBULL MAXIMUM LIKELIHOOD Y SUBSET X > 5
    JOHNSON SB MOMENTS Y
Note:
    The estimated parameters will typically be saved as internal parameters that can be used in subsequent analysis. The feedback message will specify what parameters have been saved (or you can enter STATUS PARAMETERS to see what parameters were saved).
Note:
    By default, the Gumbel case performs the maximum likelihood estimation for the maximum order statistic case. To obtain the maximum likelihood estimates for the minimum order case, enter the following command:

      SET MINMAX 1
Note:
    The beta maximum likelihood estimation uses the data minimum and maximum as the estimates for the lower and upper limits of the distribution. It then uses the maximum likelihood estimates for the shape parameters alpha and beta for the case where the lower and upper limits are assumed known.

    You can enter your own values for these lower and upper limits with the commands

      LET BETALL = <value>
      LET BETAUL = <value>
Note:
    The Johnson SB and Johnson SU moment estimators are computed using Applied Statistics algorithm 99.

    These distributions can also be estimated using the percentile method described by Slifker and Shapiro (see the Reference section below). This method is based on matching percentiles of the data with theoretical percentiles. This method first determines whether the Johnson SB or Johnson SU distribution is most appropriate. This method requires a tuning parameter that can be set with the following command:

      LET Z = <value>

    The default value is 0.54. As the sample size gets larger, the value of Z can be set closer to 1. Basically, increasing the value of Z will use more extreme percentiles in performing the estimation.

    Skifler and Shapiro do not give specific recommendations. However, using the default value for small to moderate size data sets (say a few hundred points or less) and a value of 0.8 for data sets larger than this should generate reasonable results. Alternatively, you can generate the estimates using several different values of Z between 0.5 and 1. You can perform a Kolmogorov-Smirnov goodness of fit test with the different estimates to see what value of Z results in the best fit.

Note:
    For the negative binomial distribution, a maximum likelihood estimate for P is returned assuming K is known. To specify the value of K, enter the command

      LET K = <value>

    For the hypergeometric distribution, there are four quantities of interest:

    1. N = total number of items in population
    2. n = number of items sampled
    3. K = number of defective items (or successes) in population
    4. x = number of defectives in sample

    There are two distinct cases to consider.

    1. Given that N (the population size) is known, we want to estimate the number of defectives in the population given a sample of size n with x defectives. An example is acceptance sampling where the lot size is known and a subsample is choosen for inspection. In this case, the maximum likelihood estimate of K is:

        K = MAX INTEGER ≤ x*(N+1)/n

    2. In capture/recapture problems, a sample is taken and marked. That is, K is known. Then a second sample (of size n) is taken and the number of marked items (x) are counted. In this case, the maximum likelihood estimates are:

        N = MAX INTEGER ≤ n*K/x

      We implement the refinement of Chapman (see page 263 of Johnson, Kotz, and Kemp):

        N* = (n+1)*(K+1)/(x+1) - 1

    Formulas for the variance are also given in Johnson, Kotz, and Kemp.

Note:
    For the details of maximum likelihood estimates for the Yule and Hermite distributions, enter the commands

      HELP YULPDF
      HELP HERPDF
Default:
    None
Synonyms:
    MLE is a synonymn for MAXIMUM LIKELIHOOD
Related Commands:
    FIT = Perform a least squares fit.
    PPCC PLOT = Generate a ppcc plot.
    KS PLOT = Generate a Kolmogorov-Smirnov plot.
    PROBABILITY PLOT = Generate a probability plot.
    KOLMOGOROV SMIRNOV GOODNESS OF FIT TEST = Perform a Kolmogorov Smirnov goodness of fit test.
    WILK SHAPIRO TEST = Perform a Wilks-Shapiro test for normality.
Reference:
    "Continuous Univariate Distributions: Volume I", 2nd. ed., Johnson, Kotz, and Balakrishnan, John Wiley and Sons, 1994.

    "Continuous Univariate Distributions: Volume II", 2nd. ed., Johnson, Kotz, and Balakrishnan, John Wiley and Sons, 1994.

    "Univariate Discrete Distributions", 2nd. ed., Johnson, Kotz, and Kemp, John Wiley and Sons, 1994.

    "Statistical Distributions in Engineering", Karl Bury, Cambridge University Press, 1999.

    "Statistical Distributions", Third Edition, Evans, Hastings, and Peacock, 2000.

    "Algorithm AS 99", Applied Statistics, 1976, Vol. 25, P. 180.

    "Confidence Intervals for the Parameters of the Logistic Distribution", Charles Antle, Lawrence Klimko, and William Harkness, Biometriks, (1970), pp. 397-402.

    "The Johnson System: Selection and Parameter Estimation", James F. Slifker and Samuel S. Shapiro, Technometrics, Vol. 22, No. 2, May 1980, pp. 239-246.

    "Inferences for the Cauchy Distribution Based on Maximum Likelihood Estimators", Biometrika, 1970, pp. 403-407.

Applications:
    Reliability, Data Analysis, Distributional Modeling
Implementation Date:
    1998/5
    2003/10: Gumbel case supports both minimum and maximum cases
    2003/11: Added support for logistic, uniform, and beta distributions
    2004/5: Added confidence limits (Agresti and Coull approach) for binomial case
    2004/5: Added confidence limits for lognormal case
    2004/5: Added support for the following continuous distributions
      FATIGUE LIFE
      GEOMETRIC EXTREME EXPONENTIAL
      FOLDED NORMAL
      CAUCHY
    2004/5: Added support for the following discrete distributions
      LOGARITHMIC SERIES
      GEOMETRIC
      BETA BINOMIAL
      NEGATIVE BINOMIAL
      HYPERGEOMETRIC
      HERMITE
      YULE
    2004/5: Added the JOHNSON PERCENTILE case 2004/6: Added the ASYMETRIC DOUBLE EXPONENTIAL, RAYLEIGH, and MAXWELL cases 2004/12: Rewrote the maximum likeihood output for the normal, lognormal, exponential, Weibull, gamma, Gumbel, Beta, and Pareto distributions. Added support for confidence intervals for selected percentiles for 7 distributions and support for censored data for 5 distributions.
Program:
     
        skip 25
        read vangel31.dat y
        exponential mle y
        weibull mle y
        lognormal mle y
        gamma mle y
        

    The following output is generated.

               *************************
               **  exponential mle y  **
               *************************
          
          
               EXPONENTIAL MAXIMUM LIKELIHOOD ESTIMATION: FULL SAMPLE CASE
          
         ONE-PARAMETER MODEL (LOCATION = 0)
         NUMBER OF OBSERVATIONS                =       38
         MINIMUM VALUE                         =    147.0000
          
         ML ESTIMATE OF SCALE PARAMETER        =    185.7895
         STANDARD ERROR OF SCALE PARAMETER     =    30.13903
          
         CONFIDENCE INTERVAL FOR SCALE PARAMETER
            CONFIDENCE           LOWER         UPPER
            VALUE (%)            LIMIT         LIMIT
         -------------------------------------------
              50.000           168.269       209.615
              75.000           156.300       227.404
              90.000           145.042       248.068
              95.000           138.432       262.541
              99.000           126.642       294.188
              99.900           114.602       337.472
          
         THE MINIMUM VALUE WILL BE SAVED AS THE INTERNAL PARAMETER U1
         THE SCALE PARAMETER WILL BE SAVED AS THE INTERNAL PARAMETER B1
          
          
         TWO-PARAMETER MODEL (LOCATION UNKNOWN)
         NUMBER OF OBSERVATIONS =       38
          
         ESTIMATE OF LOCATION PARAMETER                      =    147.0000
         STANDARD ERROR OF LOCATION PARAMETER                =    1.034478
         BIAS CORRECTED ESTIMATE OF LOCATION PARAMETER       =    145.9516
         STANDARD ERROR OF BIAS CORRECTED LOCATION PARAMETER =    1.062437
         ESTIMATE OF SCALE PARAMETER                         =    38.78947
         STANDARD ERROR OF SCALE PARAMETER                   =    6.376950
         BIAS CORRECTED ESTIMATE OF SCALE PARAMETER          =    39.83784
         STANDARD ERROR OF BIAS CORRECTED SCALE PARAMETER    =    6.549300
          
         CONFIDENCE INTERVAL FOR LOCATION PARAMETER
            CONFIDENCE           LOWER         UPPER
            VALUE (%)            LIMIT         LIMIT
         -------------------------------------------
              50.000           145.519       146.697
              75.000           144.758       146.860
              90.000           143.729       146.946
              95.000           142.933       146.973
              99.000           141.028       146.995
              99.900           138.154       146.999
          
         CONFIDENCE INTERVAL FOR SCALE PARAMETER
            CONFIDENCE           LOWER         UPPER
            VALUE (%)            LIMIT         LIMIT
         -------------------------------------------
              50.000           36.0380       45.0267
              75.000           33.4428       48.9045
              90.000           31.0050       53.4162
              95.000           29.5751       56.5804
              99.000           27.0274       63.5113
              99.900           24.4298       73.0145
          
          
          
         THE LOCATION PARAMETER WILL BE SAVED AS THE INTERNAL PARAMETER U2
         THE SCALE PARAMETER WILL BE SAVED AS THE INTERNAL PARAMETER B2
          
          
               *********************
               **  weibull mle y  **
               *********************
          
          
               WEIBULL MAXIMUM LIKELIHOOD ESTIMATION: FULL SAMPLE CASE
          
         TWO-PARAMETER MODEL (LOCATION = 0)
         NUMBER OF OBSERVATIONS                            =       38
         MINIMUM VALUE                                     =    147.0000
         SAMPLE MEAN VALUE                                 =    185.7895
         SAMPLE STANDARD DEVIATION VALUE                   =    18.59549
          
         ESTIMATE OF SCALE PARAMETER                       =    194.2046
         STANDARD ERROR OF SCALE PARAMETER                 =    3.137330
          
         ESTIMATE OF SHAPE PARAMETER                       =    10.57322
         STANDARD ERROR OF SHAPE PARAMETER                 =    1.337343
         BIAS CORRECTED ESTIMATE OF SHAPE PARAMETER        =    10.20502
         STANDARD ERROR OF BIAS CORRECTED SHAPE PARAMETER  =    1.290772
          
         STANDARD ERROR OF SHAPE/SCALE COVARIANCE          =    1.146094
         STD ERR OF BIAS CORRECTED SHAPE/SCALE COVARIANCE  =    1.125962
          
         CONFIDENCE INTERVAL FOR SCALE PARAMETER
          
                                NORMAL APPROXIMATION            LIKELIHOOD RATIO
            CONFIDENCE           LOWER         UPPER         LOWER         UPPER
            VALUE (%)            LIMIT         LIMIT         LIMIT         LIMIT
         -----------------------------------------------------------------------
              50.000           192.089       196.321       192.063       196.335
              75.000           190.596       197.814       190.529       197.852
              90.000           189.044       199.365       188.901       199.457
              95.000           188.056       200.354       187.840       200.504
              99.000           186.123       202.286       185.704       202.633
              99.900           183.881       204.528       183.090       205.301
          
         CONFIDENCE INTERVAL FOR SHAPE PARAMETER
         (BASED ON NO BIAS CORRECTION ESTIMATES)
          
                                NORMAL APPROXIMATION            LIKELIHOOD RATIO
            CONFIDENCE           LOWER         UPPER         LOWER         UPPER
            VALUE (%)            LIMIT         LIMIT         LIMIT         LIMIT
         -----------------------------------------------------------------------
              50.000           9.67119       11.4752       9.73924       11.4358
              75.000           9.03480       12.1116       9.16853       12.0612
              90.000           8.37348       12.7729       8.59130       12.7256
              95.000           7.95207       13.1944       8.23208       13.1567
              99.000           7.12845       14.0180       7.54982       14.0162
              99.900           6.17267       14.9738       6.79193       15.0417
          
         THE FOLLOWING INTERNAL PARAMETERS ARE SAVED:
               ALPHAML, ALPHASE, GAMMAML, GAMMASE, CAMMABC, GAMMABCSE,COVSE,COVBCSE
          
          
          
               ***********************
               **  lognormal mle y  **
               ***********************
          
          
               LOGNORMAL MAXIMUM LIKELIHOOD ESTIMATION:
               FULL SAMPLE CASE
          
         TWO-PARAMETER MODEL (LOCATION = 0)
         NUMBER OF OBSERVATIONS                       =       38
         SAMPLE MINIMUM                               =    147.0000
         SAMPLE MEAN                                  =    185.7895
         SAMPLE MEDIAN                                =    185.5000
         SAMPLE STANDARD DEVIATION                    =    18.59549
          
         ML ESTIMATE OF SHAPE PARAMETER (SIGMA)       =   0.1002546
         STANDARD ERROR OF SHAPE PARAMETER            =   0.1165436E-01
         ML ESTIMATE OF SCALE PARAMETER               =    184.8847
         ML ESTIMATE OF MU (= LOG(SCALE))             =    5.219732
         STANDARD ERROR OF SCALE/MU                   =   0.1626344E-01
          
         CONFIDENCE INTERVAL FOR SCALE PARAMETER
          
                                   SCALE PARAMETER               MU PARAMETER
            CONFIDENCE           LOWER         UPPER         LOWER         UPPER
            VALUE (%)            LIMIT         LIMIT         LIMIT         LIMIT
         -----------------------------------------------------------------------
              50.000           184.874       184.896       5.20865       5.23081
              75.000           184.866       184.904       5.20073       5.23874
              90.000           184.857       184.912       5.19229       5.24717
              95.000           184.852       184.918       5.18678       5.25269
              99.000           184.841       184.929       5.17557       5.26389
              99.900           184.827       184.943       5.16161       5.27785
          
         CONFIDENCE INTERVAL FOR SHAPE PARAMETER
          
            CONFIDENCE           LOWER         UPPER
            VALUE (%)            LIMIT         LIMIT
         -------------------------------------------
              50.000          0.936715E-01  0.109717
              75.000          0.889265E-01  0.116492
              90.000          0.844115E-01  0.124286
              95.000          0.817339E-01  0.129704
              99.000          0.769019E-01  0.141454
              99.900          0.718825E-01  0.157350
          
         THE FOLLOWING INTERNAL PARAMETERS ARE SAVED:
               SIGMAML, SIGMASE, SCALEML, UHATML, UHATSE
          
          
          
               *******************
               **  gamma mle y  **
               *******************
          
          
               GAMMA MAXIMUM LIKELIHOOD ESTIMATION: FULL SAMPLE CASE
          
         TWO-PARAMETER MODEL (LOCATION = 0)
         NUMBER OF OBSERVATIONS                            =       38
         MINIMUM VALUE                                     =    147.0000
         SAMPLE MEAN VALUE                                 =    185.7895
         SAMPLE STANDARD DEVIATION VALUE                   =    18.59549
         SAMPLE GEOMETRIC MEAN VALUE                       =    184.8847
          
          
         MOMENT ESTIMATE OF SCALE PARAMETER                =    1.861205
         MOMENT ESTIMATE OF SHAPE PARAMETER                =    99.82214
          
         ML ESTIMATE OF SCALE PARAMETER                    =    1.811020
         STANDARD ERROR OF SCALE PARAMETER                 =   0.4158159
         ML ESTIMATE OF SHAPE PARAMETER                    =    102.5883
         STANDARD ERROR OF SHAPE PARAMETER                 =    23.49724
         COVARIANCE OF THE SHAPE AND SCALE PARAMETERS      =   -9.746725
          
         CONFIDENCE INTERVAL FOR SCALE PARAMETER
          
                                NORMAL APPROXIMATION            LIKELIHOOD RATIO
            CONFIDENCE           LOWER         UPPER         LOWER         UPPER
            VALUE (%)            LIMIT         LIMIT         LIMIT         LIMIT
         -----------------------------------------------------------------------
              50.000           1.53056       2.09148       1.55723       2.12304
              75.000           1.33269       2.28935       1.40621       2.38733
              90.000           1.12706       2.49498       1.26945       2.70996
              95.000          0.996035       2.62600       1.19156       2.94588
              99.000          0.739949       2.88209       1.05707       3.49022
              99.900          0.442772       3.17927      0.925650       4.29764
          
         CONFIDENCE INTERVAL FOR SHAPE PARAMETER
          
                                NORMAL APPROXIMATION            LIKELIHOOD RATIO
            CONFIDENCE           LOWER         UPPER         LOWER         UPPER
            VALUE (%)            LIMIT         LIMIT         LIMIT         LIMIT
         -----------------------------------------------------------------------
              50.000           86.7397       118.437       87.5479       119.267
              75.000           75.5583       129.618       77.8833       132.049
              90.000           63.9388       141.238       68.6408       146.247
              95.000           56.5346       148.642       63.1638       155.791
              99.000           42.0635       163.113       53.3517       175.581
              99.900           25.2704       179.906       43.3751       200.473
          
         THE FOLLOWING INTERNAL PARAMETERS ARE SAVED:
               GAMMAML, GAMMASE, SCALEML, SCALESE, GAMMAMOM, SCALEMOM,COVSE
      
        

Date created: 6/5/2001
Last updated: 12/05/2005
Please email comments on this WWW page to alan.heckert@nist.gov.