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Dataplot Vol 1 Vol 2

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 widely used because they have excellent statistical properties. The likelihood equations have to be derived for each specific distribution (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.

    For some distributions, maximum likelihood methods may have theoretical issues (e.g., the maximum likelihood solution may not exist) or numerical issues (e.g., non-convergence).

    As maximum likelihood methods are well documented in the statistical literature, we will not discuss them here.

    CONTINUOUS DISTRIBUTIONS

    For some distributions, Dataplot will also generate estimates based on other methods. Specifically, for continuous distributions the following methods may also be used:

    1. ML: maximum likelihood
    2. MOM: moments
    3. MODMOM: modified moments
    4. LMOM: L-moments
    5. PERC: Percentile methods
    6. ELEM PERC: Elemental Percentiles
    7. OS: Order Statistics
    8. WOS: Weighted Order Statistics

    In some cases, these methods are used to obtain starting values. In other cases, they are used where maximum likehood methods are known to have performance issues.

    We distinguish the following types of data.

    1. Ungrouped, uncensored data
    2. Grouped, uncensored data
    3. Ungrouped, censored data
    4. Grouped, censored data

    The Dataplot MAXIMUM LIKELIHOOD command primarily supports case 1 (i.e., ungrouped, uncensored data). Censored data is supported for some distributions commonly used in reliability/lifetime applications.

    There is a distinction between censored and truncated data. The distinction is that for censored data the number of censored points is known while for truncated data the number of censored points is unknown. Censoring is common in life testing where we test a fixed known number of units. In this case, the censored data are those units that have not failed when the test is ended. On the other hand, an example of truncated data might be a sensor where we have a limit of detection (that is, there is a minimum level of something that must be present before the instrument can detect its presence). That is, for truncated data the number of truncated units is unknown.

    The following types of information may be reported by the maximum likelihood command:

    1. Point estimates for the parameters

    2. Standard errors for the parameter estimates and the associated confidence intervals for the estimated parameters

    3. Values for the log-likelihood and AIC/BIC/AICC information crition statistics

    4. Standard errors and confidence intervals for select percentiles

    At a minimum, point estimates will be reported. Items 2 - 4 depend on the specific distribution. For distributions where only point estimates are generated, the DISTRIBUTIONAL BOOTSTRAP command may be used to generate confidence intervals for the parameter estimates and for selected percentiles.

    The following distributions are currently supported.

      Ungrouped, Uncensored Data: Location/Scale Distributions
      Name Methods Point
      Estimates
      Parameter
      CI
      AIC/BIC
      Statistic
      Percentile
      CI
      Normal ML Yes Yes Yes Yes
      Logistic ML Yes 90%/95% Yes No
      Double Exponential ML Yes Yes Yes No
      Cauchy OS/WOS/ML Yes 90%/95% Yes No
      Slash ML Yes No Yes No
      Uniform MOM/ML Yes No Yes No
      1-par Exponential ML Yes Yes Yes Yes
      2-par Exponential ML Yes Yes Yes Lower
      1-par Rayleigh ML Yes Yes Yes Yes
      2-par Rayleigh Percentile/
      LMOM/MOM/
      MOD MOM/
      ML
      Yes Yes Yes No
      1-par Maxwell ML Yes No Yes No
      2-par Maxwell ML Yes No Yes No
      Gumbel MOM/ML Yes Yes Yes Yes



      Ungrouped, Uncensored Data: One Shape Parameter Distributions
      Name Methods Point
      Estimates
      Parameter
      CI
      AIC/BIC
      Statistic
      Percentile
      CI
      Topp and Leone ML Yes No Yes No
      Triangular ML Yes No No No
      Power ML Shape Shape Yes No
      Reflected Power ML Shape Shape Yes No
      Von Mises ML Yes No No No
      Generalized Logistic Type 5 LMOM Yes No No No
      Pearson Type 3 LMOM Yes No No No
      2-par Weibull ML Yes Yes Yes Yes
      3-par Weibull PERC/LMOM/
      MOM/MODMOM
      ML
      Yes Yes Yes Yes
      2-parameter Inverted Weibull
      (= maximum Frechet)
      ML Yes Yes Yes Yes
      2-par Gamma MOM/ML Yes Yes Yes Yes
      3-par Gamma MOM/MODMOM/ML Yes Yes Yes No
      2-par Inverted Gamma MOM/ML Yes Yes No Yes
      2-par Lognormal ML Yes Yes Yes Yes
      3-par Lognormal MOM/MODMOM/ML Yes Yes Yes Yes
      2-par Fatigue Life MOM/ML Yes No No No
      2-par Burr Type 10 ML Yes No Yes No
      2-par Logistic Expopnential ML Yes No No No
      2-par Frechet ML Yes Yes No Yes
      2-par Geometric Extreme Exponential ML Yes No No No
      2-par Inverse Gaussian ML Yes Yes Yes No
      3-par Inverse Gaussian MMOM/ML Yes Yes Yes No
      2-par Alpha MOM/ML Yes No No No
      2-par Exponential Power
      (needs work)
      ML Yes No No No
      Pareto MODMOM/MOD-ML Yes Yes Yes No
      Generalized Pareto MOM/LMOM/ELEM PERC/ML Yes Yes Yes Yes
      Asymmetric Lapalce ML Yes No No No



      Ungrouped, Uncensored Data: Two or More Shape Parameter Distributions
      Name Methods Point
      Estimates
      Parameter
      CI
      AIC/BIC
      Statistic
      Percentile
      CI
      2-Par Beta MOM/ML Yes Yes Yes Yes
      4-Par Beta MOM/ML Yes No Yes No
      Gompertz ML Yes No No No
      Reflected Generalized Topp and Leone ML Yes No Yes No
      Two-Sided Power ML Yes No No No
      Johnson SB MOM/PERC Yes No No No
      Johnson SU MOM/PERC Yes No No No
      Log Beta ML Yes No No No
      Kappa LMOM Yes No No No
      Beta Normal ML Yes No No No
      Wakeby LMOM Yes No No No
      Folded Normal ML Yes No Yes No



      Ungrouped, Censored Data: Location/Scale Distributions
      Name Methods Point
      Estimates
      Parameter
      CI
      AIC/BIC
      Statistic
      Percentile
      CI
      Normal ML Yes Yes Yes Yes
      2-par Exponential ML Yes Yes Yes Lower



      Ungrouped, Censored Data: One Shape Parameter Distributions
      Name Methods Point
      Estimates
      Parameter
      CI
      AIC/BIC
      Statistic
      Percentile
      CI
      2-par Weibull ML Yes Yes Yes Yes
      2-par Inverted Weibull (= Frechet) ML Yes Yes Yes Yes
      2-par Lognormal ML Yes Yes Yes Yes
      2-par Gamma MOM/ML Yes Yes Yes Yes
      2-par Inverted Gamma MOM/ML Yes Yes No Yes



      Grouped, Uncensored Data: Location/Scale Distributions
      Name Methods Point
      Estimates
      Parameter
      CI
      AIC/BIC
      Statistic
      Percentile
      CI
      2-par Exponential ML Yes Yes No No



      Grouped, Uncensored Data: One Shape Parameter Distributions
      Name Methods Point
      Estimates
      Parameter
      CI
      AIC/BIC
      Statistic
      Percentile
      CI
      Power ML Shape Shape Yes No
      Reflected Power ML Shape Shape Yes No



      Grouped, Uncensored Data: Two or More Shape Parameter Distributions
      Name Methods Point
      Estimates
      Parameter
      CI
      AIC/BIC
      Statistic
      Percentile
      CI
      Gompertz ML Yes No No No
      Johnson SB MOM/PERC Yes No No No
      Johnson SU MOM/PERC Yes No No No
      Reflected Generalized Topp and Leone ML Yes No Yes No



    DISCRETE DISTRIBUTIONS

    Data for discrete data can be either raw data or in the form of a frequency table. Dataplot currently requires that frequency tables have equal group sizes (data will sometimes be reported with frequency tables with unequal group sizes, typically due to groups in the upper tail being combined).

    The following methods may be used to compute point estimates

    1. ML: maximum likelihood
    2. MOM: moments
    3. EP: method of even points
    4. ZF: method of zero frequency and mean
    5. FF: method of first frequency and mean
    6. RF: method of ratio of frequencies
    7. WD: method of weighted discrepancies (a modification of the ML estimates)

      Discrete Distributions
      Name Methods Point
      Estimates
      Parameter
      CI
      Unequal
      Group Sizes
      Binomial ML Yes Yes No
      Borel Tanner ML=MOM Yes No No
      Consul (generalized geometric) FF/MOM/ML Yes No No
      *Generalized Lost Games MOM/ML Yes No No
      Geeta FF/MOM/ML Yes No No
      Geometric ML Yes Yes No
      *Hermite EP/MOM/ML Yes No No
      Katz MOM Yes No No
      Lagrange Poisson ZF/MOM/WD Yes No No
      Logarithmic Series ML=MOM Yes Yes No
      Lost Games ML Yes No No
      *Negative Binomial ML Yes Conditional No
      Poisson ML Yes Yes No
      Yule ML/MOM Yes No No
      *Zeta RF/ML/MOM Yes No No

Syntax 1:
    <DIST> MAXIMUM LIKELIHOOD <y> <SUBSET/EXCEPT/FOR qualification>
    where <DIST> is one of the supported distributions;
                <y> is the response variable;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This generates maximum likelihood estimates for the raw data (no grouping) case with no censoring.

Syntax 2:
    <DIST> MAXIMUM LIKELIHOOD <y> <x>
                            <SUBSET/EXCEPT/FOR qualification> where <DIST> is one of the supported distributions;
                <y> is a variable containing frequencies;
                <x> is a variable containing the bin mid-points;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax is used for grouped (frequency table) data. The bins are assumed to have equal width.

Syntax 3:
    <DIST> MAXIMUM LIKELIHOOD <y> <xlow> <xhigh>
                            <SUBSET/EXCEPT/FOR qualification>
    where <DIST> is one of the supported distributions;
                <y> is a variable containing frequencies;
                <xlow> is a variable containing the lower value for the bins;
                <xhigh> is a variable containing the upper value for the bins;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax is used for grouped (frequency table) data where the bins do not have equal width.

Syntax 4:
    <DIST> CENSORED MAXIMUM LIKELIHOOD <y> <x>
                            <SUBSET/EXCEPT/FOR qualification>
    where <DIST> is one of the supported distributions;
                <y> is the response variable;
                <x> is the censoring variable; and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This is for the raw data (ungrouped) case with censoring. 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> MULTIPLE MAXIMUM LIKELIHOOD <y1> ... <yk>
                            <SUBSET/EXCEPT/FOR qualification>
    where <DIST> is one of the supported distributions;
                <y1> ... <yk> is a list of 1 to 30 response variables;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax will perform maximum likelihood estimation for each of the listed response variables. Censoring is not supported for this syntax.

    The TO keyword can be used with this sytax (see Examples below).

Examples:
    NORMAL MAXIMUM LIKELIHOOD Y
    LOGNORMAL MAXIMUM LIKELIHOOD Y
    EXPONENTIAL MAXIMUM LIKELIHOOD Y
    WEIBULL CENSORED MAXIMUM LIKELIHOOD Y X
    WEIBULL MAXIMUM LIKELIHOOD Y SUBSET TAG = 1
    WEIBULL MULTIPLE MAXIMUM LIKELIHOOD Y1 Y2 Y3
    WEIBULL MULTIPLE MAXIMUM LIKELIHOOD Y1 TO Y5
Note:
    Typically after a maximum likelihood analysis, many parameters will automatically be saved. For example, the point estimates and standard errors (if computed) are saved. To see what parameters have been saved, enter STATUS PARAMETERS.
Note:
    The Weibull, Gumbel, Frechet, generalized extreme value, and generalized Pareto distributions support parameterizations based on either the minimum order statistic or the maximum order statistic.

    To specify the minimum order statistic case, enter

      SET MINMAX 1

    To specify the maximum order statistic case, enter

      SET MINMAX 2

    The default is the minimum order statistic case for the Weibull distribution and the maximum order statistic case for the other distributions.

Note:
    By default, confidence intervals for percentiles are not computed. If you want these computed, enter the command

      SET MAXIMUM LIKELIHOOD PERCENTILES DEFAULT

    This will compute 17 select percentiles. If you would like to specify the specific percentiles, do something like the following

      LET YPERC = DATA 0.01 0.05 0.10
      SET MAXIMUM LIKELIHOOD PERCENTILES YPERC

    To turn off the percentile confidence limits, enter

      SET MAXIMUM LIKELIHOOD PERCENTILES NONE

    By default, two-sided confidence intervals are generated for the percentiles. To specify lower one-sided intervals, enter

      SET DISTRIBUTIONAL PERCENTILE LOWER

    To specify upper one-sided intervals, enter

      SET DISTRIBUTIONAL PERCENTILE UPPER

    To reset two-sided intervals, enter

      SET DISTRIBUTIONAL PERCENTILE TWO SIDED

    Note that one-sided percentiles are also one-sided tolerance intervals.

    See the tables above to determine which distributions support percentile confidence intervals.

    Confidence intervals for percentiles are by default 95% confidence intervals. To change the confidence level, enter the command

      LET ALPHA = <value>

    where <value> is typically 0.10, 0.05, or 0.01.

Note:
    In some cases, you may be able to specify starting values or to fix certain parameters to known values.

    If available for a particular distribution, these will typically be documented in the PDF routine (e.g., NBPDF).

Note:
    Dataplot also supports several graphical methods for estimating the distribution parameters.

    The PPCC PLOT and PROBABILITY PLOT commands document fitting parameters by maximizing the probability plot correlation coefficient. The PPCC PLOT has variants where you can minimize the Anderson-Darling, Kolmogorov-Smirnov, and chi-square goodness of fit statistic.

    The NORMAL PLOT, WEIBULL PLOT and FRECHET PLOT can be used for the normal, 2-parameter Weibull (minimum case), and 2-parameter Frechet (maximum case).

Note:
    You can use the fit command to perform a least squares fit to the percentiles of the data. For example, if you have a response variable Y and you would like to fit it to a 2-parameter Weibull distribution, do something like the following

      let y = sort y
      let n = size y
      let p = uniform order statistic medians for i = 1 1 n
      let gamma = 3.5; . Specify a starting value for the shape
      let scale = 10; . Specify a starting value for the scale
      fit y = weippf(p,gamma,0,scale)

    Be aware that the standard indpendence assumptions for least squares fitting are not satisfied. However, this method can often give a reasonable fit. It can also sometimes be used to provide better starting values for the maximum likelihood fit.

Note:
    The bootstrap can be used to obtain confidence intervals for parameters and selected percentiles. Enter HELP DISTRIBUTIONAL BOOTSTRAP for details.
Note:
    After fitting the data, the GOODNESS OF FIT command can be used to perform various goodness of fit tests.
Note: Default:
    None
Synonyms:
    MLE is a synonymn for MAXIMUM LIKELIHOOD
Related Commands: Reference:
    Johnson, Kotz, and Balakrishnan (1994), "Continuous Univariate Distributions: Volume I", 2nd. ed., John Wiley and Sons.

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

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

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

    Cohen and Whitten (1988), "Parameter Estimation in Reliability and Life Span Models", Marcel Dekker, p. 31 and pp. 341-344.

    Evans, Hastings, and Peacock (2000), "Statistical Distributions", Third Edition, John Wiley and Sons.

Applications:
    Reliability, Data Analysis, Distributional Modeling
Implementation Date:
    1998/5
    New distributions have been continually added since the original implementation
Program 1:
     
    skip 25
    read vangel31.dat y
    .
    set write decimals 4
    exponential mle y
    weibull mle y
    lognormal mle y
    gamma mle y
        
    The following output is generated.
                 2-Parameter Exponential Parameter Estimation
                          (without Bias Correction)
      
     Summary Statistics:
     Number of Observations:                              38
     Sample Mean:                                   185.7895
     Sample Standard Deviation:                      18.5955
     Sample Minimum:                                147.0000
     Sample Maximum:                                231.0000
      
     Maximum Likelihood:
     Estimate of Location:                          147.0000
     Standard Error of Location:                      1.0344
     Estimate of Scale:                              38.7894
     Standard Error of Scale:                         6.3769
     Log-likelihood:                          -0.1770097E+03
     AIC:                                      0.3580193E+03
     AICc:                                     0.3583622E+03
     BIC:                                      0.3612945E+03
      
     Confidence Interval for Location Parameter
     ---------------------------------------------
          Confidence          Lower          Upper
         Coefficient          Limit          Limit
     ---------------------------------------------
               50.00       145.5191       146.6972
               75.00       144.7575       146.8598
               90.00       143.7287       146.9462
               95.00       142.9334       146.9733
               99.00       141.0280       146.9946
               99.90       138.1539       146.9995
     ---------------------------------------------
      
      
     Confidence Interval for Scale Parameter
     ---------------------------------------------
          Confidence          Lower          Upper
         Coefficient          Limit          Limit
     ---------------------------------------------
               50.00        36.0379        45.0266
               75.00        33.4427        48.9044
               90.00        31.0049        53.4162
               95.00        29.5750        56.5803
               99.00        27.0274        63.5112
               99.90        24.4297        73.0144
     ---------------------------------------------
      
      
                 Two-Parameter Weibull (Minimum) Parameter Estimation:
                                   Full Sample Case
      
     Summary Statistics:
     Number of Observations:                              38
     Sample Mean:                                   185.7895
     Sample Standard Deviation:                      18.5955
     Sample Minimum:                                147.0000
     Sample Maximum:                                231.0000
      
     Maximum Likelihood:
     Estimate of Scale:                             194.2045
     Estimate of Shape (Gamma):                      10.5731
     Standard Error of Scale:                         3.1373
     Standard Error of Shape:                         1.3372
     Shape/Scale Covariance:                          1.1460
     Log-likelihood:                          -0.1664930E+03
     AIC:                                      0.3369861E+03
     AICc:                                     0.3373289E+03
     BIC:                                      0.3402612E+03
      
     Maximum Likelihood (Bias Corrected):
     Estimate of Scale:                             194.2045
     Estimate of Shape (Gamma):                      10.2050
     Standard Error of Scale:                         3.1373
     Standard Error of Shape:                         1.2458
     Shape/Scale Covariance:                          1.1460
     Log-likelihood:                          -0.1665415E+03
     AIC:                                      0.3370830E+03
     AICc:                                     0.3374258E+03
     BIC:                                      0.3403581E+03
      
                 Confidence Interval for Scale Parameter
      
     ---------------------------------------------------------------------------
                          Normal  Approximation  Likelihood Ratio  Approximation
       Confidence          Lower          Upper             Lower          Upper
      Coefficient          Limit          Limit             Limit          Limit
     ---------------------------------------------------------------------------
            50.00       192.0885       196.3206          192.0630       196.3353
            75.00       190.5954       197.8136          190.5286       197.8516
            90.00       189.0440       199.3650          188.9007       199.4568
            95.00       188.0554       200.3536          187.8404       200.5043
            99.00       186.1233       202.2857          185.7038       202.6331
            99.90       183.8811       204.5280          183.0903       205.3014
     ---------------------------------------------------------------------------
      
      
                 Confidence Interval for Shape Parameter
      
     ---------------------------------------------------------------------------
                          Normal  Approximation  Likelihood Ratio  Approximation
       Confidence          Lower          Upper             Lower          Upper
      Coefficient          Limit          Limit             Limit          Limit
     ---------------------------------------------------------------------------
            50.00         9.6712        11.4751            9.7392        11.4358
            75.00         9.0348        12.1115            9.1684        12.0611
            90.00         8.3734        12.7728            8.5913        12.7256
            95.00         7.9520        13.1943            8.2321        13.1567
            99.00         7.1284        14.0180            7.5498        14.0161
            99.90         6.1726        14.9738            6.7919        15.0417
     ---------------------------------------------------------------------------
      
      
                 Two-Parameter Lognormal Parameter Estimation:
                               Full Sample Case
      
     Summary Statistics:
     Number of Observations:                              38
     Sample Mean:                                   185.7895
     Sample Standard Deviation:                      18.5955
     Sample Minimum:                                147.0000
     Sample Maximum:                                231.0000
     Sample Median:                                 185.5000
      
     Maximum Likelihood:
     Estimate of Shape (Sigma):                       0.1003
     Standard Error of Shape:                         0.0117
     Estimate of Scale:                             184.8847
     Standard Error of Scale:                         3.0314
     Estimate of MU (= LOG(Scale)):                   5.2196
     Standard Error of MU:                            0.0162
     Log-likelihood:                          -0.1643679E+03
     AIC:                                      0.3327358E+03
     AICc:                                     0.3330786E+03
     BIC:                                      0.3360109E+03
      
      
                 Confidence Interval for Scale Parameter
      
     ---------------------------------------------------------------------------
                           Scale      Parameter                MU      Parameter
       Confidence          Lower          Upper             Lower          Upper
      Coefficient          Limit          Limit             Limit          Limit
     ---------------------------------------------------------------------------
            50.00       182.8478       186.9442            5.2087         5.2308
            75.00       181.4037       188.4324            5.2007         5.2386
            90.00       179.8807       190.0278            5.1923         5.2472
            95.00       178.8915       191.0786            5.1867         5.2526
            99.00       176.8975       193.2324            5.1756         5.2638
            99.90       174.4455       195.9487            5.1615         5.2778
     ---------------------------------------------------------------------------
      
      
                 Confidence Interval for Shape Parameter
      
     ---------------------------------------------------------------------------
       Confidence          Limit          Upper
      Coefficient          Lower          Limit
     ---------------------------------------------------------------------------
            50.00         0.0937         0.1097
            75.00         0.0889         0.1165
            90.00         0.0844         0.1243
            95.00         0.0817         0.1297
            99.00         0.0769         0.1414
            99.90         0.0719         0.1574
     ---------------------------------------------------------------------------
      
      
                 Two-Parameter Gamma Parameter Estimation:
                             Full Sample Case
      
     Summary Statistics:
     Number of Observations:                       38
     Sample Mean:                                   185.7895
     Sample Standard Deviation:                      18.5955
     Sample Minimum:                                147.0000
     Sample Maximum:                                231.0000
     Sample Geometric Mean:                         184.8847
      
     Method of Moments:
     Estimate of Shape (Gamma):                      99.8221
     Estimate of Scale:                               1.8612
      
     Maximum Likelihood:
     Estimate of Shape (Gamma):                     102.5883
     Standard Error of Shape:                        23.4971
     Estimate of Scale:                               1.8109
     Standard Error of Scale:                         0.4158
     Shape/Scale Covariance:                         -9.7466
      
                 Confidence Interval for Scale Parameter
      
     ---------------------------------------------------------------------------
                          Normal  Approximation  Likelihood Ratio  Approximation
       Confidence          Lower          Upper             Lower          Upper
      Coefficient          Limit          Limit             Limit          Limit
     ---------------------------------------------------------------------------
            50.00         1.5306         2.0914            1.5571         2.1230
            75.00         1.3327         2.2894            1.4061         2.3873
            90.00         1.1271         2.4950            1.2695         2.7100
            95.00         0.9960         2.6259            1.1916         2.9459
            99.00         0.7399         2.8820            1.0570         3.4902
            99.90         0.4428         3.1793            0.9257         4.2976
     ---------------------------------------------------------------------------
      
      
                 Confidence Interval for Shape Parameter
      
     ---------------------------------------------------------------------------
                          Normal  Approximation  Likelihood Ratio  Approximation
       Confidence          Lower          Upper             Lower          Upper
      Coefficient          Limit          Limit             Limit          Limit
     ---------------------------------------------------------------------------
            50.00        86.7396       118.4368           87.5478       119.2668
            75.00        75.5583       129.6183           77.8833       132.0490
            90.00        63.9388       141.2377           68.6407       146.2472
            95.00        56.5345       148.6419           63.1638       155.7909
            99.00        42.0634       163.1132           53.3517       175.5810
            99.90        25.2699       179.9065           43.3751       200.4731
     ---------------------------------------------------------------------------
        
Program 2:
     
    . Purpose:  Example of fitting 2-parameter Weibull using
    .           maximum likelihood
    .
    . Step 1:   Read the data
    .
    skip 25
    read weibbury.dat y
    skip 0
    .
    . Step 2:   Maximum likelihood estimates
    .
    set write decimals 5
    set maximum likelihood percentiles default
    set distributional percentile two-sided
    feedback off
    capture screen on
    capture wei2.out
    weibull mle y
    let ksloc = 0
    let ksscale = alphaml
    let gamma = gammaml
    .
    . Step 3:   Goodness of fit via Anderson-Darling and by
    .           probability plot
    .
    set goodness of fit fully specified on
    set anderson darling critical value simulation
    weibull anderson darling goodness of fit y
    end of capture
    .
    let pploc   = ksloc
    let ppscale = ksscale
    .
    title case asis
    label case asis
    title Weibull Probability Plot
    y1label Sorted Data
    x1label Percentiles for Fitted Weibull Distribution
    character circle
    character hw 1 0.75
    character fill on
    line blank
    .
    weibull probability plot y
    .
    let gamma = round(gamma,2)
    move 20 85
    text Gamma = ^gamma
    let ksscale = round(ksscale,2)
    move 20 82
    text Scale = ^ksscale
    let ppcc = round(ppcc,3)
    move 20 79
    text PPCC = ^ppcc
        
    The following output is generated.
                 Two-Parameter Weibull (Minimum) Parameter Estimation:
                                   Full Sample Case
      
     Summary Statistics:
     Number of Observations:                              20
     Sample Mean:                                   52.64000
     Sample Standard Deviation:                      7.48517
     Sample Minimum:                                40.70000
     Sample Maximum:                                66.09999
      
     Maximum Likelihood:
     Estimate of Scale:                             55.83881
     Estimate of Shape (Gamma):                      8.11761
     Standard Error of Scale:                        1.61954
     Standard Error of Shape:                        1.41527
     Shape/Scale Covariance:                         0.84710
     Log-likelihood:                          -0.6835433E+02
     AIC:                                      0.1407087E+03
     AICc:                                     0.1414145E+03
     BIC:                                      0.1427001E+03
      
     Maximum Likelihood (Bias Corrected):
     Estimate of Scale:                             55.83881
     Estimate of Shape (Gamma):                      7.55464
     Standard Error of Scale:                        1.61954
     Standard Error of Shape:                        1.22578
     Shape/Scale Covariance:                         0.84710
     Log-likelihood:                          -0.6844503E+02
     AIC:                                      0.1408901E+03
     AICc:                                     0.1415959E+03
     BIC:                                      0.1428815E+03
      
                 Confidence Interval for Scale Parameter
      
     ---------------------------------------------------------------------------
                          Normal  Approximation  Likelihood Ratio  Approximation
       Confidence          Lower          Upper             Lower          Upper
      Coefficient          Limit          Limit             Limit          Limit
     ---------------------------------------------------------------------------
            50.00       54.74645       56.93119          54.73232       56.93923
            75.00       53.97578       57.70186          53.92918       57.73490
            90.00       53.17490       58.50274          53.06172       58.60202
            95.00       52.66458       59.01306          52.48611       59.18824
            99.00       51.66716       60.01048          51.29590       60.44785
            99.90       50.50966       61.16798          49.77662       62.19297
     ---------------------------------------------------------------------------
      
      
                 Confidence Interval for Shape Parameter
      
     ---------------------------------------------------------------------------
                          Normal  Approximation  Likelihood Ratio  Approximation
       Confidence          Lower          Upper             Lower          Upper
      Coefficient          Limit          Limit             Limit          Limit
     ---------------------------------------------------------------------------
            50.00        7.16302        9.07219           7.19360        9.10074
            75.00        6.48955        9.74566           6.57700        9.83032
            90.00        5.78969       10.44552           5.96703       10.62030
            95.00        5.34372       10.89148           5.59468       11.14076
            99.00        4.47210       11.76310           4.90364       12.19666
            99.90        3.46061       12.77459           4.16282       13.48682
     ---------------------------------------------------------------------------
      
      
                 Confidence Intervals for Select Percentiles (alpha = 0.050)
      
     (Based on Normal Approximation)
     ---------------------------------------------------------------------------
                              Point       Standard          Lower          Upper
          Percentile       Estimate          Error          Limit          Limit
     ---------------------------------------------------------------------------
               0.500       29.08095        3.66048       21.90653       36.25536
               1.000       31.68303        3.52762       24.76901       38.59707
               5.000       38.72843        3.01713       32.81494       44.64191
              10.000       42.31954        2.69485       37.03772       47.60134
              20.000       46.41828        2.30418       41.90216       50.93437
              30.000       49.17916        2.04763       45.16586       53.19248
              40.000       51.40420        1.86149       47.75574       55.05266
              50.000       53.37375        1.72632       49.99020       56.75730
              60.000       55.24069        1.63762       52.03099       58.45040
              70.000       57.13040        1.60039       53.99371       60.26711
              80.000       59.21016        1.63240       56.01073       62.40959
              90.000       61.88098        1.79006       58.37254       65.38943
              95.000       63.91991        1.98932       60.02091       67.81892
              97.500       65.58001        2.19282       61.28213       69.87788
              99.000       67.39704        2.44995       62.59519       72.19889
              99.500       68.57125        2.63203       63.41255       73.72996
     ---------------------------------------------------------------------------
      
      
                 Anderson-Darling Goodness of Fit Test
                        (Fully Specified Model)
      
     Response Variable: Y
      
     H0: The distribution fits the data
     Ha: The distribution does not fit the data
      
     Distribution: WEIBULL
     Location Parameter:                                0.00000
     Scale Parameter:                                  55.83881
     Shape Parameter 1:                                 8.11761
      
     Summary Statistics:
     Number of Observations:                                 20
     Sample Minimum:                                   40.70000
     Sample Maximum:                                   66.09999
     Sample Mean:                                      52.64000
     Sample SD:                                         7.48517
      
     Anderson-Darling Test Statistic Value:             0.29226
     Number of Monte Carlo Simulations:             10000.00000
     CDF Value:                                         0.05700
     P-Value                                            0.94300
      
      
      
     Percent Points of the Reference Distribution
     -----------------------------------
       Percent Point               Value
     -----------------------------------
                 0.0    =          0.000
                50.0    =          0.774
                75.0    =          1.237
                90.0    =          1.927
                95.0    =          2.513
                97.5    =          3.033
                99.0    =          3.758
                99.5    =          4.357
      
     Conclusions (Upper 1-Tailed Test)
     ----------------------------------------------
       Alpha    CDF   Critical Value     Conclusion
     ----------------------------------------------
         10%    90%            1.927      Accept H0
          5%    95%            2.513      Accept H0
        2.5%  97.5%            3.033      Accept H0
          1%    99%            3.758      Accept H0
        
    plot generated by sample program

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Date created: 12/17/2014
Last updated: 12/17/2014

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