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Dataplot Vol 2 Vol 1

LEXPDF

Name:
    LEXPDF (LET)
Type:
    Library Function
Purpose:
    Compute the logistic-exponential probability density function with shape parameter beta.
Description:
    The standard logistic-exponential distribution has the following probability density function:

      f(x;beta) = beta*EXP(x)*(EXP(x)-1)**(beta-1)/(1+(EXP(x)-1)**beta)**2} beta, x > 0

    with beta denoting the shape parameter.

    This distribution can be generalized with location and scale parameters in the usual way using the relation

      f(x;beta,mu,alpha) = (1/alpha)*f((x-mu)/alpha;beta,0,1)

    with mu and alpha denoting the location and scale parameters, respectively.

Syntax:
    LET <y> = LEXPDF(<x>,<beta>,<loc>,<scale>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a number, parameter, or variable;
                <y> is a variable or a parameter (depending on what <x> is) where the computed logistic-exponential pdf value is stored;
                <beta> is a number, parameter, or variable that specifies the shape parameter;
                <loc> is a number, parameter, or variable that specifies the location parameter;
                <scale> is a positive number, parameter, or variable that specifies the scale parameter;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    If <loc> and <scale> are omitted, they default to 0 and 1, respectively.

Examples:
    LET A = LEXPDF(0.3,0.2)
    LET Y = LEXPDF(X,0.5,0,5)
    PLOT LEXPDF(X,0.7,0,3) FOR X = 0 0.01 5
Note:
    Logistic-exponential random numbers, probability plots, and goodness of fit tests can be generated with the commands:

      LET BETA = <value>
      LET Y = LOGISTIC EXPONENTIAL RANDOM NUMBERS FOR I = 1 1 N
      LOGISTIC EXPONENTIAL PROBABILITY PLOT Y
      LOGISTIC EXPONENTIAL PROBABILITY PLOT Y2 X2
      LOGISTIC EXPONENTIAL PROBABILITY PLOT Y3 XLOW XHIGH
      LOGISTIC EXPONENTIAL KOLMOGOROV SMIRNOV ...
                  GOODNESS OF FIT Y
      LOGISTIC EXPONENTIAL CHI-SQUARE GOODNESS OF FIT Y2 X2
      LOGISTIC EXPONENTIAL CHI-SQUARE GOODNESS OF FIT ...
                  Y3 XLOW XHIGH

    The following commands can be used to estimate the beta shape parameter for the logistic-exponential distribution:

      LET BETA1 = <value>
      LET BETA2 = <value>
      LOGISTIC EXPONENTIAL PPCC PLOT Y
      LOGISTIC EXPONENTIAL PPCC PLOT Y2 X2
      LOGISTIC EXPONENTIAL PPCC PLOT Y3 XLOW XHIGH
      LOGISTIC EXPONENTIAL KS PLOT Y
      LOGISTIC EXPONENTIAL KS PLOT Y2 X2
      LOGISTIC EXPONENTIAL KS PLOT Y3 XLOW XHIGH

    The default values for BETA1 and BETA2 are 0.1 and 10.

    The probability plot can then be used to estimate the location and scale (location = PPA0, scale = PPA1).

    The 2-parameter logistic-exponential maximum likelihood estimates can be obtained using the command

      LOGISTIC EXPONENTIAL MAXIMUM LIKELIHOOD Y

    The maximum likelihood estimates for the full sample case are obtained as the solution of the following simultaneous equations (from Lan and Leemis):

      (n/beta) + SUM[i=1 to n][LOG(EXP(alpha*X(i)) - 1) - 2*SUM[i=1 to n][(EXP(alpha*X(I) - 1)**beta* LOG(EXP(alpha*X(i)) -1)/{1 + (EXP(alpha*X(i)) - 1)**beta} = 0

      (n/alpha) + SUM[i=1 to n][(beta-1)*X(i)*EXP(alpha*X(i))/ (EXP(ALHA*X(i)) - 1) + SUM[i=1 to n][X(i)] - 2*SUM[i=1 to n][beta*(EXP(alpha*X(I) - 1)**(beta - 1)* X(I)*EXP(alpha*X(i))/{1 + (EXP(alpha*X(i)) - 1)**beta} = 0

    with beta and alpha denoting the shape and scale parameters, respectively.

    If the maximum likelihood estimates do not converge to reasonable values, you can try specifying starting values with the commands

      LET RSV = <value>
      LET SCALESV = <value>

    For example, the estimates obtained from the ppcc plot or ks plot method can be used as starting values for the maximum likelihood.

    The BOOTSTRAP DISTRIBUTION command can be used to find uncertainty intervals for the ppcc plot, the ks plot, and the maximum likelihood estimates.

Default:
    None
Synonyms:
    None
Related Commands:
    LEXCDF = Compute the logistic-exponential cumulative distribution function.
    LEXCHAZ = Compute the logistic-exponential cumulative hazard function.
    LEXHAZ = Compute the logistic-exponential hazard function.
    LEXPPF = Compute the logistic-exponential percent point function.
    RAYPDF = Compute the Rayleigh probability density function.
    WEIPDF = Compute the Weibull probability density function.
    LGNPDF Compute the lognormal probability density function.
    EXPPDF = Compute the exponential probability density function.
    LOGPDF = Compute the logistic probability density function.
    GAMPDF = Compute the gamma probability density function.
    EWEPDF = Compute the exponentiated Weibull probability density function.
    B10PDF = Compute the Burr type 10 probability density function.
Reference:
    Leemis and McQuestion (2008), "Univariate Distribution Relationships", The American Statistician, Vol. 62, No. 1, pp. 45-53.

    Lan and Leemis (2008), "The Logistic-Exponential Survival Distribution", Naval Research Logistics, to appear.

Applications:
    Distributional Modeling
Implementation Date:
    2008/2
Program 1: 
     
LABEL CASE ASIS
TITLE CASE ASIS
TITLE OFFSET 2
.
MULTIPLOT 2 2
MULTIPLOT CORNER COORDINATES 0 0 100 95
MULTIPLOT SCALE FACTOR 2
.
LET BETA  = 0.5
TITLE BETA = ^BETA
PLOT LEXPDF(X,BETA) FOR X = 0.01  0.01  5
.
LET BETA  = 1
TITLE BETA = ^BETA
PLOT LEXPDF(X,BETA) FOR X = 0.01  0.01  5
.
LET BETA  = 2
TITLE BETA = ^BETA
PLOT LEXPDF(X,BETA) FOR X = 0.01  0.01  5
.
LET BETA  = 5
TITLE BETA = ^BETA
PLOT LEXPDF(X,BETA) FOR X = 0.01  0.01  5
.
END OF MULTIPLOT
.
JUSTIFICATION CENTER
MOVE 50 97
TEXT Logistic-Exponential Probability Density Functions
    plot generated by sample program
Program 2: 
     
let beta = 2.1
let betasav = beta
.
let y = logistic exponential random numbers for i = 1 1 200
let y = 10*y
let amax = maximum y
.
label case asis
title case asis
.
y1label Correlation Coefficient
x1label Beta
logistic exponential ppcc plot y
let beta = shape
justification center
move 50 6
text Betahat = ^beta (BETA = ^betasav)
move 50 2
text Maximum PPCC = ^maxppcc
let beta1 = shape - 1
let beta1 = max(0.1,beta1)
let beta2 = shape + 1
logistic exponential ppcc plot y
let beta = shape
justification center
move 50 6
text Betahat = ^beta (BETA = ^betasav)
move 50 2
text Maximum PPCC = ^maxppcc
.
y1label Data
x1label Theoretical
char x
line bl
logistic exponential prob plot y
move 50 6
text Location = ^ppa0, Scale = ^ppa1
char bl
line so
.
y1label Relative Frequency
x1label
relative hist y
limits freeze
pre-erase off
line color blue
plot lexpdf(x,beta,ppa0,ppa1) for x = 0.01 .01 amax
line color black
limits 
pre-erase on
.
let ksloc = ppa0
let ksscale = ppa1
logistic exponential kolmogorov smirnov goodness of fit y
.
logistic exponential mle y
let beta = betaml
let ksloc = 0
let ksscale = alphaml
logistic exponential kolmogorov smirnov goodness of fit y


    plot generated by sample program

    plot generated by sample program

    plot generated by sample program

    plot generated by sample program 

                       KOLMOGOROV-SMIRNOV GOODNESS-OF-FIT TEST
  
 NULL HYPOTHESIS H0:      DISTRIBUTION FITS THE DATA
 ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA
 DISTRIBUTION:            LOGISTIC-EXPONENTIAL
    NUMBER OF OBSERVATIONS              =      200
  
 TEST:
 KOLMOGOROV-SMIRNOV TEST STATISTIC      =   0.3711542E-01
  
    ALPHA LEVEL         CUTOFF              CONCLUSION
            10%       0.086*              ACCEPT H0
                      0.085**
             5%       0.096*              ACCEPT H0
                      0.095**
             1%       0.115*              ACCEPT H0
                      0.114**
  
     *  - STANDARD LARGE SAMPLE APPROXIMATION  ( C/SQRT(N) )
    ** - MORE ACCURATE LARGE SAMPLE APPROXIMATION  ( C/SQRT(N + SQRT(N/10)) )
  

  
       LOGISTIC-EXPONENTIAL PARAMETER ESTIMATION: FULL SAMPLE CASE
             TWO-PARAMETER MODEL (LOCATION = 0)
  
 SUMMARY STATISTICS:
 NUMBER OF OBSERVATIONS                            =      200
 SAMPLE MEAN                                       =    8.171047
 SAMPLE STANDARD DEVIATION                         =    4.589424
 SAMPLE MINIMUM                                    =   0.4414481
 SAMPLE MAXIMUM                                    =    26.37044
  
 MAXIMUM LIKELIHOOD ESTIMATES:
 SHAPE PARAMETER (BETA)                           =    2.118914
 SCALE (1/ALPHA) PARAMETER                         =    10.54572
  
                   KOLMOGOROV-SMIRNOV GOODNESS-OF-FIT TEST
  
 NULL HYPOTHESIS H0:      DISTRIBUTION FITS THE DATA
 ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA
 DISTRIBUTION:            LOGISTIC-EXPONENTIAL
    NUMBER OF OBSERVATIONS              =      200
  
 TEST:
 KOLMOGOROV-SMIRNOV TEST STATISTIC      =   0.3192246E-01
  
    ALPHA LEVEL         CUTOFF              CONCLUSION
            10%       0.086*              ACCEPT H0
                      0.085**
             5%       0.096*              ACCEPT H0
                      0.095**
             1%       0.115*              ACCEPT H0
                      0.114**
  
     *  - STANDARD LARGE SAMPLE APPROXIMATION  ( C/SQRT(N) )
    ** - MORE ACCURATE LARGE SAMPLE APPROXIMATION  ( C/SQRT(N + SQRT(N/10)) )

Date created: 2/14/2008
Last updated: 2/14/2008
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