
LOSPDFName:
with p and r denoting the shape parameters. The r parameter is restricted to nonnegative integers. This distribution is used to model the "gamblers ruin" problem. For this problem, p is the probability that the gambler loses one unit (1  p is the probability that the gambler wins one unit). The value of r is the number of units the gambler starts with. The lost games distribution is then the distribution of the number of games lost until the gambler loses all of his fortune. This problem is referred to as the gambler's ruin since if the probability of winning is less than 0.5, the gambler will eventually lose all of his fortune with probability 1. Although this distribution was developed to model gambling, Kemp and Kemp demonstrated its applicability to a number of other important applications. For example, Haight used it to model the queue with r initial customers, where new customers arrive according to a homogeneous Poisson process with shape parameter , and the service time follows an exponential distribution with shape parameter ( ≥ ). The p parameter in our formula can be expressed as
Note that Haight use the parameterization Assuming a constant service time (rather than an exponential service time) results in the BorelTanner distribution.
<SUBSET/EXCEPT/FOR qualification> where <x> is a positive integer variable, number, or parameter; <p> is a number or parameter in the range (0.5,1) that specifies the first shape parameter; <r> is a number or parameter denoting a positive integer that specifies the second shape parameter; <y> is a variable or a parameter where the computed lost games pdf value is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET Y = LOSPDF(X1,0.7,2) PLOT LOSPDF(X,0.6,5) FOR X = 5 1 50
LET R = <value> LET P = <value> LET Y = LOST GAMES RANDOM NUMBERS FOR I = 1 1 N
LOST GAMES PROBABILITY PLOT Y
LOST GAMES CHISQUARE GOODNESS OF FIT Y To obtain the maximum likelihood estimate of p assuming that r is known, enter the command
LOST GAMES MAXIMUM LIKELIHOOD Y2 X2 The maximum likelihood estimate of p is
with denoting the sample mean. For a given value of r, generate an estimate of p based on the maximum ppcc value or the minimum chisquare goodness of fit with the commands
LET P1 = <value> LET P2 = <value> LOST GAMES KS PLOT Y LOST GAMES KS PLOT Y2 X2 LOST GAMES KS PLOT Y3 XLOW XHIGH LOST GAMES PPCC PLOT Y LOST GAMES PPCC PLOT Y2 X2 LOST GAMES PPCC PLOT Y3 XLOW XHIGH The default values of P1 and P2 are 0.51 and 0.95, respectively. The value of R should typically be set to the minimum value of the data. Due to the discrete nature of the percent point function for discrete distributions, the ppcc plot will not be smooth. For that reason, if there is sufficient sample size the KS PLOT (i.e., the minimum chisquare value) is typically preferred. Also, since the data is integer values, one of the binned forms is preferred for these commands.
Kemp and Kemp (1968), "On a Distribution Associated with Certain Stochastic Processes", Journal of the Royal Statistical Society, Series B, 30, pp. 401410. Haight (1961), "A Distribution Analogous to the BorelTanner Distribution", Biometrika, 48, pp. 167173. Johnson, Kotz, and Kemp (1992), "Univariate Discrete Distributions", Second Edition, Wiley, pp. 445447.
let r = 3 let p = 0.6 let y = lost games random numbers for i = 1 1 500 . let y3 xlow xhigh = integer frequency table y class lower 1.5 class width 1 let amax = maximum y let amax2 = amax + 0.5 class upper amax2 let y2 x2 = binned y . let k = minimum y lost games mle y let p = pml lost games chisquare goodness of fit y3 xlow xhigh relative histogram y2 x2 limits freeze preerase off line color blue title Lost Games MLE FIt: Phat = ^pml (r = ^r) plot lospdf(x,pml,r) for x = r 1 amax title limits preerase on line color black . label case asis x1label P y1label Minimum ChiSquare let p1 = 0.5 let p2 = 0.9 lost games ks plot y3 xlow xhigh let p = shape case asis justification center move 50 5 text P = ^p lost games chisquare goodness of fit y3 xlow xhigh LOST GAMES MAXIMUM LIKELIHOOD ESTIMATION: NUMBER OF OBSERVATIONS = 500 SAMPLE MEAN = 8.892000 SAMPLE STANDARD DEVIATION = 8.214822 SAMPLE MINIMUM = 3.000000 SAMPLE MAXIMUM = 62.00000 ESTIMATE OF R = 3.000000 MAXIMUM LIKELIHOOD ESTIMATE OF P = 0.6014611 THE MAXIMUM LIKELIHOOD ESTIMATES FOR R AND P ARE SAVED IN THE INTERNAL PARAMETERS RML AND PML THE COMPUTED VALUE OF THE CONSTANT P = 0.6014611E+00 CHISQUARED GOODNESSOFFIT TEST NULL HYPOTHESIS H0: DISTRIBUTION FITS THE DATA ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA DISTRIBUTION: LOST GAMES SAMPLE: NUMBER OF OBSERVATIONS = 500 NUMBER OF NONEMPTY CELLS = 24 NUMBER OF PARAMETERS USED = 2 TEST: CHISQUARED TEST STATISTIC = 22.46355 DEGREES OF FREEDOM = 21 CHISQUARED CDF VALUE = 0.626767 ALPHA LEVEL CUTOFF CONCLUSION 10% 29.61509 ACCEPT H0 5% 32.67057 ACCEPT H0 1% 38.93217 ACCEPT H0 CELL NUMBER, LOWER BIN POINT, UPPER BIN POINT, OBSERVED FREQUENCY, AND EXPECTED FREQUENCY WRITTEN TO FILE DPST1F.DAT CHISQUARED GOODNESSOFFIT TEST NULL HYPOTHESIS H0: DISTRIBUTION FITS THE DATA ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA DISTRIBUTION: LOST GAMES SAMPLE: NUMBER OF OBSERVATIONS = 500 NUMBER OF NONEMPTY CELLS = 24 NUMBER OF PARAMETERS USED = 2 TEST: CHISQUARED TEST STATISTIC = 21.82713 DEGREES OF FREEDOM = 21 CHISQUARED CDF VALUE = 0.590470 ALPHA LEVEL CUTOFF CONCLUSION 10% 29.61509 ACCEPT H0 5% 32.67057 ACCEPT H0 1% 38.93217 ACCEPT H0 CELL NUMBER, LOWER BIN POINT, UPPER BIN POINT, OBSERVED FREQUENCY, AND EXPECTED FREQUENCY WRITTEN TO FILE DPST1F.DAT
Date created: 6/20/2006 