Estimate Probability Parameter of a Geometric Distribution
Estimate the probability parameter of a geometric distribution.
egeom(x, method = "mle/mme")
x |
vector of non-negative integers indicating the number of trials that took place
before the first “success” occurred. (The total number of trials
that took place is |
method |
character string specifying the method of estimation. Possible values are |
If x contains any missing (NA), undefined (NaN) or
infinite (Inf, -Inf) values, they will be removed prior to
performing the estimation.
Let \underline{x} = (x_1, x_2, …, x_n) be a vector of n
independent observations from a geometric distribution
with parameter prob=p.
It can be shown (e.g., Forbes et al., 2011) that if X is defined as:
X = ∑^n_{i = 1} x_i
then X is an observation from a
negative binomial distribution with
parameters prob=p and size=n.
Estimation
The maximum likelihood and method of moments estimator (mle/mme) of
p is given by:
\hat{p}_{mle} = \frac{n}{X + n}
and the minimum variance unbiased estimator (mvue) of p is given by:
\hat{p}_{mvue} = \frac{n - 1}{X + n - 1}
(Forbes et al., 2011). Note that the mvue of p is not defined for n=1.
a list of class "estimate" containing the estimated parameters and other information.
See estimate.object for details.
The geometric distribution with parameter
prob=p is a special case of the
negative binomial distribution with parameters
size=1 and prob=p.
The negative binomial distribution has its roots in a gambling game where participants would bet on the number of tosses of a coin necessary to achieve a fixed number of heads. The negative binomial distribution has been applied in a wide variety of fields, including accident statistics, birth-and-death processes, and modeling spatial distributions of biological organisms.
Steven P. Millard (EnvStats@ProbStatInfo.com)
Forbes, C., M. Evans, N. Hastings, and B. Peacock. (2011). Statistical Distributions. Fourth Edition. John Wiley and Sons, Hoboken, NJ.
Johnson, N. L., S. Kotz, and A. Kemp. (1992). Univariate Discrete Distributions. Second Edition. John Wiley and Sons, New York, Chapter 5.
# Generate an observation from a geometric distribution with parameter # prob=0.2, then estimate the parameter prob. # (Note: the call to set.seed simply allows you to reproduce this example.) set.seed(250) dat <- rgeom(1, prob = 0.2) dat #[1] 4 egeom(dat) #Results of Distribution Parameter Estimation #-------------------------------------------- # #Assumed Distribution: Geometric # #Estimated Parameter(s): prob = 0.2 # #Estimation Method: mle/mme # #Data: dat # #Sample Size: 1 #---------- # Generate 3 observations from a geometric distribution with parameter # prob=0.2, then estimate the parameter prob with the mvue. # (Note: the call to set.seed simply allows you to reproduce this example.) set.seed(200) dat <- rgeom(3, prob = 0.2) dat #[1] 0 1 2 egeom(dat, method = "mvue") #Results of Distribution Parameter Estimation #-------------------------------------------- # #Assumed Distribution: Geometric # #Estimated Parameter(s): prob = 0.4 # #Estimation Method: mvue # #Data: dat # #Sample Size: 3 #---------- # Clean up #--------- rm(dat)
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