EnvStats (version 2.3.1)

egeom: Estimate Probability Parameter of a Geometric Distribution

Description

Estimate the probability parameter of a geometric distribution.

Usage

egeom(x, method = "mle/mme")

Arguments

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 x+1). Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are allowed but will be removed. If length(x)=n and n is greater than 1, it is assumed that x represents observations from n separate geometric experiments that all had the same probability of success (prob).

method

character string specifying the method of estimation. Possible values are "mle/mme" (maximum likelihood and method of moments; the default) and "mvue" (minimum variance unbiased). You cannot use method="mvue" if length(x)=1. See the DETAILS section for more information on these estimation methods.

Value

a list of class "estimate" containing the estimated parameters and other information. See estimate.object for details.

Details

If x contains any missing (NA), undefined (NaN) or infinite (Inf, -Inf) values, they will be removed prior to performing the estimation.

Let \(\underline{x} = (x_1, x_2, \ldots, x_n)\) 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: $$X = \sum^n_{i = 1} x_i$$ 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 \(n=1\).

References

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.

See Also

Geometric, enbinom, NegBinomial.

Examples

# NOT RUN {
  # 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)
# }