enbinom: Estimate Probability Parameter of a Negative Binomial Distribution

Description

Estimate the probability parameter of a negative binomial distribution.

Usage

enbinom(x, size, method = "mle/mme")

Arguments

vector of non-negative integers indicating the number of trials that took place before size successes occurred. (The total number of trials that took place is x+1). Missing (NA), u

size

vector of positive integers indicating the number of successes that must be observed before the trials are stopped. Missing (NA), undefined (NaN), and infinite (Inf, -Inf) value

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"

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 negative binomial distributions with parameters prob=$p$ and size=$\underline{k}$, where where $\underline{k} = c(k_1, k_2, \ldots, k_n)$ is a vector of $n$ (possibly different) values. It can be shown (e.g., Forbes et al., 2011) that if $X$ is defined as:

X = \sum_{i = 1}^{n} x_{i}

then $X$ is an observation from a negative binomial distribution with parameters prob=$p$ and size=$K$, where

K = \sum_{i = 1}^{n} k_{i}

Estimation The maximum likelihood and method of moments estimator (mle/mme) of $p$ is given by:

{\hat{p}}_{m l e} = \frac{K}{X + K}

and the minimum variance unbiased estimator (mvue) of $p$ is given by:

{\hat{p}}_{m v u e} = \frac{K - 1}{X + K - 1}

(Forbes et al., 2011). Note that the mvue of $p$ is not defined for $K=1$.

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.

Examples

Run this code

# Generate an observation from a negative binomial distribution with 
  # parameters size=2 and prob=0.2, then estimate the parameter prob. 
  # Note: the call to set.seed simply allows you to reproduce this example. 
  # Also, the only parameter that is estimated is prob; the parameter 
  # size is supplied in the call to enbinom.  The parameter size is printed in 
  # order to show all of the parameters associated with the distribution.

  set.seed(250) 
  dat <- rnbinom(1, size = 2, prob = 0.2) 
  dat
  #[1] 5

  enbinom(dat, size = 2)
  #Results of Distribution Parameter Estimation
  #--------------------------------------------
  #
  #Assumed Distribution:            Negative Binomial
  #
  #Estimated Parameter(s):          size = 2.0000000
  #                                 prob = 0.2857143
  #
  #Estimation Method:               mle/mme for 'prob'
  #
  #Data:                            dat, 2
  #
  #Sample Size:                     1

  #----------

  # Generate 3 observations from negative binomial distributions with 
  # parameters size=c(2,3,4) and prob=0.2, then estimate the parameter 
  # prob using the mvue. 
  # (Note: the call to set.seed simply allows you to reproduce this example.)

  size.vec <- 2:4 
  set.seed(250) 
  dat <- rnbinom(3, size = size.vec, prob = 0.2) 
  dat 
  #[1]  5 19 12 

  enbinom(dat, size = size.vec, method = "mvue") 
  #Results of Distribution Parameter Estimation
  #--------------------------------------------
  #
  #Assumed Distribution:            Negative Binomial
  #
  #Estimated Parameter(s):          size = 9.0000000
  #                                 prob = 0.1818182
  #
  #Estimation Method:               mvue for 'prob'
  #
  #Data:                            dat, size.vec
  #
  #Sample Size:                     3

  #----------

  # Clean up
  #---------
  rm(dat)

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