# eqnbinom: Estimate Quantiles of a Negative Binomial Distribution

### Description

Estimate quantiles of a negative binomial distribution.

### Usage

 `1` ``` eqnbinom(x, size = NULL, p = 0.5, method = "mle/mme", digits = 0) ```

### Arguments

 `x` 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`), or an object resulting from a call to an estimating function that assumes a negative binomial distribution (e.g., `enbinom`). If `x` is a vector of non-negative integers, then 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 negative binomial experiments that all had the same probability of success (`prob`), but possibly different values of `size`. `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`) values are allowed but will be removed. The length of `size` must be 1 or else the same length as `x`. `p` numeric vector of probabilities for which quantiles will be estimated. All values of `p` must be between 0 and 1. The default value is `p=0.5`. `method` character string specifying the method of estimating the probability parameter. Possible values are `"mle/mme"` (maximum likelihood and method of moments; the default) and `"mvue"` (minimum variance unbiased). You cannot use `method="mvue"` if the sum of the elements in `size` is 1. See the DETAILS section of the help file for `enbinom` for more information on these estimation methods. `digits` an integer indicating the number of decimal places to round to when printing out the value of `100*p`. The default value is `digits=0`.

### Details

The function `eqnbinom` returns estimated quantiles as well as estimates of the `prob` parameter.

Quantiles are estimated by 1) estimating the prob parameter by calling `enbinom`, and then 2) calling the function `qnbinom` and using the estimated value for `prob`.

### Value

If `x` is a numeric vector, `eqnbinom` returns a list of class `"estimate"` containing the estimated quantile(s) and other information. See `estimate.object` for details.

If `x` is the result of calling an estimation function, `eqnbinom` returns a list whose class is the same as `x`. The list contains the same components as `x`, as well as components called `quantiles` and `quantile.method`.

### Note

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.

The geometric distribution with parameter `prob=`p is a special case of the negative binomial distribution with parameters `size=1` and `prob=`p.

### Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

### 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.

`enbinom`, NegBinomial, `egeom`, Geometric, `estimate.object`.

### Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39``` ``` # Generate an observation from a negative binomial distribution with # parameters size=2 and prob=0.2, then estimate the parameter prob # and the 90th percentile. # 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 eqnbinom(dat, size = 2, p = 0.9) #Results of Distribution Parameter Estimation #-------------------------------------------- # #Assumed Distribution: Negative Binomial # #Estimated Parameter(s): size = 2.0000000 # prob = 0.2857143 # #Estimation Method: mle/mme for 'prob' # #Estimated Quantile(s): 90'th %ile = 11 # #Quantile Estimation Method: Quantile(s) Based on # mle/mme for 'prob' Estimators # #Data: dat, 2 # #Sample Size: 1 #---------- # Clean up rm(dat) ```

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