eexp: Estimate Rate Parameter of an Exponential Distribution

Description

Estimate the rate parameter of an exponential distribution, and optionally construct a confidence interval for the rate parameter.

Usage

 1 2  eexp(x, method = "mle/mme", ci = FALSE, ci.type = "two-sided", ci.method = "exact", conf.level = 0.95) 

Arguments

 x numeric vector of observations. method character string specifying the method of estimation. Currently the only possible value is "mle/mme" (maximum likelihood/method of moments; the default). See the DETAILS section for more information. ci logical scalar indicating whether to compute a confidence interval for the location or scale parameter. The default value is FALSE. ci.type character string indicating what kind of confidence interval to compute. The possible values are "two-sided" (the default), "lower", and "upper". This argument is ignored if ci=FALSE. ci.method character string indicating what method to use to construct the confidence interval for the location or scale parameter. Currently, the only possible value is "exact" (the default). See the DETAILS section for more information. This argument is ignored if ci=FALSE. conf.level a scalar between 0 and 1 indicating the confidence level of the confidence interval. The default value is conf.level=0.95. This argument is ignored if ci=FALSE.

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, …, x_n) be a vector of n observations from an exponential distribution with parameter rate=λ.

Estimation
The maximum likelihood estimator (mle) of λ is given by:

\hat{λ}_{mle} = \frac{1}{\bar{x}}

where

\bar{x} = \frac{1}{n}∑^n_{i=1} x_i

(Forbes et al., 2011). That is, the mle is the reciprocal of the sample mean.

Sometimes the exponential distribution is parameterized with a scale parameter instead of a rate parameter. The scale parameter is the reciprocal of the rate parameter, and the sample mean is both the mle and the minimum variance unbiased estimator (mvue) of the scale parameter.

Confidence Interval
When ci=TRUE, an exact (1-α)100\% confidence intervals for λ can be constructed based on the relationship between the exponential distribution, the gamma distribution, and the chi-square distribution. An exponential distribution with parameter rate=λ is equivalent to a gamma distribution with parameters shape=1 and scale=1/λ. The sum of n iid gamma random variables with parameters shape=1 and scale=1/λ is a gamma random variable with parameters shape=n and scale=1/λ. Finally, a gamma distribution with parameters shape=n and scale=1/λ is equivalent to 0.5 times a chi-square distribution with degrees of freedom df=2n. Thus, the quantity 2n\bar{x} has a chi-square distribution with degrees of freedom df=2n.

A two-sided (1-α)100\% confidence interval for λ is therefore constructed as:

[\frac{χ^2(2n, α/2)}{2n\bar{x}}, \; \frac{chi^2(2n, 1 - α/2)}{2n\bar{x}} ]

where χ^2(ν,p) is the p'th quantile of a chi-square distribution with ν degrees of freedom.

One-sided confidence intervals are computed in a similar fashion.

Value

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

Note

The exponential distribution is a special case of the gamma distribution, and takes on positive real values. A major use of the exponential distribution is in life testing where it is used to model the lifetime of a product, part, person, etc.

The exponential distribution is the only continuous distribution with a “lack of memory” property. That is, if the lifetime of a part follows the exponential distribution, then the distribution of the time until failure is the same as the distribution of the time until failure given that the part has survived to time t.

The exponential distribution is related to the double exponential (also called Laplace) distribution, and to the extreme value distribution.

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 N. Balakrishnan. (1994). Continuous Univariate Distributions, Volume 1. Second Edition. John Wiley and Sons, New York.

Exponential.

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  # Generate 20 observations from an exponential distribution with parameter # rate=2, then estimate the parameter and construct a 90% confidence interval. # (Note: the call to set.seed simply allows you to reproduce this example.) set.seed(250) dat <- rexp(20, rate = 2) eexp(dat, ci=TRUE, conf = 0.9) #Results of Distribution Parameter Estimation #-------------------------------------------- # #Assumed Distribution: Exponential # #Estimated Parameter(s): rate = 2.260587 # #Estimation Method: mle/mme # #Data: dat # #Sample Size: 20 # #Confidence Interval for: rate # #Confidence Interval Method: Exact # #Confidence Interval Type: two-sided # #Confidence Level: 90% # #Confidence Interval: LCL = 1.498165 # UCL = 3.151173 #---------- # Clean up #--------- rm(dat) 

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