Description

Estimate the shape and scale parameters (or the mean and coefficient of variation) of a Gamma distribution.

Usage

Arguments

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 denote a random sample of n observations from a gamma distribution with parameters shape=κ and scale=θ. The relationship between these parameters and the mean (mean=μ) and coefficient of variation (cv=τ) of this distribution is given by:

κ = τ^{-2} \;\;\;\;\;\; (1)

θ = μ/κ \;\;\;\;\;\; (2)

μ = κ \; θ \;\;\;\;\;\; (3)

τ = κ^{-1/2} \;\;\;\;\;\; (4)

The function egamma returns estimates of the shape and scale parameters. The function egammaAlt returns estimates of the mean (μ) and coefficient of variation (cv) based on the estimates of the shape and scale parameters.

Estimation

Maximum Likelihood Estimation (method="mle")
The maximum likelihood estimators (mle's) of the shape and scale parameters κ and θ are solutions of the simultaneous equations:

\hat{κ}_{mle} = \frac{1}{n}∑_{i=1}^n log(x_i) - log(\bar{x}) = ψ(\hat{κ}_{mle}) - log(\hat{κ}_{mle}) \ \;\;\;\;\;\; (5)

\hat{θ}_{mle} = \bar{x} / \hat{κ}_{mle} \;\;\;\;\;\; (6)

where ψ denotes the digamma function, and \bar{x} denotes the sample mean:

\bar{x} = \frac{1}{n}∑_{i=1}^n x_i \;\;\;\;\;\; (7)

(Forbes et al., 2011, chapter 22; Johnson et al., 1994, chapter 17).

Bias-Corrected Maximum Likelihood Estimation (method="bcmle")
The “bias-corrected” maximum likelihood estimator of the shape parameter is based on the suggestion of Anderson and Ray (1975; see also Johnon et al., 1994, p.366 and Singh et al., 2010b, p.48), who noted that the bias of the maximum likelihood estimator of the shape parameter can be considerable when the sample size is small. This estimator is given by:

\hat{κ}_{bcmle} = \frac{n-3}{n}\hat{κ}_{mle} + \frac{2}{3n} \;\;\;\;\;\; (8)

When method="bcmle", Equation (6) above is modified so that the estimate of the scale paramter is based on the “bias-corrected” maximum likelihood estimator of the shape parameter:

\hat{θ}_{bcmle} = \bar{x} / \hat{κ}_{bcmle} \;\;\;\;\;\; (9)

Method of Moments Estimation (method="mme")
The method of moments estimators (mme's) of the shape and scale parameters κ and θ are:

\hat{κ}_{mme} = (\bar{x}/s_m)^2 \;\;\;\;\;\; (10)

\hat{θ}_{mme} = s_m^2 / \bar{x} \;\;\;\;\;\; (11)

where s_m^2 denotes the method of moments estimator of variance:

s_m^2 = \frac{1}{n} ∑_{i=1}^n (x_i - \bar{x})^2 \;\;\;\;\;\; (12)

Method of Moments Estimation Based on the Unbiased Estimator of Variance (method="mmue")
The method of moments estimators based on the unbiased estimator of variance are exactly the same as the method of moments estimators, except that the method of moments estimator of variance is replaced with the unbiased estimator of variance:

\hat{κ}_{mmue} = (\bar{x}/s)^2 \;\;\;\;\;\; (13)

\hat{θ}_{mmue} = s^2 / \bar{x} \;\;\;\;\;\; (14)

where s^2 denotes the unbiased estimator of variance:

s^2 = \frac{1}{n-1} ∑_{i=1}^n (x_i - \bar{x})^2 \;\;\;\;\;\; (15)

Confidence Intervals
This section discusses how confidence intervals for the mean μ are computed.

Normal Approximation (ci.method="normal.approx")
The normal approximation method is based on the method of Kulkarni and Powar (2010), who use a power transformation of the the original data to approximate a sample from a normal distribuiton, compute the confidence interval for the mean on the transformed scale using the usual formula for a confidence interval for the mean of a normal distribuiton, and then tranform the limits back to the original space using equations based on the expected value of a gamma random variable raised to a power.

The particular power used for the normal approximation is defined by the argument
normal.approx.transform. The value normal.approx.transform="cube.root" uses the cube root transformation suggested by Wilson and Hilferty (1931), and the value
"fourth.root" uses the fourth root transformation suggested by Hawkins and Wixley (1986). The default value "kulkarni.powar" uses the “Optimum Power Normal Approximation Method” of Kulkarni and Powar (2010), who show this method performs the best in terms of maintining coverage and minimizing confidence interval width compared to eight other methods. The “optimum” power p is determined by:

where \hat{κ} denotes the estimate of the shape parameter. Kulkarni and Powar (2010) derived this equation by determining what power transformation yields a skew closest to 0 and a kurtosis closest to 3 for a gamma random variable with a given shape parameter. Although Kulkarni and Powar (2010) use the maximum likelihood estimate of shape to determine the power to use to induce approximate normality, for the functions egamma and egammaAlt the power is based on whatever estimate of shape is used (e.g., method="mle", method="bcmle", etc.).

Likelihood Profile (ci.method="profile.likelihood")
This method was proposed by Cox (1970, p.88), and Venzon and Moolgavkar (1988) introduced an efficient method of computation. This method is also discussed by Stryhn and Christensen (2003) and Royston (2007). The idea behind this method is to invert the likelihood-ratio test to obtain a confidence interval for the mean μ while treating the coefficient of variation τ as a nuisance parameter.

The likelihood function is given by:

L(μ, τ | \underline{x}) = ∏_{i=1}^n \frac{x_i^{κ-1} e^{-x_i/θ}}{θ^κ Γ(κ)} \;\;\;\;\;\; (17)

where κ, θ, μ, and τ are defined in Equations (1)-(4) above, and Γ(t) denotes the Gamma function evaluated at t.

Following Stryhn and Christensen (2003), denote the maximum likelihood estimates of the mean and coefficient of variation by (μ^*, τ^*). The likelihood ratio test statistic (G^2) of the hypothesis H_0: μ = μ_0 (where μ_0 is a fixed value) equals the drop in 2 log(L) between the “full” model and the reduced model with μ fixed at μ_0, i.e.,

G^2 = 2 \{log[L(μ^*, τ^*)] - log[L(μ_0, τ_0^*)]\} \;\;\;\;\;\; (18)

where τ_0^* is the maximum likelihood estimate of τ for the reduced model (i.e., when μ = μ_0). Under the null hypothesis, the test statistic G^2 follows a chi-squared distribution with 1 degree of freedom.

Alternatively, we may express the test statistic in terms of the profile likelihood function L_1 for the mean μ, which is obtained from the usual likelihood function by maximizing over the parameter τ, i.e.,

L_1(μ) = max_{τ} L(μ, τ) \;\;\;\;\;\; (19)

Then we have

G^2 = 2 \{log[L_1(μ^*)] - log[L_1(μ_0)]\} \;\;\;\;\;\; (20)

A two-sided (1-α)100\% confidence interval for the mean μ consists of all values of μ_0 for which the test is not significant at level alpha:

μ_0: G^2 ≤ χ^2_{1, {1-α}} \;\;\;\;\;\; (21)

where χ^2_{ν, p} denotes the p'th quantile of the chi-squared distribution with ν degrees of freedom. One-sided lower and one-sided upper confidence intervals are computed in a similar fashion, except that the quantity 1-α in Equation (21) is replaced with 1-2α.

Chi-Square Approximation (ci.method="chisq.approx")
This method is based on the relationship between the sample mean of the gamma distribution and the chi-squared distribution (Grice and Bain, 1980):

\frac{2n\bar{x}}{θ} \sim χ^2_{2nκ} \;\;\;\;\;\; (22)

Therefore, an exact one-sided upper (1-α)100\% confidence interval for the mean μ is given by:

[0, \, \frac{2n\bar{x}κ}{χ^2_{2nκ, α}}] \;\;\;\;\;\; (23)

an exact one-sided lower (1-α)100\% confidence interval is given by:

[\frac{2n\bar{x}κ}{χ^2_{2nκ, 1-α}}, \, ∞] \;\;\;\;\;\; (24)

and a two-sided (1-α)100\% confidence interval is given by:

[\frac{2n\bar{x}κ}{χ^2_{2nκ, 1-α/2}}, \, \frac{2n\bar{x}κ}{χ^2_{2nκ, α/2}}] \;\;\;\;\;\; (25)

Because this method is exact only when the shape parameter κ is known, the method used here is called the “chi-square approximation” method because the estimate of the shape parameter, \hat{κ}, is used in place of κ in Equations (23)-(25) above. The Chi-Square Approximation method is not the method proposed by Grice and Bain (1980) in which the confidence interval is adjusted based on adjusting for the fact that the shape parameter κ is estimated (see the explanation of the Chi-Square Adjusted method below). The Chi-Square Approximation method used by egamma and egammaAlt is equivalent to the “approximate gamma” method of ProUCL (USEPA, 2015, equation (2-34), p.62).

Chi-Square Adjusted (ci.method="chisq.adj")
This is the same method as the Chi-Square Approximation method discussed above, execpt that the value of α is adjusted to account for the fact that the shape parameter κ is estimated rather than known. Grice and Bain (1980) performed Monte Carlo simulations to determine how to adjust α and the values in their Table 2 are given in the matrix Grice.Bain.80.mat. This method requires that the sample size n is at least 5 and the confidence level is between 75% and 99.5% (except when n=5, in which case the confidence level must be less than 99%). For values of the sample size n and/or α that are not listed in the table, linear interpolation is used (when the sample size n is greater than 40, linear interpolation on 1/n is used, as recommended by Grice and Bain (1980)). The Chi-Square Adjusted method used by egamma and egammaAlt is equivalent to the “adjusted gamma” method of ProUCL (USEPA, 2015, equation (2-35), p.63).

Value

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

Warning

When ci=TRUE and ci.method="normal.approx", it is possible for the lower confidence limit based on the transformed data to be less than 0. In this case, the lower confidence limit on the original scale is set to 0 and a warning is issued stating that the normal approximation is not accurate in this case.

Note

The gamma distribution takes values on the positive real line. Special cases of the gamma are the exponential distribution and the chi-square distributions. Applications of the gamma include life testing, statistical ecology, queuing theory, inventory control, and precipitation processes. A gamma distribution starts to resemble a normal distribution as the shape parameter a tends to infinity.

Some EPA guidance documents (e.g., Singh et al., 2002; Singh et al., 2010a,b) strongly recommend against using a lognormal model for environmental data and recommend trying a gamma distribuiton instead.

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

Anderson, C.W., and W.D. Ray. (1975). Improved Maximum Likelihood Estimators for the Gamma Distribution. Communications in Statistics, 4, 437–448.

Forbes, C., M. Evans, N. Hastings, and B. Peacock. (2011). Statistical Distributions, Fourth Edition. John Wiley and Sons, Hoboken, NJ.

Grice, J.V., and L.J. Bain. (1980). Inferences Concerning the Mean of the Gamma Distribution. Journal of the American Statistician, 75, 929-933.

Hawkins, D. M., and R.A.J. Wixley. (1986). A Note on the Transformation of Chi-Squared Variables to Normality. The American Statistician, 40, 296–298.

Johnson, N.L., S. Kotz, and N. Balakrishnan. (1994). Continuous Univariate Distributions, Volume 1. Second Edition. John Wiley and Sons, New York, Chapter 17.

Kulkarni, H.V., and S.K. Powar. (2010). A New Method for Interval Estimation of the Mean of the Gamma Distribution. Lifetime Data Analysis, 16, 431–447.

Singh, A., A.K. Singh, and R.J. Iaci. (2002). Estimation of the Exposure Point Concentration Term Using a Gamma Distribution. EPA/600/R-02/084. October 2002. Technology Support Center for Monitoring and Site Characterization, Office of Research and Development, Office of Solid Waste and Emergency Response, U.S. Environmental Protection Agency, Washington, D.C.

USEPA. (2015). ProUCL Version 5.1.002 Technical Guide. EPA/600/R-07/041, October 2015. Office of Research and Development. U.S. Environmental Protection Agency, Washington, D.C.

Wilson, E.B., and M.M. Hilferty. (1931). The Distribution of Chi-Squares. Proceedings of the National Academy of Sciences, 17, 684–688.

See Also

GammaDist, estimate.object, eqgamma, predIntGamma, tolIntGamma.

Examples