Description

Estimate the mean and standard deviation of a zero-modified normal distribution, and optionally construct a confidence interval for the mean.

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) be a vector of n observations from a zero-modified normal distribution with parameters mean=μ, sd=σ, and p.zero=p. Let r denote the number of observations in \underline{x} that are equal to 0, and order the observations so that x_1, x_2, …, x_r denote the r zero observations, and x_{r+1}, x_{r+2}, …, x_n denote the n-r non-zero observations.

Note that μ is not the mean of the zero-modified normal distribution; it is the mean of the normal part of the distribution. Similarly, σ is not the standard deviation of the zero-modified normal distribution; it is the standard deviation of the normal part of the distribution.

Let γ and δ denote the mean and standard deviation of the overall zero-modified normal distribution. Aitchison (1955) shows that:

γ = (1 - p) μ \;\;\;\; (1)

δ^2 = (1 - p) σ^2 + p (1 - p) μ^2 \;\;\;\; (2)

Estimation

Minimum Variance Unbiased Estimation (method="mvue")
Aitchison (1955) shows that the minimum variance unbiased estimators (mvue's) of γ and δ are:

\hat{γ}_{mvue} = \bar{x} \;\;\;\; (3)

where

\bar{x} = \frac{1}{n} ∑_{i=1}^n x_i \;\;\;\; (5)

\bar{x}^* = \frac{1}{n-r} ∑_{i=r+1}^n x_i \;\;\;\; (6)

(s^*)^2 = \frac{1}{n-r-1} ∑_{i=r+1}^n (x_i - \bar{x}^*)^2 \;\;\;\; (7)

Note that the quantity in equation (5) is the sample mean of all observations (including 0 values), the quantity in equation (6) is the sample mean of all non-zero observations, and the quantity in equation (7) is the sample variance of all non-zero observations. Also note that for r=n-1 or r=n, the estimator of δ^2 is the sample variance for all observations (including 0 values).

Confidence Intervals

Based on Normal Approximation (ci.method="normal.approx")
An approximate (1-α)100\% confidence interval for γ is constructed based on the assumption that the estimator of γ is approximately normally distributed. Aitchison (1955) shows that

Var(\hat{γ}_{mvue}) = Var(\bar{x}) = \frac{δ^2}{n} \;\;\;\; (8)

Thus, an approximate two-sided (1-α)100\% confidence interval for γ is constructed as:

[ \hat{γ}_{mvue} - t_{n-2, 1-α/2} \frac{\hat{δ}_{mvue}}{√{n}}, \; \hat{γ}_{mvue} + t_{n-2, 1-α/2} \frac{\hat{δ}_{mvue}}{√{n}} ] \;\;\;\; (9)

where t_{ν, p} is the p'th quantile of Student's t-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.

The component called parameters is a numeric vector with the following estimated parameters:

Note

The zero-modified normal distribution is sometimes used to model chemical concentrations for which some observations are reported as “Below Detection Limit”. See, for example USEPA (1992c, pp.27-34). In most cases, however, the zero-modified lognormal (delta) distribution will be more appropriate, since chemical concentrations are bounded below at 0 (e.g., Gilliom and Helsel, 1986; Owen and DeRouen, 1980).

Once you estimate the parameters of the zero-modified normal distribution, it is often useful to characterize the uncertainty in the estimate of the mean. This is done with a confidence interval.

One way to try to assess whether a zero-modified lognormal (delta), zero-modified normal, censored normal, or censored lognormal is the best model for the data is to construct both censored and detects-only probability plots (see qqPlotCensored).

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

Aitchison, J. (1955). On the Distribution of a Positive Random Variable Having a Discrete Probability Mass at the Origin. Journal of the American Statistical Association 50, 901–908.

Gilliom, R.J., and D.R. Helsel. (1986). Estimation of Distributional Parameters for Censored Trace Level Water Quality Data: 1. Estimation Techniques. Water Resources Research 22, 135–146.

Owen, W., and T. DeRouen. (1980). Estimation of the Mean for Lognormal Data Containing Zeros and Left-Censored Values, with Applications to the Measurement of Worker Exposure to Air Contaminants. Biometrics 36, 707–719.

USEPA (1992c). Statistical Analysis of Ground-Water Monitoring Data at RCRA Facilities: Addendum to Interim Final Guidance. Office of Solid Waste, Permits and State Programs Division, US Environmental Protection Agency, Washington, D.C.

See Also

ZeroModifiedNormal, Normal, ezmlnorm, ZeroModifiedLognormal, estimate.object.

Examples