elogis.RdEstimate the location and scale parameters of a logistic distribution, and optionally construct a confidence interval for the location parameter.
elogis(x, method = "mle", ci = FALSE, ci.type = "two-sided",
ci.method = "normal.approx", conf.level = 0.95)numeric vector of observations.
character string specifying the method of estimation. Possible values are
"mle" (maximum likelihood; the default), "mme" (methods of moments),
and "mmue" (method of moments based on the unbiased estimator of variance).
See the DETAILS section for more information on these estimation methods.
logical scalar indicating whether to compute a confidence interval for the
location or scale parameter. The default value is FALSE.
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.
character string indicating what method to use to construct the confidence interval
for the location or scale parameter. Currently, the only possible value is
"normal.approx" (the default). See the DETAILS section for more information.
This argument is ignored if ci=FALSE.
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.
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
\(n\) observations from an logistic distribution with
parameters location=\(\eta\) and scale=\(\theta\).
Estimation
Maximum Likelihood Estimation (method="mle")
The maximum likelihood estimators (mle's) of \(\eta\) and \(\theta\) are
the solutions of the simultaneous equations (Forbes et al., 2011):
$$\sum_{i=1}^{n} \frac{1}{1 + e^{z_i}} = \frac{n}{2} \;\;\;\; (1)$$
$$\sum_{i=1}^{n} z_i \, [\frac{1 - e^{z_i}}{1 + e^{z_i}} = n \;\;\;\; (2)$$
where
$$z_i = \frac{x_i - \hat{eta}_{mle}}{\hat{\theta}_{mle}} \;\;\;\; (3)$$
Method of Moments Estimation (method="mme")
The method of moments estimators (mme's) of \(\eta\) and \(\theta\) are
given by:
$$\hat{\eta}_{mme} = \bar{x} \;\;\;\; (4)$$
$$\hat{\theta}_{mme} = \frac{\sqrt{3}}{\pi} s_m \;\;\;\; (5)$$
where
$$\bar{x} = \sum_{i=1}^n x_i \;\;\;\; (6)$$
$$s_m^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2 \;\;\;\; (7)$$
that is, \(s_m\) denotes the square root of the method of moments estimator
of variance.
Method of Moments Estimators Based on the Unbiased Estimator of Variance (method="mmue")
These estimators are exactly the same as the method of moments estimators given in
equations (4-7) above, except that the method of moments estimator of variance in
equation (7) is replaced with the unbiased estimator of variance:
$$s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 \;\;\;\; (8)$$
Confidence Intervals
When ci=TRUE, an approximate \((1-\alpha)\)100% confidence intervals
for \(\eta\) can be constructed assuming the distribution of the estimator of
\(\eta\) is approximately normally distributed. A two-sided confidence
interval is constructed as:
$$[\hat{\eta} - t(n-1, 1-\alpha/2) \hat{\sigma}_{\hat{\eta}}, \, \hat{\eta} + t(n-1, 1-\alpha/2) \hat{\sigma}_{\hat{\eta}}]$$
where \(t(\nu, p)\) is the \(p\)'th quantile of
Student's t-distribution with
\(\nu\) degrees of freedom, and the quantity
$$\hat{\sigma}_{\hat{\eta}} = \frac{\pi \hat{\theta}}{\sqrt{3n}} \;\;\;\; (9)$$
denotes the estimated asymptotic standard deviation of the estimator of \(\eta\).
One-sided confidence intervals for \(\eta\) and \(\theta\) are computed in a similar fashion.
a list of class "estimate" containing the estimated parameters and other information.
See estimate.object for details.
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. (1995). Continuous Univariate Distributions, Volume 2. Second Edition. John Wiley and Sons, New York.
The logistic distribution is defined on the real line and is unimodal and symmetric about its location parameter (the mean). It has longer tails than a normal (Gaussian) distribution. It is used to model growth curves and bioassay data.
# Generate 20 observations from a logistic distribution with
# parameters location=0 and scale=1, then estimate the parameters
# and construct a 90% confidence interval for the location parameter.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(250)
dat <- rlogis(20)
elogis(dat, ci = TRUE, conf.level = 0.9)
#>
#> Results of Distribution Parameter Estimation
#> --------------------------------------------
#>
#> Assumed Distribution: Logistic
#>
#> Estimated Parameter(s): location = -0.2181845
#> scale = 0.8152793
#>
#> Estimation Method: mle
#>
#> Data: dat
#>
#> Sample Size: 20
#>
#> Confidence Interval for: location
#>
#> Confidence Interval Method: Normal Approximation
#> (t Distribution)
#>
#> Confidence Interval Type: two-sided
#>
#> Confidence Level: 90%
#>
#> Confidence Interval: LCL = -0.7899382
#> UCL = 0.3535693
#>
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Logistic
#
#Estimated Parameter(s): location = -0.2181845
# scale = 0.8152793
#
#Estimation Method: mle
#
#Data: dat
#
#Sample Size: 20
#
#Confidence Interval for: location
#
#Confidence Interval Method: Normal Approximation
# (t Distribution)
#
#Confidence Interval Type: two-sided
#
#Confidence Level: 90%
#
#Confidence Interval: LCL = -0.7899382
# UCL = 0.3535693
#----------
# Clean up
#---------
rm(dat)