Description Usage Arguments Details Value Note Author(s) References See Also Examples
Estimate the mean, standard deviation, and threshold parameters for a threeparameter lognormal distribution, and optionally construct a confidence interval for the threshold or the median of the distribution.
1 2 3 
x 
numeric vector of observations. 
method 
character string specifying the method of estimation. Possible values are:
See the DETAILS section for more information. 
ci 
logical scalar indicating whether to compute a confidence interval for either
the threshold or median of the distribution. The default value is 
ci.parameter 
character string indicating the parameter for which the confidence interval is
desired. The possible values are 
ci.method 
character string indicating the method to use to construct the confidence interval.
The possible values are 
ci.type 
character string indicating what kind of confidence interval to compute. The
possible values are 
conf.level 
a scalar between 0 and 1 indicating the confidence level of the confidence interval.
The default value is 
threshold.lb.sd 
a positive numeric scalar specifying the range over which to look for the
local maximum likelihood ( 
evNormOrdStats.method 
character string indicating which method to use in the call to

If x
contains any missing (NA
), undefined (NaN
) or
infinite (Inf
, Inf
) values, they will be removed prior to
performing the estimation.
Let X denote a random variable from a
threeparameter lognormal distribution with
parameters meanlog=
μ, sdlog=
σ, and
threshold=
γ. Let \underline{x} denote a vector of
n observations from this distribution. Furthermore, let x_{(i)} denote
the i'th order statistic in the sample, so that x_{(1)} denotes the
smallest value and x_{(n)} denote the largest value in \underline{x}.
Finally, denote the sample mean and variance by:
\bar{x} = \frac{1}{n} ∑_{i=1}^n x_i \;\;\;\; (1)
s^2 = \frac{1}{n1} ∑_{i=1}^n (x_i  \bar{x})^2 \;\;\;\; (2)
Note that the sample variance is the unbiased version. Denote the method of moments estimator of variance by:
s^2_m = \frac{1}{n} ∑_{i=1}^n (x_i  \bar{x})^2 \;\;\;\; (3)
Estimation
Local Maximum Likelihood Estimation (method="lmle"
)
Hill (1963) showed that the likelihood function approaches infinity as γ
approaches x_{(1)}, so that the global maximum likelihood estimators of
(μ, σ, γ) are (∞, ∞, x_{(1)}), which are
inadmissible, since γ must be smaller than x_{(1)}. Cohen (1951)
suggested using local maximum likelihood estimators (lmle's), derived by equating
partial derivatives of the loglikelihood function to zero. These estimators were
studied by Harter and Moore (1966), Calitz (1973), Cohen and Whitten (1980), and
Griffiths (1980), and appear to possess most of the desirable properties ordinarily
associated with maximum likelihood estimators.
Cohen (1951) showed that the lmle of γ is given by the solution to the following equation:
[∑_{i=1}^n \frac{1}{w_i}] \, \{∑_{i=1}^n y_i  ∑_{i=1}^n y_i^2 + \frac{1}{n}[∑_{i=1}^n y_i]^2 \}  n ∑_{i=1}^n \frac{y_i}{w_i} = 0 \;\;\;\; (4)
where
w_i = x_i  \hat{γ} \;\;\;\; (5)
y_i = log(x_i  \hat{γ}) = log(w_i) \;\;\;\; (6)
and that the lmle's of μ and σ then follow as:
\hat{μ} = \frac{1}{n} ∑_{i=1}^n y_i = \bar{y} \;\;\;\; (7)
\hat{σ}^2 = \frac{1}{n} ∑_{i=1}^n (y_i  \bar{y})^2 \;\;\;\; (8)
Unfortunately, while equation (4) simplifies the task of computing the lmle's, for certain data sets there still may be convergence problems (Calitz, 1973), and occasionally multiple roots of equation (4) may exist. When multiple roots to equation (4) exisit, Cohen and Whitten (1980) recommend using the one that results in closest agreement between the mle of μ (equation (7)) and the sample mean (equation (1)).
On the other hand, Griffiths (1980) showed that for a given value of the threshold parameter γ, the maximized value of the loglikelihood (the “profile likelihood” for γ) is given by:
log[L(γ)] = \frac{n}{2} [1 + log(2π) + 2\hat{μ} + log(\hat{σ}^2) ] \;\;\;\; (9)
where the estimates of μ and σ are defined in equations (7) and (8), so the lmle of γ reduces to an iterative search over the values of γ. Griffiths (1980) noted that the distribution of the lmle of γ is far from normal and that log[L(γ)] is not quadratic near the lmle of γ. He suggested a better parameterization based on
η = log(x_{(1)}  γ) \;\;\;\; (10)
Thus, once the lmle of η is found using equations (9) and (10), the lmle of γ is given by:
\hat{γ} = x_{(1)}  exp(\hat{η}) \;\;\;\; (11)
When method="lmle"
, the function elnorm3
uses the function
nlminb
to search for the minimum of 2log[L(η)], using the
modified method of moments estimator (method="mmme"
; see below) as the
starting value for γ. Equation (11) is then used to solve for the
lmle of γ, and equation (4) is used to “fine tune” the estimated
value of γ. The lmle's of μ and σ are then computed
using equations (6)(8).
Method of Moments Estimation (method="mme"
)
Denote the r'th sample central moment by:
m_r = \frac{1}{n} ∑_{i=1}^n (x_i  \bar{x})^r \;\;\;\; (12)
and note that
s^2_m = m_2 \;\;\;\; (13)
Equating the sample first moment (the sample mean) with its population value (the population mean), and equating the second and third sample central moments with their population values yields (Johnson et al., 1994, p.228):
\bar{x} = γ + β √{ω} \;\;\;\; (14)
m_2 = s^2_m = β^2 ω (ω  1) \;\;\;\; (15)
m_3 = β^3 ω^{3/2} (ω  1)^2 (ω + 2) \;\;\;\; (16)
where
β = exp(μ) \;\;\;\; (17)
ω = exp(σ^2) \;\;\;\; (18)
Combining equations (15) and (16) yields:
b_1 = \frac{m_3}{m_2^{3/2}} = (ω + 2) √{ω  1} \;\;\;\; (19)
The quantity on the lefthand side of equation (19) is the usual estimator of skewness. Solving equation (19) for ω yields:
\hat{ω} = (d + h)^{1/3} + (d  h)^{1/3}  1 \;\;\;\; (20)
where
d = 1 + \frac{b_1}{2} \;\;\;\; (21)
h = sqrt{d^2  1} \;\;\;\; (22)
Using equation (18), the method of moments estimator of σ is then computed as:
\hat{σ}^2 = log(\hat{ω}) \;\;\;\; (23)
Combining equations (15) and (17), the method of moments estimator of μ is computed as:
\hat{μ} = \frac{1}{2} log[\frac{s^2_m}{\hat{omega}(\hat{ω}  1)}] \;\;\;\; (24)
Finally, using equations (14), (17), and (18), the method of moments estimator of γ is computed as:
\bar{x}  exp(\hat{mu} + \frac{\hat{σ}^2}{2}) \;\;\;\; (25)
There are two major problems with using method of moments estimators for the
threeparameter lognormal distribution. First, they are subject to very large
sampling error due to the use of second and third sample moments
(Cohen, 1988, p.121; Johnson et al., 1994, p.228). Second, Heyde (1963) showed
that the lognormal distribution is not uniquely determined by its moments.
Method of Moments Estimators Using an Unbiased Estimate of Variance (method="mmue"
)
This method of estimation is exactly the same as the method of moments
(method="mme"
), except that the unbiased estimator of variance (equation (3))
is used in place of the method of moments one (equation (4)). This modification is
given in Cohen (1988, pp.119120).
Modified Method of Moments Estimation (method="mmme"
)
This method of estimation is described by Cohen (1988, pp.125132). It was
introduced by Cohen and Whitten (1980; their MMEII with r=1) and was further
investigated by Cohen et al. (1985). It is motivated by the fact that the first
order statistic in the sample, x_{(1)}, contains more information about
the threshold parameter γ than any other observation and often more
information than all of the other observations combined (Cohen, 1988, p.125).
The first two sets of equations are the same as for the modified method of moments
estimators (method="mmme"
), i.e., equations (14) and (15) with the
unbiased estimator of variance (equation (3)) used in place of the method of
moments one (equation (4)). The third equation replaces equation (16)
by equating a function of the first order statistic with its expected value:
log(x_{(1)}  γ) = μ + σ E[Z_{(1,n)}] \;\;\;\; (26)
where E[Z_{(i,n)}] denotes the expected value of the i'th order
statistic in a random sample of n observations from a standard normal
distribution. (See the help file for evNormOrdStats
for information
on how E[Z_{(i,n)}] is computed.) Using equations (17) and (18),
equation (26) can be rewritten as:
x_{(1)} = γ + β exp\{√{log(ω)} \, E[Z_{(i,n)}] \} \;\;\;\; (27)
Combining equations (14), (15), (17), (18), and (27) yields the following equation for the estimate of ω:
\frac{s^2}{[\bar{x}  x_{(1)}]^2} = \frac{\hat{ω}(\hat{ω}  1)}{[√{\hat{ω}}  exp\{√{log(ω)} \, E[Z_{(i,n)}] \} ]^2} \;\;\;\; (28)
After equation (28) is solved for \hat{ω}, the estimate of σ
is again computed using equation (23), and the estimate of μ is computed
using equation (24), where the unbiased estimate of variaince is used in place of
the biased one (just as for method="mmue"
).
ZeroSkewness Estimation (method="zero.skew"
)
This method of estimation was introduced by Griffiths (1980), and elaborated upon
by Royston (1992b). The idea is that if the threshold parameter γ were
known, then the distribution of:
Y = log(X  γ) \;\;\;\; (29)
is normal, so the skew of Y is 0. Thus, the threshold parameter γ is estimated as that value that forces the sample skew (defined in equation (19)) of the observations defined in equation (6) to be 0. That is, the zeroskewness estimator of γ is the value that satisfies the following equation:
0 = \frac{\frac{1}{n} ∑_{i=1}^n (y_i  \bar{y})^3}{[\frac{1}{n} ∑_{i=1}^n (y_i  \bar{y})^2]^{3/2}} \;\;\;\; (30)
where
y_i = log(x_i  \hat{γ}) \;\;\;\; (31)
Note that since the denominator in equation (30) is always positive (assuming there are at least two unique values in \underline{x}), only the numerator needs to be used to determine the value of \hat{γ}.
Once the value of \hat{γ} has been determined, μ and σ are estimated using equations (7) and (8), except the unbiased estimator of variance is used in equation (8).
Royston (1992b) developed a modification of the ShaprioWilk goodnessoffit test
for normality based on tranforming the data using equation (6) and the zeroskewness
estimator of γ (see gofTest
).
Estimators Based on Royston's Index of Skewness (method="royston.skew"
)
This method of estimation is discussed by Royston (1992b), and is similar to the
zeroskewness method discussed above, except a different measure of skewness is used.
Royston's (1992b) index of skewness is given by:
q = \frac{y_{(n)}  \tilde{y}}{\tilde{y}  y_{(1)}} \;\;\;\; (32)
where y_{(i)} denotes the i'th order statistic of y and y is defined in equation (31) above, and \tilde{y} denotes the median of y. Royston (1992b) shows that the value of γ that yields a value of q=0 is given by:
\hat{γ} = \frac{y_{(1)}y_{(n)}  \tilde{y}^2}{y_{(1)} + y_{(n)}  2\tilde{y}} \;\;\;\; (33)
Again, as for the zeroskewness method, once the value of \hat{γ} has been determined, μ and σ are estimated using equations (7) and (8), except the unbiased estimator of variance is used in equation (8).
Royston (1992b) developed this estimator as a quick way to estimate γ.
Confidence Intervals
This section explains three different methods for constructing confidence intervals
for the threshold parameter γ, or the median of the threeparameter
lognormal distribution, which is given by:
Med[X] = γ + exp(μ) = γ + β \;\;\;\; (34)
Normal Approximation Based on Asymptotic Variances and Covariances (ci.method="avar"
)
Formulas for asymptotic variances and covariances for the threeparameter lognormal
distribution, based on the information matrix, are given in Cohen (1951), Cohen and
Whitten (1980), Cohen et al., (1985), and Cohen (1988). The relevant quantities for
γ and the median are:
Var(\hat{γ}) = σ^2_{\hat{γ}} = \frac{σ^2}{n} \, (\frac{β^2}{ω}) H \;\;\;\; (35)
Var(\hat{β}) = σ^2_{\hat{β}} = \frac{σ^2}{n} \, β^2 (1 + H) \;\;\;\; (36)
Cov(\hat{γ}, \hat{β}) = σ_{\hat{γ}, \hat{β}} = \frac{σ^3}{n} \, (\frac{β^2}{√{ω}}) H \;\;\;\; (37)
where
H = [ω (1 + σ^2)  2σ^2  1]^{1} \;\;\;\; (38)
A twosided (1α)100\% confidence interval for γ is computed as:
\hat{γ}  t_{n2, 1α/2} \hat{σ}_{\hat{γ}}, \, \hat{γ} + t_{n2, 1α/2} \hat{σ}_{\hat{γ}} \;\;\;\; (39)
where t_{ν, p} denotes the p'th quantile of Student's tdistribution with n degrees of freedom, and the quantity \hat{σ}_{\hat{γ}} is computed using equations (35) and (38) and substituting estimated values of β, ω, and σ. Onesided confidence intervals are computed in a similar manner.
A twosided (1α)100\% confidence interval for the median (see equation (34) above) is computed as:
\hat{γ} + \hat{β}  t_{n2, 1α/2} \hat{σ}_{\hat{γ} + \hat{β}}, \, \hat{γ} + \hat{β} + t_{n2, 1α/2} \hat{σ}_{\hat{γ} + \hat{β}} \;\;\;\; (40)
where
\hat{σ}^2_{\hat{γ} + \hat{β}} = \hat{σ}^2_{\hat{γ}} + \hat{σ}^2_{\hat{β}} + \hat{σ}_{\hat{γ}, \hat{β}} \;\;\;\; (41)
is computed using equations (35)(38) and substituting estimated values of β, ω, and σ. Onesided confidence intervals are computed in a similar manner.
This method of constructing confidence intervals is analogous to using the Wald test (e.g., Silvey, 1975, pp.115118) to test hypotheses on the parameters.
Because of the regularity problems associated with the global maximum likelihood estimators, it is questionble whether the asymptotic variances and covariances shown above apply to local maximum likelihood estimators. Simulation studies, however, have shown that these estimates of variance and covariance perform reasonably well (Harter and Moore, 1966; Cohen and Whitten, 1980).
Note that this method of constructing confidence intervals can be used with
estimators other than the lmle's. Cohen and Whitten (1980) and Cohen et al. (1985)
found that the asymptotic variances and covariances are reasonably close to
corresponding simulated variances and covariances for the modified method of moments
estimators (method="mmme"
).
Likelihood Profile (ci.method="likelihood.profile"
)
Griffiths (1980) suggested constructing confidence intervals for the threshold
parameter γ based on the profile likelihood function given in equations
(9) and (10). Royston (1992b) further elaborated upon this procedure. A
twosided (1α)100\% confidence interval for η is constructed as:
[η_{LCL}, η_{UCL}] \;\;\;\; (42)
by finding the two values of η (one larger than the lmle of η and one smaller than the lmle of η) that satisfy:
log[L(η)] = log[L(\hat{η}_{lmle})]  \frac{1}{2} χ^2_{1, α/2} \;\;\;\; (43)
where χ^2_{ν, p} denotes the p'th quantile of the chisquare distribution with ν degrees of freedom. Once these values are found, the twosided confidence for γ is computed as:
[γ_{LCL}, γ_{UCL}] \;\;\;\; (44)
where
γ_{LCL} = x_{(1)}  exp(η_{LCL}) \;\;\;\; (45)
γ_{UCL} = x_{(1)}  exp(η_{UCL}) \;\;\;\; (46)
Onesided intervals are construced in a similar manner.
This method of constructing confidence intervals is analogous to using the likelihoodratio test (e.g., Silvey, 1975, pp.108115) to test hypotheses on the parameters.
To construct a twosided (1α)100\% confidence interval for the median (see equation (34)), Royston (1992b) suggested the following procedure:
Construct a confidence interval for γ using the likelihood profile procedure.
Construct a confidence interval for β as:
[β_{LCL}, β_{UCL}] = [exp(\hat{μ}  t_{n2, 1α/2} \frac{\hat{σ}}{n}), \, exp(\hat{μ} + t_{n2, 1α/2} \frac{\hat{σ}}{n})] \;\;\;\; (47)
Construct the confidence interval for the median as:
[γ_{LCL} + β_{LCL}, γ_{UCL} + β_{UCL}] \;\;\;\; (48)
Royston (1992b) actually suggested using the quantile from the standard normal
distribution instead of Student's tdistribution in step 2 above. The function
elnorm3
, however, uses the Student's t quantile.
Note that this method of constructing confidence intervals can be used with
estimators other than the lmle's.
Royston's Confidence Interval Based on Significant Skewness (ci.method="skewness"
)
Royston (1992b) suggested constructing confidence intervals for the threshold
parameter γ based on the idea behind the zeroskewness estimator
(method="zero.skew"
). A twosided (1α)100\% confidence interval
for γ is constructed by finding the two values of γ that yield
a pvalue of α/2 for the test of zeroskewness on the observations
\underline{y} defined in equation (6) (see gofTest
). Onesided
confidence intervals are constructed in a similar manner.
To construct (1α)100\% confidence intervals for the median
(see equation (34)), the exact same procedure is used as for
ci.method="likelihood.profile"
, except that the confidence interval for
γ is based on the zeroskewness method just described instead of the
likelihood profile method.
a list of class "estimate"
containing the estimated parameters and other information.
See
estimate.object
for details.
The problem of estimating the parameters of a threeparameter lognormal distribution has been extensively discussed by Aitchison and Brown (1957, Chapter 6), Calitz (1973), Cohen (1951), Cohen (1988), Cohen and Whitten (1980), Cohen et al. (1985), Griffiths (1980), Harter and Moore (1966), Hill (1963), and Royston (1992b). Stedinger (1980) and Hoshi et al. (1984) discuss fitting the threeparameter lognormal distribution to hydrologic data.
The global maximum likelihood estimates are inadmissible. In the past, several researchers have found that the local maximum likelihood estimates (lmle's) occasionally fail because of convergence problems, but they were not using the likelihood profile and reparameterization of Griffiths (1980). Cohen (1988) recommends the modified methods of moments estimators over lmle's because they are easy to compute, they are unbiased with respect to μ and σ^2 (the mean and standard deviation on the logscale), their variances are minimal or near minimal, and they do not suffer from regularity problems.
Because the distribution of the lmle of the threshold parameter γ is far
from normal for moderate sample sizes (Griffiths, 1980), it is questionable whether
confidence intervals for γ or the median based on asymptotic variances
and covariances will perform well. Cohen and Whitten (1980) and Cohen et al. (1985),
however, found that the asymptotic variances and covariances are reasonably close to
corresponding simulated variances and covariances for the modified method of moments
estimators (method="mmme"
). In a simulation study (5000 monte carlo trials),
Royston (1992b) found that the coverage of confidence intervals for γ
based on the likelihood profile (ci.method="likelihood.profile"
) was very
close the nominal level (94.1% for a nominal level of 95%), although not
symmetric. Royston (1992b) also found that the coverage of confidence intervals
for γ based on the skewness method (ci.method="skewness"
) was also
very close (95.4%) and symmetric.
Steven P. Millard ([email protected])
Aitchison, J., and J.A.C. Brown (1957). The Lognormal Distribution (with special references to its uses in economics). Cambridge University Press, London, Chapter 5.
Calitz, F. (1973). Maximum Likelihood Estimation of the Parameters of the ThreeParameter Lognormal Distribution–a Reconsideration. Australian Journal of Statistics 15(3), 185–190.
Cohen, A.C. (1951). Estimating Parameters of LogarithmicNormal Distributions by Maximum Likelihood. Journal of the American Statistical Association 46, 206–212.
Cohen, A.C. (1988). ThreeParameter Estimation. In Crow, E.L., and K. Shimizu, eds. Lognormal Distributions: Theory and Applications. Marcel Dekker, New York, Chapter 4.
Cohen, A.C., and B.J. Whitten. (1980). Estimation in the ThreeParameter Lognormal Distribution. Journal of the American Statistical Association 75, 399–404.
Cohen, A.C., B.J. Whitten, and Y. Ding. (1985). Modified Moment Estimation for the ThreeParameter Lognormal Distribution. Journal of Quality Technology 17, 92–99.
Crow, E.L., and K. Shimizu. (1988). Lognormal Distributions: Theory and Applications. Marcel Dekker, New York, Chapter 2.
Griffiths, D.A. (1980). Interval Estimation for the ThreeParameter Lognormal Distribution via the Likelihood Function. Applied Statistics 29, 58–68.
Harter, H.L., and A.H. Moore. (1966). LocalMaximumLikelihood Estimation of the Parameters of ThreeParameter Lognormal Populations from Complete and Censored Samples. Journal of the American Statistical Association 61, 842–851.
Heyde, C.C. (1963). On a Property of the Lognormal Distribution. Journal of the Royal Statistical Society, Series B 25, 392–393.
Hill, .B.M. (1963). The ThreeParameter Lognormal Distribution and Bayesian Analysis of a PointSource Epidemic. Journal of the American Statistical Association 58, 72–84.
Hoshi, K., J.R. Stedinger, and J. Burges. (1984). Estimation of LogNormal Quantiles: Monte Carlo Results and FirstOrder Approximations. Journal of Hydrology 71, 1–30.
Johnson, N. L., S. Kotz, and N. Balakrishnan. (1994). Continuous Univariate Distributions, Volume 1. Second Edition. John Wiley and Sons, New York.
Royston, J.P. (1992b). Estimation, Reference Ranges and Goodness of Fit for the ThreeParameter LogNormal Distribution. Statistics in Medicine 11, 897–912.
Stedinger, J.R. (1980). Fitting Lognormal Distributions to Hydrologic Data. Water Resources Research 16(3), 481–490.
Lognormal3, Lognormal, LognormalAlt, Normal.
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 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168  # Generate 20 observations from a 3parameter lognormal distribution
# with parameters meanlog=1.5, sdlog=1, and threshold=10, then use
# Cohen and Whitten's (1980) modified moments estimators to estimate
# the parameters, and construct a confidence interval for the
# threshold based on the estimated asymptotic variance.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(250)
dat < rlnorm3(20, meanlog = 1.5, sdlog = 1, threshold = 10)
elnorm3(dat, method = "mmme", ci = TRUE)
#Results of Distribution Parameter Estimation
#
#
#Assumed Distribution: 3Parameter Lognormal
#
#Estimated Parameter(s): meanlog = 1.5206664
# sdlog = 0.5330974
# threshold = 9.6620403
#
#Estimation Method: mmme
#
#Data: dat
#
#Sample Size: 20
#
#Confidence Interval for: threshold
#
#Confidence Interval Method: Normal Approximation
# Based on Asymptotic Variance
#
#Confidence Interval Type: twosided
#
#Confidence Level: 95%
#
#Confidence Interval: LCL = 6.985258
# UCL = 12.338823
#
# Repeat the above example using the other methods of estimation
# and compare.
round(elnorm3(dat, "lmle")$parameters, 1)
#meanlog sdlog threshold
# 1.3 0.7 10.5
round(elnorm3(dat, "mme")$parameters, 1)
#meanlog sdlog threshold
# 2.1 0.3 6.0
round(elnorm3(dat, "mmue")$parameters, 1)
#meanlog sdlog threshold
# 2.2 0.3 5.8
round(elnorm3(dat, "mmme")$parameters, 1)
#meanlog sdlog threshold
# 1.5 0.5 9.7
round(elnorm3(dat, "zero.skew")$parameters, 1)
#meanlog sdlog threshold
# 1.3 0.6 10.3
round(elnorm3(dat, "royston")$parameters, 1)
#meanlog sdlog threshold
# 1.4 0.6 10.1
#
# Compare methods for computing a twosided 95% confidence interval
# for the threshold:
# modified method of moments estimator using asymptotic variance,
# lmle using asymptotic variance,
# lmle using likelihood profile, and
# zeroskewness estimator using the skewness method.
elnorm3(dat, method = "mmme", ci = TRUE,
ci.method = "avar")$interval$limits
# LCL UCL
# 6.985258 12.338823
elnorm3(dat, method = "lmle", ci = TRUE,
ci.method = "avar")$interval$limits
# LCL UCL
# 9.017223 11.980107
elnorm3(dat, method = "lmle", ci = TRUE,
ci.method="likelihood.profile")$interval$limits
# LCL UCL
# 3.699989 11.266029
elnorm3(dat, method = "zero.skew", ci = TRUE,
ci.method = "skewness")$interval$limits
# LCL UCL
#25.18851 11.18652
#
# Now construct a confidence interval for the median of the distribution
# based on using the modified method of moments estimator for threshold
# and the asymptotic variances and covariances. Note that the true median
# is given by threshold + exp(meanlog) = 10 + exp(1.5) = 14.48169.
elnorm3(dat, method = "mmme", ci = TRUE, ci.parameter = "median")
#Results of Distribution Parameter Estimation
#
#
#Assumed Distribution: 3Parameter Lognormal
#
#Estimated Parameter(s): meanlog = 1.5206664
# sdlog = 0.5330974
# threshold = 9.6620403
#
#Estimation Method: mmme
#
#Data: dat
#
#Sample Size: 20
#
#Confidence Interval for: median
#
#Confidence Interval Method: Normal Approximation
# Based on Asymptotic Variance
#
#Confidence Interval Type: twosided
#
#Confidence Level: 95%
#
#Confidence Interval: LCL = 11.20541
# UCL = 17.26922
#
# Compare methods for computing a twosided 95% confidence interval
# for the median:
# modified method of moments estimator using asymptotic variance,
# lmle using asymptotic variance,
# lmle using likelihood profile, and
# zeroskewness estimator using the skewness method.
elnorm3(dat, method = "mmme", ci = TRUE, ci.parameter = "median",
ci.method = "avar")$interval$limits
# LCL UCL
#11.20541 17.26922
elnorm3(dat, method = "lmle", ci = TRUE, ci.parameter = "median",
ci.method = "avar")$interval$limits
# LCL UCL
#12.28326 15.87233
elnorm3(dat, method = "lmle", ci = TRUE, ci.parameter = "median",
ci.method = "likelihood.profile")$interval$limits
# LCL UCL
# 6.314583 16.165525
elnorm3(dat, method = "zero.skew", ci = TRUE, ci.parameter = "median",
ci.method = "skewness")$interval$limits
# LCL UCL
#22.38322 16.33569
#
# Clean up
#
rm(dat)

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