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
three-parameter lognormal distribution with
parameters meanlog=\(\mu\), sdlog=\(\sigma\), and
threshold=\(\gamma\). 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:
$$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i \;\;\;\; (1)$$
$$s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 \;\;\;\; (2)$$
Note that the sample variance is the unbiased version. Denote the method of
moments estimator of variance by:
$$s^2_m = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2 \;\;\;\; (3)$$
Estimation
Local Maximum Likelihood Estimation (method="lmle")
Hill (1963) showed that the likelihood function approaches infinity as \(\gamma\)
approaches \(x_{(1)}\), so that the global maximum likelihood estimators of
\((\mu, \sigma, \gamma)\) are \((-\infty, \infty, x_{(1)})\), which are
inadmissible, since \(\gamma\) must be smaller than \(x_{(1)}\). Cohen (1951)
suggested using local maximum likelihood estimators (lmle's), derived by equating
partial derivatives of the log-likelihood 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 \(\gamma\) is given by the solution to the
following equation:
$$[\sum_{i=1}^n \frac{1}{w_i}] \, \{\sum_{i=1}^n y_i - \sum_{i=1}^n y_i^2 + \frac{1}{n}[\sum_{i=1}^n y_i]^2 \} - n \sum_{i=1}^n \frac{y_i}{w_i} = 0 \;\;\;\; (4)$$
where
$$w_i = x_i - \hat{\gamma} \;\;\;\; (5)$$
$$y_i = log(x_i - \hat{\gamma}) = log(w_i) \;\;\;\; (6)$$
and that the lmle's of \(\mu\) and \(\sigma\) then follow as:
$$\hat{\mu} = \frac{1}{n} \sum_{i=1}^n y_i = \bar{y} \;\;\;\; (7)$$
$$\hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^n (y_i - \bar{y})^2 \;\;\;\; (8)$$
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 \(\mu\) (equation (7)) and the sample
mean (equation (1)).
On the other hand, Griffiths (1980) showed that for a given value of the threshold
parameter \(\gamma\), the maximized value of the log-likelihood (the
“profile likelihood” for \(\gamma\)) is given by:
$$log[L(\gamma)] = \frac{-n}{2} [1 + log(2\pi) + 2\hat{\mu} + log(\hat{\sigma}^2) ] \;\;\;\; (9)$$
where the estimates of \(\mu\) and \(\sigma\) are defined in equations (7)
and (8), so the lmle of \(\gamma\) reduces to an iterative search over the values
of \(\gamma\). Griffiths (1980) noted that the distribution of the lmle of
\(\gamma\) is far from normal and that \(log[L(\gamma)]\) is not quadratic
near the lmle of \(\gamma\). He suggested a better parameterization based on
$$\eta = -log(x_{(1)} - \gamma) \;\;\;\; (10)$$
Thus, once the lmle of \(\eta\) is found using equations (9) and (10), the lmle of
\(\gamma\) is given by:
$$\hat{\gamma} = x_{(1)} - exp(-\hat{\eta}) \;\;\;\; (11)$$
When method="lmle", the function elnorm3 uses the function
nlminb to search for the minimum of \(-2log[L(\eta)]\), using the
modified method of moments estimator (method="mmme"; see below) as the
starting value for \(\gamma\). Equation (11) is then used to solve for the
lmle of \(\gamma\), and equation (4) is used to “fine tune” the estimated
value of \(\gamma\). The lmle's of \(\mu\) and \(\sigma\) are then computed
using equations (6)-(8).
Method of Moments Estimation (method="mme")
Denote the \(r\)'th sample central moment by:
$$m_r = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^r \;\;\;\; (12)$$
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):
$$\bar{x} = \gamma + \beta \sqrt{\omega} \;\;\;\; (14)$$
$$m_2 = s^2_m = \beta^2 \omega (\omega - 1) \;\;\;\; (15)$$
$$m_3 = \beta^3 \omega^{3/2} (\omega - 1)^2 (\omega + 2) \;\;\;\; (16)$$
where
$$\beta = exp(\mu) \;\;\;\; (17)$$
$$\omega = exp(\sigma^2) \;\;\;\; (18)$$
Combining equations (15) and (16) yields:
$$b_1 = \frac{m_3}{m_2^{3/2}} = (\omega + 2) \sqrt{\omega - 1} \;\;\;\; (19)$$
The quantity on the left-hand side of equation (19) is the usual estimator of
skewness. Solving equation (19) for \(\omega\) yields:
$$\hat{\omega} = (d + h)^{1/3} + (d - h)^{1/3} - 1 \;\;\;\; (20)$$
where
$$d = 1 + \frac{b_1}{2} \;\;\;\; (21)$$
$$h = sqrt{d^2 - 1} \;\;\;\; (22)$$
Using equation (18), the method of moments estimator of \(\sigma\) is then
computed as:
$$\hat{\sigma}^2 = log(\hat{\omega}) \;\;\;\; (23)$$
Combining equations (15) and (17), the method of moments estimator of \(\mu\)
is computed as:
$$\hat{\mu} = \frac{1}{2} log[\frac{s^2_m}{\hat{omega}(\hat{\omega} - 1)}] \;\;\;\; (24)$$
Finally, using equations (14), (17), and (18), the method of moments estimator of
\(\gamma\) is computed as:
$$\bar{x} - exp(\hat{mu} + \frac{\hat{\sigma}^2}{2}) \;\;\;\; (25)$$
There are two major problems with using method of moments estimators for the
three-parameter 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.119-120).
Modified Method of Moments Estimation (method="mmme")
This method of estimation is described by Cohen (1988, pp.125-132). It was
introduced by Cohen and Whitten (1980; their MME-II 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 \(\gamma\) 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:
$$log(x_{(1)} - \gamma) = \mu + \sigma E[Z_{(1,n)}] \;\;\;\; (26)$$
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:
$$x_{(1)} = \gamma + \beta exp\{\sqrt{log(\omega)} \, E[Z_{(i,n)}] \} \;\;\;\; (27)$$
Combining equations (14), (15), (17), (18), and (27) yields the following equation
for the estimate of \(\omega\):
$$\frac{s^2}{[\bar{x} - x_{(1)}]^2} = \frac{\hat{\omega}(\hat{\omega} - 1)}{[\sqrt{\hat{\omega}} - exp\{\sqrt{log(\omega)} \, E[Z_{(i,n)}] \} ]^2} \;\;\;\; (28)$$
After equation (28) is solved for \(\hat{\omega}\), the estimate of \(\sigma\)
is again computed using equation (23), and the estimate of \(\mu\) is computed
using equation (24), where the unbiased estimate of variaince is used in place of
the biased one (just as for method="mmue").
Zero-Skewness 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 \(\gamma\) were
known, then the distribution of:
$$Y = log(X - \gamma) \;\;\;\; (29)$$
is normal, so the skew of \(Y\) is 0. Thus, the threshold parameter \(\gamma\)
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 zero-skewness
estimator of \(\gamma\) is the value that satisfies the following equation:
$$0 = \frac{\frac{1}{n} \sum_{i=1}^n (y_i - \bar{y})^3}{[\frac{1}{n} \sum_{i=1}^n (y_i - \bar{y})^2]^{3/2}} \;\;\;\; (30)$$
where
$$y_i = log(x_i - \hat{\gamma}) \;\;\;\; (31)$$
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 \(\hat{\gamma}\).
Once the value of \(\hat{\gamma}\) has been determined, \(\mu\) and \(\sigma\)
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 Shaprio-Wilk goodness-of-fit test
for normality based on tranforming the data using equation (6) and the zero-skewness
estimator of \(\gamma\) (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
zero-skewness 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 \(\gamma\) that yields a value of
\(q=0\) is given by:
$$\hat{\gamma} = \frac{y_{(1)}y_{(n)} - \tilde{y}^2}{y_{(1)} + y_{(n)} - 2\tilde{y}} \;\;\;\; (33)$$
Again, as for the zero-skewness method, once the value of \(\hat{\gamma}\) has
been determined, \(\mu\) and \(\sigma\) 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 \(\gamma\).
Confidence Intervals
This section explains three different methods for constructing confidence intervals
for the threshold parameter \(\gamma\), or the median of the three-parameter
lognormal distribution, which is given by:
$$Med[X] = \gamma + exp(\mu) = \gamma + \beta \;\;\;\; (34)$$
Normal Approximation Based on Asymptotic Variances and Covariances (ci.method="avar")
Formulas for asymptotic variances and covariances for the three-parameter 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
\(\gamma\) and the median are:
$$Var(\hat{\gamma}) = \sigma^2_{\hat{\gamma}} = \frac{\sigma^2}{n} \, (\frac{\beta^2}{\omega}) H \;\;\;\; (35)$$
$$Var(\hat{\beta}) = \sigma^2_{\hat{\beta}} = \frac{\sigma^2}{n} \, \beta^2 (1 + H) \;\;\;\; (36)$$
$$Cov(\hat{\gamma}, \hat{\beta}) = \sigma_{\hat{\gamma}, \hat{\beta}} = \frac{-\sigma^3}{n} \, (\frac{\beta^2}{\sqrt{\omega}}) H \;\;\;\; (37)$$
where
$$H = [\omega (1 + \sigma^2) - 2\sigma^2 - 1]^{-1} \;\;\;\; (38)$$
A two-sided \((1-\alpha)100\%\) confidence interval for \(\gamma\) is computed as:
$$\hat{\gamma} - t_{n-2, 1-\alpha/2} \hat{\sigma}_{\hat{\gamma}}, \, \hat{\gamma} + t_{n-2, 1-\alpha/2} \hat{\sigma}_{\hat{\gamma}} \;\;\;\; (39)$$
where \(t_{\nu, p}\) denotes the \(p\)'th quantile of
Student's t-distribution with \(n\) degrees of freedom, and the
quantity \(\hat{\sigma}_{\hat{\gamma}}\) is computed using equations (35) and (38)
and substituting estimated values of \(\beta\), \(\omega\), and \(\sigma\).
One-sided confidence intervals are computed in a similar manner.
A two-sided \((1-\alpha)100\%\) confidence interval for the median (see equation
(34) above) is computed as:
$$\hat{\gamma} + \hat{\beta} - t_{n-2, 1-\alpha/2} \hat{\sigma}_{\hat{\gamma} + \hat{\beta}}, \, \hat{\gamma} + \hat{\beta} + t_{n-2, 1-\alpha/2} \hat{\sigma}_{\hat{\gamma} + \hat{\beta}} \;\;\;\; (40)$$
where
$$\hat{\sigma}^2_{\hat{\gamma} + \hat{\beta}} = \hat{\sigma}^2_{\hat{\gamma}} + \hat{\sigma}^2_{\hat{\beta}} + \hat{\sigma}_{\hat{\gamma}, \hat{\beta}} \;\;\;\; (41)$$
is computed using equations (35)-(38) and substituting estimated values of
\(\beta\), \(\omega\), and \(\sigma\). One-sided 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.115-118) 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 \(\gamma\) based on the profile likelihood function given in equations
(9) and (10). Royston (1992b) further elaborated upon this procedure. A
two-sided \((1-\alpha)100\%\) confidence interval for \(\eta\) is constructed as:
$$[\eta_{LCL}, \eta_{UCL}] \;\;\;\; (42)$$
by finding the two values of \(\eta\) (one larger than the lmle of \(\eta\) and
one smaller than the lmle of \(\eta\)) that satisfy:
$$log[L(\eta)] = log[L(\hat{\eta}_{lmle})] - \frac{1}{2} \chi^2_{1, \alpha/2} \;\;\;\; (43)$$
where \(\chi^2_{\nu, p}\) denotes the \(p\)'th quantile of the
chi-square distribution with \(\nu\) degrees of freedom.
Once these values are found, the two-sided confidence for \(\gamma\) is computed as:
$$[\gamma_{LCL}, \gamma_{UCL}] \;\;\;\; (44)$$
where
$$\gamma_{LCL} = x_{(1)} - exp(-\eta_{LCL}) \;\;\;\; (45)$$
$$\gamma_{UCL} = x_{(1)} - exp(-\eta_{UCL}) \;\;\;\; (46)$$
One-sided intervals are construced in a similar manner.
This method of constructing confidence intervals is analogous to using the
likelihood-ratio test (e.g., Silvey, 1975, pp.108-115) to test hypotheses on the
parameters.
To construct a two-sided \((1-\alpha)100\%\) confidence interval for the median
(see equation (34)), Royston (1992b) suggested the following procedure:
Construct a confidence interval for \(\gamma\) using the likelihood
profile procedure.
Construct a confidence interval for \(\beta\) as:
$$[\beta_{LCL}, \beta_{UCL}] = [exp(\hat{\mu} - t_{n-2, 1-\alpha/2} \frac{\hat{\sigma}}{n}), \, exp(\hat{\mu} + t_{n-2, 1-\alpha/2} \frac{\hat{\sigma}}{n})] \;\;\;\; (47)$$
Construct the confidence interval for the median as:
$$[\gamma_{LCL} + \beta_{LCL}, \gamma_{UCL} + \beta_{UCL}] \;\;\;\; (48)$$
Royston (1992b) actually suggested using the quantile from the standard normal
distribution instead of Student's t-distribution 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 \(\gamma\) based on the idea behind the zero-skewness estimator
(method="zero.skew"). A two-sided \((1-\alpha)100\%\) confidence interval
for \(\gamma\) is constructed by finding the two values of \(\gamma\) that yield
a p-value of \(\alpha/2\) for the test of zero-skewness on the observations
\(\underline{y}\) defined in equation (6) (see gofTest). One-sided
confidence intervals are constructed in a similar manner.
To construct \((1-\alpha)100\%\) 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
\(\gamma\) is based on the zero-skewness method just described instead of the
likelihood profile method.