Three-parameter lognormal distribution

Distribution function and quantile function of the three-parameter lognormal distribution.

cdfln3(x, para = c(0, 0, 1))
qualn3(f, para = c(0, 0, 1))

Arguments

x

Vector of quantiles.

f

Vector of probabilities.

para

Numeric vector containing the parameters of the distribution, in the order \(\zeta, \mu, \sigma\) (lower bound, mean on log scale, standard deviation on log scale).

Details

The three-parameter lognormal distribution with lower bound \(\zeta\), mean on log scale \(\mu\), and standard deviation on log scale \(\sigma\) has distribution function $$F(x)=\Phi(y),$$ \(x>0\), where $$y=\lbrace\log(x - \zeta)-\mu\rbrace/\sigma$$ and \(\Phi(y)\) is the distribution function of the standard normal distribution.

Value

cdfln3 gives the distribution function; qualn3 gives the quantile function.

Note

The functions expect the distribution parameters in a vector, rather than as separate arguments as in the standard R distribution functions pnorm, qnorm, etc.

See also

cdfgno for the generalized normal distribution, a more general form of the three-parameter lognormal distribution.

qlnorm for the standard R version of the two-parameter lognormal distribution.

Examples

# Random sample from three-parameter lognormal distribution
# with parameters zeta=0, mu=1, sigma=0.5.
qualn3(runif(100), c(0,1,0.5))
#>   [1] 1.2004313 0.6669662 2.7041279 1.9211434 1.9576542 2.9030473 1.9449442
#>   [8] 3.2023204 2.1397853 5.5737930 5.2578814 1.3988337 3.7369934 2.9679065
#>  [15] 3.5154372 3.5370909 1.1323696 2.4936568 4.2494891 2.9560231 3.0389716
#>  [22] 4.8952558 3.6342701 5.8283526 2.4870302 2.0497568 3.9372533 3.1523521
#>  [29] 1.6508305 2.3998194 0.6253221 2.4508875 3.8942762 4.1001695 1.9174525
#>  [36] 3.8644490 1.0543633 2.5121592 2.8239867 2.4882962 4.2108067 5.5359761
#>  [43] 3.4106869 1.4072618 1.7764936 2.6361806 3.6078095 2.8513655 4.8360873
#>  [50] 2.0720320 1.8624658 3.4876453 2.1284006 2.6619562 1.5173628 3.4831443
#>  [57] 2.1386302 4.2651010 2.8796654 6.1085520 1.1229687 3.2853841 1.3447129
#>  [64] 2.4981270 3.8751688 2.0725539 1.4616684 1.3140011 2.2610659 2.9747937
#>  [71] 3.4247896 3.4866620 2.3130880 1.4239299 1.4281681 3.9644884 1.6241423
#>  [78] 2.3781759 5.7952850 3.7473176 3.2331335 2.2063439 6.6175208 1.4791505
#>  [85] 3.2979975 7.8946256 1.7171267 1.5765808 1.2310061 2.8813088 2.6064015
#>  [92] 5.1068922 0.9583565 4.5736448 2.7231356 1.6741036 2.2753061 3.3047575
#>  [99] 4.8219571 3.6834445

## Functions for the three-parameter lognormal distribution can
## also be used with the two-parameter lognormal distribution
# Generate a random sample from a standard lognormal distribution
xx <- qualn3(runif(50))
# Fit 2-parameter LN distribution
pelln3(samlmu(xx), bound=0)
#>       zeta         mu      sigma 
#>  0.0000000 -0.1069368  0.8889428 
# Fit 2-parameter LN distribution "in log space",
# i.e. fit normal distribution to log-transformed data
pelnor(samlmu(log(xx)))
#>         mu      sigma 
#> -0.1220053  0.9843600