Lognormal inverse cumulative distribution function

`X = logninv(P,mu,sigma)`

[X,XLO,XUP] = logninv(P,mu,sigma,pcov,alpha)

`X = logninv(P,mu,sigma)`

returns
values at `P`

of the inverse lognormal cdf with distribution
parameters `mu`

and `sigma`

. `mu`

and `sigma`

are
the mean and standard deviation, respectively, of the associated normal
distribution. `mu`

and `sigma`

can
be vectors, matrices, or multidimensional arrays that all have the
same size, which is also the size of `X`

.
A scalar input for `P`

, `mu`

, or `sigma`

is
expanded to a constant array with the same dimensions as the other
inputs.

`[X,XLO,XUP] = logninv(P,mu,sigma,pcov,alpha)`

returns
confidence bounds for `X`

when the input parameters `mu`

and `sigma`

are
estimates. `pcov`

is the covariance matrix of the
estimated parameters. `alpha`

specifies 100(1 - `alpha`

)% confidence
bounds. The default value of `alpha`

is 0.05. `XLO`

and `XUP`

are
arrays of the same size as `X`

containing the lower
and upper confidence bounds.

`logninv`

computes confidence bounds for `P`

using
a normal approximation to the distribution of the estimate

$$\widehat{\mu}+\widehat{\sigma}q$$

where *q* is the `P`

th quantile
from a normal distribution with mean 0 and standard deviation 1. The
computed bounds give approximately the desired confidence level when
you estimate `mu`

, `sigma`

, and `pcov`

from
large samples, but in smaller samples other methods of computing the
confidence bounds might be more accurate.

The lognormal inverse function is defined in terms of the lognormal cdf as

$$x={F}^{-1}(p|\mu ,\sigma )=\{x:F(x|\mu ,\sigma )=p\}$$

where

$$p=F(x|\mu ,\sigma )=\frac{1}{\sigma \sqrt{2\pi}}{\displaystyle {\int}_{0}^{x}\frac{{e}^{\frac{-{(\mathrm{ln}(t)-\mu )}^{2}}{2{\sigma}^{2}}}}{t}}dt$$

[1] Evans, M., N. Hastings, and B. Peacock. *Statistical
Distributions*. Hoboken, NJ: Wiley-Interscience, 2000.
pp. 102–105.

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