# Documentation

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# logninv

Lognormal inverse cumulative distribution function

## Syntax

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

## Description

`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

`$\stackrel{^}{\mu }+\stackrel{^}{\sigma }q$`

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

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

where

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

## Examples

collapse all

Compute the inverse cdf of a lognormal distribution with `mu = 0` and `sigma = 0.5`.

```p = (0.005:0.01:0.995); crit = logninv(p,1,0.5); ```

Plot the inverse cdf.

```figure; plot(p,crit) xlabel('Probability'); ylabel('Critical Value'); grid ```

## References

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