The logitnormal distribution is useful as a prior density for variables that are bounded between 0 and 1, such as proportions. The following figure displays its density for various combinations of parameters mu (panels) and sigma (lines).

Example: Plot the cumulative distribution

```
x <- seq(0,1, length.out=81)
d <- plogitnorm(x, mu=0.5, sigma=0.5)
plot(d~x,type="l")
```

The moments have no analytical solution. This package estimates them by numerical integration:

Example: estimate mean and standard deviation.

```
(theta <- momentsLogitnorm(mu=0.6,sigma=0.5))
```

```
## mean var
## 0.63812093 0.01208171
```

The mode is found by setting derivatives to zero and optimizing the resulting equation: \(logit(x) = \sigma^2(2x-1)+\mu\).

Example: estimate the mode

```
(mle <- modeLogitnorm(mu=0.6,sigma=0.5))
```

```
## [1] 0.6641416
```

from upper quantile and

- mode (Maximum Likelihood Estimate)
- mean (Expected value)
- median

Example: estimate the parameters, with mode 0.7 and upper quantile 0.9

```
(theta <- twCoefLogitnormMLE(0.7,0.9))
```

```
## mu sigma
## [1,] 0.7608886 0.464783
```

```
x <- seq(0,1, length.out=81)
d <- dlogitnorm(x, mu=theta[1,"mu"], sigma=theta[1,"sigma"])
plot(d~x,type="l")
abline(v=c(0.7,0.9), col="grey")
```

When increasing the \(\sigma\) parameter, the distribution becomes
eventually becomes bi-model, i.e. has two maxima. The unimodal distribution for
a given mode with widest confidence intervals is obtained by
function `twCoefLogitnormMLEFlat`

.

```
(theta <- twCoefLogitnormMLEFlat(0.7))
```

```
## mu sigma
## [1,] 0.0335529 1.426311
```

```
x <- seq(0,1, length.out=81)
d <- dlogitnorm(x, mu=theta[1,"mu"], sigma=theta[1,"sigma"])
plot(d~x,type="l")
abline(v=c(0.7), col="grey")
```