When dealing with univariate data you want to do one or more of

- Find a good model for the data.
- Estimate parameters for your candidate models.
- Get an idea about the uncertainty of your estimates.

The `unvariateML`

package has a fast and reliable
functions to help you with these tasks. The core of the package are more
than 20 functions for fast and thoroughly tested calculation of maximum
likelihood estimates for univariate models.

- Compare the fit of your candidate models with
`AIC`

or`BIC`

. - Look at QQ plots or PP plots of your data.
- Plot the data together with density estimates.
- Compute confidence intervals using parametric bootstrap.

This vignette shows you how to use the tools of
`univariateML`

to do exploratory data analysis.

## Mortality in Ancient Egypt

The dataset `egypt`

contains contains the age at death of
141 Roman era Egyptian mummies. Our first task is to find a univariate
model that fits this data.

```
library("univariateML")
head(egypt)
```

```
## age sex
## 1 1.50 male
## 2 1.83 male
## 3 2.00 male
## 4 2.00 male
## 5 3.00 male
## 6 3.00 male
```

`hist(egypt$age, main = "Mortality in Ancient Egypt", freq = FALSE)`

### Comparing Many Models with AIC

The AIC
is a handy and easy to use model selection tool, as it only depends on
the log-likelihood and number of parameters of the models. The generic
in `R`

can take multiple models, and the lower the the
better.

Since all the data is positive we will only try densities support on the positive half-line.

```
AIC(
mlbetapr(egypt$age),
mlexp(egypt$age),
mlinvgamma(egypt$age),
mlgamma(egypt$age),
mllnorm(egypt$age),
mlrayleigh(egypt$age),
mlinvgauss(egypt$age),
mlweibull(egypt$age),
mlinvweibull(egypt$age),
mllgamma(egypt$age)
)
```

```
## df AIC
## mlbetapr(egypt$age) 2 1312.464
## mlexp(egypt$age) 1 1249.553
## mlinvgamma(egypt$age) 2 1322.949
## mlgamma(egypt$age) 2 1234.772
## mllnorm(egypt$age) 2 1263.874
## mlrayleigh(egypt$age) 1 1260.217
## mlinvgauss(egypt$age) 2 1287.124
## mlweibull(egypt$age) 2 1230.229
## mlinvweibull(egypt$age) 2 1319.120
## mllgamma(egypt$age) 2 1314.187
```

The Weibull and Gamma models stand out with an AIC far below the other candidate models.

To see the parameter estimates of `mlweibull(egypt$age)`

just print it:

`mlweibull(egypt$age)`

```
## Maximum likelihood estimates for the Weibull model
## shape scale
## 1.404 33.564
```

`mlweibull(egypt$age)`

is a `univariateML`

object. For more details about it call `summary`

:

```
##
## Maximum likelihood for the Weibull model
##
## Call: mlweibull(x = egypt$age)
##
## Estimates:
## shape scale
## 1.404158 33.563564
##
## Data: egypt$age (141 obs.)
## Support: (0, Inf)
## Density: stats::dweibull
## Log-likelihood: -613.1144
```

### Automatically select the best model

The model selection process can be automatized with
`model_select(egypt$age)`

:

`model_select(egypt$age, models = c("gamma", "weibull"))`

```
## Maximum likelihood estimates for the Weibull model
## shape scale
## 1.404 33.564
```

### Quantile-quantile Plots

Now we will investigate how the two models differ with quantile-quantile plots, or Q-Q plots for short.

```
qqmlplot(egypt$age, mlweibull, datax = TRUE, main = "QQ Plot for Ancient Egypt")
# Can also use qqmlplot(mlweibull(egypt$age), datax = TRUE) directly.
qqmlpoints(egypt$age, mlgamma, datax = TRUE, col = "red")
qqmlline(egypt$age, mlweibull, datax = TRUE)
qqmlline(egypt$age, mlgamma, datax = TRUE, col = "red")
```

The Q-Q plot shows that neither Weibull nor Gamma fits the data very well.

If you prefer P-P plots to Q-Q plots take a look at
`?ppplotml`

instead.

### Confidence Intervals with Parametric Bootstrap

Now we want to get an idea about the uncertainties of our model
parameters. Do to this we can do a parametric bootstrap to calculate
confidence intervals using either `bootstrapml`

or
`confint`

. While `bootstrapml`

allows you to
calculate any functional of the parameters and manipulate them
afterwards, `confint`

is restricted to the main parameters of
the model.

```
# Calculate two-sided 95% confidence intervals for the two Gumbel parameters.
bootstrapml(mlweibull(egypt$age)) # same as confint(mlweibull(egypt$age))
```

```
## 2.5% 97.5%
## shape 1.242941 1.625041
## scale 29.812896 38.039005
```

`bootstrapml(mlgamma(egypt$age))`

```
## 2.5% 97.5%
## shape 1.33737775 2.03223380
## rate 0.04120681 0.06822472
```

These confidence intervals are not directly comparable. That is, the
`scale`

parameter in the Weibull model is not directly
comparable to the `rate`

parameter in the gamma model. So let
us take a look at a a parameter with a familiar interpretation, namely
the mean.

The mean of the Weibull distribution with parameters
`shape`

and `scale`

is
`scale*gamma(1 + 1/shape)`

. On the other hand, the mean of
the Gamma distribution with parameters `shape`

and
`rate`

is `shape/rate`

.

The `probs`

argument can be used to modify the limits of
confidence interval. Now we will calculate two 90% confidence intervals
for the mean.

```
# Calculate two-sided 90% confidence intervals for the mean of a Weibull.
bootstrapml(mlweibull(egypt$age),
map = function(x) x[2] * gamma(1 + 1 / x[1]),
probs = c(0.05, 0.95)
)
```

```
## 5% 95%
## 27.66788 33.67035
```

```
# Calculate two-sided 90% confidence intervals for the mean of a Gamma.
bootstrapml(mlgamma(egypt$age),
map = function(x) x[1] / x[2],
probs = c(0.05, 0.95)
)
```

```
## 5% 95%
## 27.23868 33.80332
```

We are be interested in the quantiles of the underlying distribution, for instance the median:

```
# Calculate two-sided 90% confidence intervals for the two Gumbel parameters.
bootstrapml(mlweibull(egypt$age),
map = function(x) qweibull(0.5, x[1], x[2]),
probs = c(0.05, 0.95)
)
```

```
## 5% 95%
## 23.03794 29.20766
```

```
bootstrapml(mlgamma(egypt$age),
map = function(x) qgamma(0.5, x[1], x[2]),
probs = c(0.05, 0.95)
)
```

```
## 5% 95%
## 21.76814 27.52931
```

We can also plot the bootstrap samples.

```
hist(
bootstrapml(mlweibull(egypt$age),
map = function(x) x[2] * gamma(1 + 1 / x[1]),
reducer = identity
),
main = "Bootstrap Samples of the Mean",
xlab = "x",
freq = FALSE
)
```

### Density, CDF, quantiles and random variate generation

The functions `dml`

, `pml`

, `qml`

and `rml`

can be used to calculate densities, cumulative
probabilities, quantiles, and generate random variables. Here are \(10\) random observations from the most
likely distribution of Egyptian mortalities given the Weibull model.

```
## [1] 25.90552 59.64456 13.36882 44.29378 12.22563 17.66144 54.57633 22.86824
## [9] 11.48328 19.94814
```

Compare the empirical distribution of the random variates to the true cumulative probability.