SMNGmoments {BayesLN} R Documentation

## SMNG Moments and Moment Generating Function

### Description

Functions that implement the mean, the generic moments (both raw and centered) and the moment generating function of the SMNG distribution.

### Usage

```SMNG_MGF(
r,
mu = 0,
delta,
gamma,
lambda,
beta = 0,
inf_sum = FALSE,
rel_tol = 1e-05
)

meanSMNG(mu, delta, gamma, lambda, beta)

SMNGmoment(j, mu, delta, gamma, lambda, beta, type = "central")
```

### Arguments

 `r` Coefficient of the MGF. Can be viewed also as the order of the logSMNG moments. `mu` Location parameter, default set to 0. `delta` Concentration parameter, must be positive. `gamma` Tail parameter, must be positive. `lambda` Shape parameter. `beta` Skewness parameter, default set to 0 (symmetric case). `inf_sum` Logical: if FALSE (default), the integral representation of the SMNG density is used, otherwise the infinite sum is employed. `rel_tol` Level of relative tolerance required for the `integrate` procedure or for the infinite sum. Default set to `1e-5`. `j` Order of the moment. `type` String that indicate the kind of moment to comupute. Could be `"central"` (default) or `"raw"`.

### Details

If the mean (i.e. the first order raw moment) of the SMNG distribution is required, then the function `meanSMNG` could be use.

On the other hand, to obtain the generic j-th moment both `"raw"` or `"centered"` around the mean, the function `momentSMNG` could be used.

Finally, to evaluate the Moment Generating Function (MGF) of the SMNG distribution in the point `r`, the function `SMNG_MGF` is provided. It is defined only for points that are lower then the parameter `gamma`, and for integer values of `r` it could also considered as the r-th raw moment of the logSMNG distribution. The last function is implemented both in the integral form, which uses the routine `integrate`, or in the infinite sum structure.

### Examples

```
### Comparisons sample quantities vs true values
sample <- rSMNG(n=1000000,mu = 0,delta = 2,gamma = 2,lambda = 1,beta = 2)
mean(sample)
meanSMNG(mu = 0,delta = 2,gamma = 2,lambda = 1,beta = 2)

var(sample)
SMNGmoment(j = 2,mu = 0,delta = 2,gamma = 2,lambda = 1,beta = 2,type = "central")
SMNGmoment(j = 2,mu = 0,delta = 2,gamma = 2,lambda = 1,beta = 2,type = "raw")-
meanSMNG(mu = 0,delta = 2,gamma = 2,lambda = 1,beta = 2)^2

mean(exp(sample))
SMNG_MGF(r = 1,mu = 0,delta = 2,gamma = 2,lambda = 1,beta = 2)

```

[Package BayesLN version 0.2.2 Index]