Description Usage Arguments Details Value Author(s) References Examples

Computes the pdf, cdf, quantile, and random numbers, draws the q-q plot, and estimates the parameters of the modified beta `G`

distribution. General form for the probability density function (pdf) of the modified beta `G`

distribution due to Nadarajah et al. (2013) is given by

*f(x,{Θ}) = \frac{{{d^a}g(x-μ,θ ){{≤ft( {G(x-μ,θ )} \right)}^{a - 1}}{{≤ft( {1 - G(x-μ,θ )} \right)}^{b - 1}}}}{{B(a,b){{≤ft[ {1 - ≤ft( {1 - d} \right)G(x-μ,θ )} \right]}^{a + b}}}},*

where *θ* is the baseline family parameter vector. Also, a>0, b>0, d>0, and *μ* are the extra parameters induced to the baseline cumulative distribution function (cdf) `G`

whose pdf is `g`

. The general form for the cdf of the modified beta `G`

distribution is given by

*F(x,{Θ}) = \frac{{\int_0^{\frac{{d\,G(x-μ,θ )}}{{1 - (1 - d)G(x-μ,θ )}}} {{y^{a - 1}}{{≤ft( {1 - y} \right)}^{b - 1}}} dy}}{{B(a,b)}}.*

Here, the baseline `G`

refers to the cdf of famous families such as: Birnbaum-Saunders, Burr type XII, Exponential, Chen, Chisquare, F, Frechet, Gamma, Gompertz, Linear failure rate (lfr), Log-normal, Log-logistic, Lomax, Rayleigh, and Weibull. The parameter vector is *Θ=(a,b,d,θ,μ)* where *θ* is the baseline `G`

family's parameter space. If *θ* consists of the shape and scale parameters, the last component of *θ* is the scale parameter (here, a, b, and d are the first, second, and the third shape parameters). Always, the location parameter *μ* is placed in the last component of *Θ*.

1 2 3 4 5 6 | ```
dmbetag(mydata, g, param, location = TRUE, log=FALSE)
pmbetag(mydata, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
qmbetag(p, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
rmbetag(n, g, param, location = TRUE)
qqmbetag(mydata, g, location = TRUE, method)
mpsmbetag(mydata, g, location = TRUE, method, sig.level)
``` |

`g` |
The name of family's pdf including: " |

`p` |
a vector of value(s) between 0 and 1 at which the quantile needs to be computed. |

`n` |
number of realizations to be generated. |

`mydata` |
Vector of observations. |

`param` |
parameter vector |

`location` |
If |

`log` |
If |

`log.p` |
If |

`lower.tail` |
If |

`method` |
The used method for maximizing the sum of log-spacing function. It will be " |

`sig.level` |
Significance level for the Chi-square goodness-of-fit test. |

It can be shown that the Moran's statistic follows a normal distribution. Also, a chi-square approximation exists for small samples whose mean and variance approximately are m(`log`

(m)+0.57722)-0.5-1/(12m) and m(*π^2*/6-1)-0.5-1/(6m), respectively, with `m=n+1`

, see Cheng and Stephens (1989). So, a hypothesis tesing can be constructed based on a sample of `n`

independent realizations at the given significance level, indicated in above as `sig.level`

.

A vector of the same length as

`mydata`

, giving the pdf values computed at`mydata`

.A vector of the same length as

`mydata`

, giving the cdf values computed at`mydata`

.A vector of the same length as

`p`

, giving the quantile values computed at`p`

.A vector of the same length as

`n`

, giving the random numbers realizations.A sequence of goodness-of-fit statistics such as: Akaike Information Criterion (

`AIC`

), Consistent Akaike Information Criterion (`CAIC`

), Bayesian Information Criterion (`BIC`

), Hannan-Quinn information criterion (`HQIC`

), Cramer-von Misses statistic (`CM`

), Anderson Darling statistic (`AD`

), log-likelihood statistic (`log`

), and Moran's statistic (`M`

). The Kolmogorov-Smirnov (`KS`

) test statistic and corresponding`p-value`

. The Chi-square test statistic, critical upper tail Chi-square distribution, related`p-value`

, and the convergence status.

Mahdi Teimouri

Cheng, R. C. H. and Stephens, M. A. (1989). A goodness-of-fit test using Moran's statistic with estimated parameters, *Biometrika*, 76 (2), 385-392.

Nadarajah, S., Teimouri, M., and Shih, S. H. (2014). Modified beta distributions, *Sankhya*, 76 (1), 19-48.

1 2 3 4 5 6 7 | ```
mydata<-rweibull(100,shape=2,scale=2)+3
dmbetag(mydata, "weibull", c(1,1,1,2,2,3))
pmbetag(mydata, "weibull", c(1,1,1,2,2,3))
qmbetag(runif(100), "weibull", c(1,1,1,2,2,3))
rmbetag(100, "weibull", c(1,1,1,2,2,3))
qqmbetag(mydata, "weibull", TRUE, "Nelder-Mead")
mpsmbetag(mydata, "weibull", TRUE, "Nelder-Mead", 0.05)
``` |

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