goftype1 {bccp} | R Documentation |

The goodness-of-fit (GOF) measures consist of Anderson-Darling (`AD`

) and Cram\'eer-von Misses (`CVM`

) statistics for progressive type-I interval censoring scheme are given, respectively, by

`AD=n\sum_{i=1}^{m}\gamma^{2}_{i}\log\left[\frac{A_{i+1}\bigl(1-A_i\bigr)}{A_i\bigl(1-A_{i+1}\bigr)}\right]+2n\sum_{i=1}^{m}\gamma_{i}\log\Bigl(\frac{1-A_{i+1}}{1-A_i}\Bigr)-n\bigl(A_{m+1}-A_1\bigr)`

`-n\log\Bigl(\frac{1-A_{m+1}}{1-A_1}\Bigr)+n\bigl(1-A_{m+1}-\log A_{m+1}\bigr),`

`{CVM}=n\sum_{i=1}^{m}\gamma^{2}_{i}\bigl(A_{i+1}-A_i\bigr)-n\sum_{i=1}^{m}\gamma_{i}\bigl(A^{2}_{i+1}-A^2_i\bigr)+\frac{n}{3}\bigl(A^{3}_{m+1}-A^{3}_{1}\bigr)+\frac{n}{3}\bigl(1-A_{m+1}\bigr)^3,`

where `R_0=0`

, `\gamma_{i}=\bigl(\sum_{j=1}^{i}{X_j}+\sum_{j=1}^{i-1}{R_j}\bigr)/n`

, and `A_i=G\bigl(T_{i-1}\big|\widehat{\Theta}\bigr)`

, for `i=1,\dots,m`

.

```
goftype1(plan, param, mle, cdf.expression = FALSE, pdf.expression = TRUE, cdf, pdf
, lb = 0)
```

`plan` |
Censoring plan for progressive type-I interval censoring scheme. It must be given as a |

`param` |
Vector of the of the family parameter's names. |

`mle` |
Vector of the estimated parameters. |

`cdf.expression` |
Logical. That is |

`pdf.expression` |
Logical. That is |

`cdf` |
Expression of the cumulative distribution function. |

`pdf` |
Expression of the probability density function. |

`lb` |
Lower bound of the family support. That is zero by default. |

We note that for lifetime distributions whose support is the positive semi-axis, i.e., `x>0`

, the cumulative distribution function (cdf) may not be differentiable. In this case, the lower bound of the support of random variable, i.e., `lb`

that is zero by default, must be chosen some positive small value to ensure the differentiability of the cdf. Theoretically, for lifetime distribution, we have `lb`

`=T_{0}=0`

.

A vector of goodness-of-fit measures consist of Anderson-Darling (`AD`

) and Cramer-von Misses (`CVM`

) statistics.

Mahdi Teimouri

M. Teimouri 2020. Bias corrected maximum likelihood estimators under progressive type-I interval censoring scheme, *Communications in Statistics-Simulation and Computation*, https://doi.org/10.10
80/03610918.2020.1819320.

```
data(plasma)
n <- 20
param <- c("alpha","beta")
mle <- c(0.4, 0.05)
cdf <- quote( 1-exp( beta*(1-exp( x^alpha )) ) )
pdf <- quote( exp( beta*(1-exp( x^alpha )) )*( beta*(exp( x^alpha )*(x^(alpha-1)*alpha) )) )
lb <- 0
plan <- rtype1(n = n, P = plasma$P, T = plasma$upper, param = param, mle = mle, cdf.expression
= FALSE, pdf.expression = TRUE, cdf = cdf, pdf = pdf, lb = lb)
goftype1(plan = plan, param = param, mle = mle, cdf.expression=TRUE, pdf.expression = FALSE, cdf =
cdf, pdf = pdf, lb = lb)
```

[Package *bccp* version 0.5.0 Index]