fitype2 {bccp} | R Documentation |

## Computing the Fisher information matrix under progressive type-II censoring scheme

### Description

Computes the Fisher information matrix under progressive type-I interval censoring scheme. The Fisher information matrix is given by

`I_{rs}=-E\Bigl(\frac{\partial^2 l(\Theta)}{\partial \theta_r \partial \theta_s}\Bigr),`

where

`l(\Theta)=\log L(\Theta) \propto C \sum_{i=1}^{m} \log f(x_{i:m:n}{{;}}\Theta) + \sum_{i=1}^{m} R_i \log \bigl[1-F(x_{i:m:n}{{;}}\Theta)\bigr],`

in which `F(.;\Theta)`

is the family cumulative distribution function for `\Theta=(\theta_1,\dots,\theta_k)^T`

and `r,s=1,\dots,k`

, and `C=n(n-R_1-1)(n-R_1-R_2-2)\dots (n-R_1-R_2-\dots R_{m-1}-m+1)`

.

### Usage

`fitype2(plan, param, mle, cdf, pdf, lb = 0, ub = Inf, N = 100)`

### Arguments

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

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

`mle` |
Vector of the maximum likelihood estimators. |

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

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

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

`ub` |
Upper bound of the family support. That is |

`N` |
An even integer value indicating the number of subdivisions for applying Simpson's integration method. |

### Details

For some families of 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.

### Value

Matrices that represent the expected and observed Fisher information matrices.

### Author(s)

Mahdi Teimouri

### References

N. Balakrishnan and AHMED Hossain 2007. Inference for the Type II generalized logistic distribution under progressive Type II censoring, *Journal of Statistical Computation and Simulation*, 77(12), 1013-1031.

M. Teimouri and S. Nadarajah 2016. Bias corrected MLEs under progressive type-II censoring scheme, *Journal of Statistical Computation and Simulation*, 86 (14), 2714-2726.

### Examples

```
n <- 20
R <- c(5, rep(0, n-6) )
param <- c("alpha","beta")
mle <- c(2,6)
pdf <- quote( alpha/beta*(x/beta)^(alpha-1)*exp( -(x/beta)^alpha ) )
cdf <- quote( 1-exp( -(x/beta)^alpha ) )
lb <- 0
ub <- Inf
N <- 100
plan <- rtype2(n = n, R = R, param = param, mle = mle, cdf = cdf, lb = lb, ub = ub)
fitype2(plan = plan, param = param, mle = mle, cdf = cdf, pdf = pdf, lb = lb, ub = ub, N = N)
```

*bccp*version 0.5.0 Index]