coxbctype1 {bccp} | R Documentation |

## Computing the bias corrected maximum likelihood estimator.

### Description

Computing the bias corrected maximum likelihood estimator (MLE) for the parameters of the general family of distributions under progressive type-I interval censoring scheme. Let `y_1,y_2,\ldots,y_n`

represent the lifetimes of `n`

items that independently follow the cumulative distribution function (cdf) `F(.,\theta_{j})`

under a progressive type-I interval censoring scheme. We use `bctypei`

to compute the bias corrected ML estimator using the method of Cox and Snell (1968). Let `[T_{0}-T_{1})`

,`[T_{1}-T_{2})`

,...,`[T_{m-1}-T_{m})`

show a number of `m`

censoring time intervals, `{\bf{X}}=(X_{1},X_{2},\ldots,X_{m})`

denotes the vector of failed items, and `{\bf{R}}=(R_{1},R_{2},\ldots,R_{m})`

represents the vector of removed alive items in each interval, from `m\geq 1`

. A schematic, given by the following, displays the progressive type-I interval censoring scheme. We note that the sample size `n`

is `n=\sum_{i=1}^{m}X_{i}+\sum_{i=1}^{m}R_{i}`

. Furthermore, `R_i`

can be determined by the pre-specified percentage of the remaining surviving items at `T_i`

or equivalently `R_i=\lfloor P_i X_i\rfloor`

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

. Here, `\lfloor z\rfloor`

denotes the largest integer less than or equal to `z`

.

row | Time interval | `{\bf{X}}` | `{\bf{R}}` |

1 | `[T_{0}, T_{1})` | `X_1` | `R_1` |

2 | `[T_{1}, T_{2})` | `X_2` | `R_2` |

. | . | . | . |

. | . | . | . |

. | . | . | . |

`m-1` | `[T_{m-2}, T_{m-1})` | `X_{m-1}` | `R_{m-1}` |

`m` | `[T_{m-1},T_{m})` | `X_m` | `R_m` |

### Usage

```
coxbctype1(plan, param, mle, cdf.expression = FALSE, pdf.expression = TRUE, cdf, pdf
, lb = 0)
```

### Arguments

`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` |
A vector that contains MLE of the 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's support. That is zero by default. |

### 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

A list of the outputs including: a matric that represents the variance-covariance matrix of the MLE, a matrix that represents the variance-covariance matrix of the bias corrected MLE, a list of three outputs including MLE, bias of MLE, and bias corrected MLE, a list of goodness-of-fit measures consists of Anderson-Darling (`AD`

), Cramer-von Misses (`CVM`

), and Kolmogorov-Smirnov (`KS`

) statistics.

### Author(s)

Mahdi Teimouri

### References

Z. Chen 2000. A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, *Statistics & Probability Letters*, 49 (2), 155-161.

D. G. Chen and Y. L. Lio 2010. Parameter estimations for generalized exponential distribution under progressive
type-I interval censoring, *Computational Statistics and Data Analysis*, 54, 1581-1591.

D. R. Cox and E. J. Snell, 1968. A general definition of residuals. *Journal of the Royal Statistical Society: Series B (Methodological)*, 30(2), 248-265.

M. Teimouri, 2020. Bias corrected maximum likelihood estimators under progressive type-I interval censoring scheme, https://doi.org/10.1080/03610918.2020.1819320

### Examples

```
data(plasma, package="bccp")
plan <- data.frame(T = plasma$upper, X = plasma$X, P = plasma$P, R = plasma$R)
param <- c("lambda","beta")
mle <- c(1.4, 0.05)
pdf <- quote( lambda*(1-exp( -(x*beta)))^(lambda-1)*beta*exp( -(x*beta)) )
cdf <- quote( (1-exp( -(x*beta)))^lambda )
lb <- 0
coxbctype1(plan = plan, param = param, mle = mle, cdf.expression = FALSE, pdf.expression = TRUE,
cdf = cdf, pdf = pdf, lb = lb)
```

*bccp*version 0.5.0 Index]