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Lognormal distribution has abundant applications in various fields. In literature, most inferences on the two parameters of the lognormal distribution are based on Type-I censored sample data. However, exact measurements are not always attainable especially when the observation is below or above the detection limits, and only the numbers of measurements falling into predetermined intervals can be recorded instead. This is the so-called grouped data. In this paper, we will show the existence and uniqueness of the maximum likelihood estimators of the two parameters of the underlying lognormal distribution with Type-I censored data and grouped data. The proof was first established under the case of normal distribution and extended to the lognormal distribution through invariance property. The results are applied to estimate the median and mean of the lognormal population.

Lognormal distribution has been used to model many skewed frequency distributions, especially to model continuous random quantities in medical, physical, chemical, biological, toxicological, economical, and environmental processes.

For example, in medicine, the red cell volume distributions; size distributions of plaques in Alzheimer’s patients; surgical procedure times; survival times of breast and ovarian cancer; all have been modeled by lognormal distribution by various researchers. Tai et al. [

It is also common to apply the lognormal distribution for fatigue life and residual strength of composite materials [

In all these studies, it is critical to estimate the parameters of a lognormal distribution. A random variable follows lognormal distribution

The paper is organized as follows. In Section

In this section, we will first show that the MLEs of the parameters

Before proceed, we present two lemmas. Please refer to the appendix for the proofs of the lemmas.

Assume

Basically, Lemma

Let

Suppose that the observed

We need only to show that the MLEs of parameters

For any given

The Hessian matrix of

Condition (i) is certainly satisfied by Lemma

To this end we should consider each of the three terms in the expression (

Let

To show

Note that (a) is equivalent to

Now denote

In the similar way as the above we can show that the matrix

Finally, let us consider

The Hessian matrix

Under the conditions of Theorem

For each

The same results as in Theorem

Suppose that the number of observable

The result can be proved in the same way as Balakrishnan and Mi [

(a) The same result holds for the case of right censoring; (b) the results of Theorem

Suppose that random variable

Denote the inverse of the matrix

In this section, we will conduct simulation studies on the MLEs and confidence intervals of

We create a population of size

We compute the MLEs of

Type-I censored data are very common in various experiments. It is widely used in life test in order to save test time. Particularly, in environmental data analysis, values are often reported simply as being below detection limit along with the stated detection limit. The data obtained in this way are Type-I left singly censored. To compare the performance of the MLEs based on the grouped data with those obtained from Type-I left singly censored data, we will use

Grouped data:

S.S | ||||||
---|---|---|---|---|---|---|

Average | A.W. | C.R. | Average | A.W. | C.R. | |

30 | 21.992 | 33.374 | 90.9% | 245.490 | 1517.674 | 70.7% |

35 | 21.665 | 31.882 | 91.7% | 229.715 | 2810.490 | 78.8% |

40 | 21.441 | 29.207 | 92.9% | 231.326 | 2356.412 | 81.4% |

50 | 21.211 | 25.697 | 93.4% | 222.684 | 1632.301 | 82.9% |

100 | 20.614 | 17.983 | 94.5% | 199.014 | 647.065 | 85.6% |

Grouped data:

S.S | ||||||
---|---|---|---|---|---|---|

Average | A.W. | C.R. | Average | A.W. | C.R. | |

30 | 21.660 | 34.091 | 92.4% | 245.569 | 1835.284 | 85.4% |

35 | 21.320 | 30.027 | 92.1% | 211.600 | 794.547 | 81.9% |

40 | 21.159 | 27.386 | 93.4% | 205.518 | 763.310 | 84.1% |

50 | 20.884 | 24.588 | 94.4% | 190.732 | 525.019 | 83.3% |

100 | 20.465 | 16.790 | 94.3% | 167.344 | 343.435 | 90.0% |

Grouped data:

S.S | ||||||
---|---|---|---|---|---|---|

Average | A.W. | C.R. | Average | A.W. | C.R. | |

30 | 21.657 | 31.826 | 92.7% | 203.855 | 845.742 | 82.2% |

35 | 21.397 | 29.168 | 93.4% | 189.163 | 679.646 | 83.8% |

40 | 21.201 | 26.989 | 93.1% | 181.097 | 558.176 | 84.6% |

50 | 20.974 | 23.931 | 93.6% | 172.924 | 429.353 | 85.1% |

100 | 20.495 | 16.577 | 94.3% | 159.906 | 274.166 | 89.2% |

Grouped data:

S.S | ||||||
---|---|---|---|---|---|---|

Average | A.W. | C.R. | Average | A.W. | C.R. | |

30 | 21.752 | 32.374 | 92.4% | 176.952 | 610.020 | 80.1% |

35 | 21.381 | 29.635 | 92.5% | 169.389 | 512.877 | 81.2% |

40 | 21.247 | 28.552 | 92.6% | 168.496 | 552.496 | 85.0% |

50 | 21.179 | 24.738 | 93.8% | 166.919 | 373.012 | 83.3% |

100 | 20.507 | 17.132 | 94.3% | 156.027 | 264.165 | 88.8% |

Grouped data:

S.S | ||||||
---|---|---|---|---|---|---|

Average | A.W. | C.R. | Average | A.W. | C.R. | |

30 | 21.688 | 32.743 | 93.1% | 183.834 | 659.789 | 79.8% |

35 | 21.496 | 29.564 | 92.3% | 180.633 | 569.356 | 82.7% |

40 | 21.141 | 27.514 | 92.5% | 175.077 | 554.142 | 84.1% |

50 | 21.093 | 24.082 | 93.1% | 170.522 | 360.508 | 82.7% |

100 | 20.562 | 16.932 | 94.3% | 158.738 | 266.687 | 89.1% |

Grouped data:

S.S | ||||||
---|---|---|---|---|---|---|

Average | A.W. | C.R. | Average | A.W. | C.R. | |

30 | 21.641 | 32.122 | 90.8% | 170.705 | 505.267 | 77.8% |

35 | 21.386 | 30.498 | 92.8% | 167.457 | 517.091 | 80.3% |

40 | 21.250 | 28.960 | 93.5% | 164.214 | 555.646 | 83.9% |

50 | 21.035 | 26.623 | 95.2% | 161.572 | 583.249 | 88.7% |

100 | 20.469 | 17.481 | 94.1% | 154.492 | 271.900 | 88.7% |

Grouped data:

Grouped data:

Grouped data:

Grouped data:

Grouped data:

Grouped data:

From these figures (grouped data: solid line, type I censoring: dotted line), it is easy to see that estimations under both data situations improved dramatically with the increasing sample size. The estimated values are very close to the true values with error less than 0.003% when

Let us consider a sample of 47 observations from the guidance document USEPA [

The normal

To apply the results in Section

Grouped data:

47 | ( | (0.354, 0.709) | (0.455, 0.645) | (0.525, 0.741) |

To prove the lemma, it is sufficient to verify the following three limits:

To see (

To show (

Note that,

Denote

Note that, if

This further implies

If

To verify (

We define function

Notice that for any

Since

In this case, the inequality

For any given

The inequality

From the expression of

We have

It is obvious that

It holds that

From all the above, we conclude that both conditions (i) and (ii) are satisfied and thus the Hessian matrix

Sample size

MLEs of

MLEs of

MLE of median

MLE of mean

The average of estimates from 5000 simulations

The average width of 5000 approximate 95% confidence intervals

The average coverage rate of 5000 approximate 95% confidence intervals.