On the Existence and Uniqueness of the Maximum Likelihood Estimators of Normal and Lognormal Population Parameters with Grouped Data

Abstract

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.

1. Introduction

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. [1] and Mould et al. [2] validated the use of the lognormal model for predicting long-term survival rates of laryngeal cancer patients using short-term follow-up data.

It is also common to apply the lognormal distribution for fatigue life and residual strength of composite materials [3], reliability analysis [4], size distributions in economics and actuarial sciences [5], cell growth [6], and many other phenomena.

In all these studies, it is critical to estimate the parameters of a lognormal distribution. A random variable follows lognormal distribution 𝐿𝑁(𝜇,𝜎) if the logarithm of the random variable follows normal distribution 𝑁(𝜇,𝜎). Thus to estimate the parameters (𝜇,𝜎), it suffices to convert the lognormal data to normal data by log-transformation. In literature, the estimation of these two parameters was considered with complete sample, or in most cases Type-I censored sample. However, estimation with grouped data has not yet been studied. We complement this literature by proposing maximum likelihood estimators (MLEs) of the two parameters that are based on grouped sample data (i.e., interval censored data).

The paper is organized as follows. In Section 2, we will show that the MLEs of the two parameters exist uniquely under mild conditions and thus the asymptotic normality of the estimators. The results are applied to derive the point and confidence interval estimation of the mean and median of the underlying lognormal distribution in Section 2.1. Section 3 provides the simulation results comparing the properties of the estimator based on grouped sample to those of type I censoring. Section 4 contains study results of a practical problem by the above method. To facilitate reading, proofs are relegated to the appendix.

2. Main Results

In this section, we will first show that the MLEs of the parameters 𝜇 and 𝜎 of a normal population 𝑁(𝜇,𝜎²) based on grouped data uniquely exist. Here, the grouped data refers to the following. Assume that a sample 𝑋₁,…,𝑋_𝑛 is drawn from a normal population, the values of 𝑋_𝑗s are unknown; however, according to 𝑘 preestablished partition points 𝜏₁<𝜏₂<⋯<𝜏_𝑘, we know 𝑛_𝑖, the number of 𝑋_𝑗s that fall into the interval [𝜏_𝑖−1,𝜏_𝑖), 1≤𝑖≤𝑘+1 where 𝜏₀≡−∞ and 𝜏_𝑘+1≡∞. Denote the density of the standard normal distribution 𝑁(0,1) as 𝜑(𝑡), then the density of 𝑁(𝜇,𝜎²) distribution is 𝑓(𝑡;𝜇,𝜎)=(1/𝜎)𝜑((𝑡−𝜇)/𝜎), −∞<𝜇<∞, 𝜎>0. In order to prove our results, we consider two new parameters 𝜃₁=𝜇/𝜎 and 𝜃₂=1/𝜎. There is a one-to-one correspondence between (𝜇,𝜎) and (𝜃₁,𝜃₂), namely, 𝜇=𝜃₁/𝜃₂ and 𝜎=1/𝜃₂. We will show that the MLEs of 𝜃₁ and 𝜃₂ based on grouped data uniquely exist. Then due to the invariance property of MLEs, the existence and uniqueness of the MLEs of (𝜇,𝜎) follow. With the new parameters (𝜃₁,𝜃₂), the CDF of 𝑁(𝜇,𝜎²) can be expressed as Φ(𝜃₂𝑡−𝜃₁) where Φ(⋅) is the CDF of the standard normal distribution, and the log-likelihood function ln𝐿 is given by

ln𝐿=𝑐+𝑛₁lnΦ(𝜃₂𝜏₁−𝜃₁)+𝑛_𝑘+1ln[1−Φ(𝜃₂𝜏_𝑘−𝜃₁+)]_𝑘𝑖=2𝑛_𝑖ln[Φ(𝜃₂𝜏_𝑖−𝜃₁)−Φ(𝜃₂𝜏_𝑖−1−𝜃₁)], (2.1) where 𝑐 is a known constant.

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

Lemma 2.1. Assume 𝑛₁+𝑛_𝑘+1<𝑛, 𝑛_𝑗−1+𝑛_𝑗<𝑛, 2≤𝑗≤𝑘+1. For any given 𝜂>0, there exists a compact subset 𝐾≡𝐾(𝜂)⊂(−∞,∞)×(0,∞) such that {(𝜃₁,𝜃₂)∶ln𝐿(𝜃₁,𝜃₂)≥−𝜂}⊂𝐾. (2.2)

Basically, Lemma 2.1 means that the log-likelihood function ln𝐿(𝜃₁,𝜃₂) will not achieve its maximum value at the boundary of its domain.

Lemma 2.2. Let 𝑔(𝑢,𝑣)≡ln(Φ(𝑢)−Φ(𝑣)) for 𝑣<𝑢. Then the Hessian matrix 𝐻^∗ of 𝑔(𝑢,𝑣), 𝐻^∗=𝜕²𝑔𝜕𝑢²𝜕²𝑔𝜕𝜕𝑢𝜕𝑣₂𝑔𝜕𝜕𝑢𝜕𝑣²𝑔𝜕𝑣², (2.3) is negative definite.

Theorem 2.3. Suppose that the observed 𝑛₁,…,𝑛_𝑘+1 satisfy 𝑛₁+𝑛_𝑘+1<𝑛 and 𝑛_𝑗−1+𝑛_𝑗<𝑛,∀2≤𝑗≤𝑘+1, then the MLEs of parameters 𝜇 and 𝜎 of normal population 𝑁(𝜇,𝜎²) uniquely exist.

Proof. We need only to show that the MLEs of parameters 𝜃₁ and 𝜃₂ uniquely exist. According to the results of Mäkeläinen et al. [7], in order to show the existence and uniqueness of the MLEs of (𝜃₁,𝜃₂), it is sufficient to verify the following two conditions.(i)For any given 𝜂>0, (2.2) holds.(ii)The Hessian matrix of ln𝐿, 𝐻(𝜃₁,𝜃₂𝜕)=(²ln𝐿𝜕𝜃_𝑖𝜕𝜃_𝑗), (2.4) is negative definite at every point (𝜃₁,𝜃₂)∈(−∞,∞)×(0,∞).
Condition (i) is certainly satisfied by Lemma 2.1. Therefore, to prove the theorem, we need only to show (ii), that is, the log-likelihood function ln𝐿 is negative definite function of 𝜽=(𝜃₁,𝜃₂)∈(−∞,∞)×(0,∞). (2.5)
To this end we should consider each of the three terms in the expression (2.1) of ln𝐿(𝜽).
Let 𝑔₁(𝜽)≡lnΦ(𝜃₂𝜏₁−𝜃₁). It is evident that the Hessian matrix of 𝑔₁ is 𝐻₁≡𝜕²𝑔₁𝜕𝜃²₁𝜕²𝑔₁𝜕𝜃₁𝜕𝜃₂𝜕²𝑔₁𝜕𝜃₁𝜕𝜃₂𝜕²𝑔₁𝜕𝜃²₂=𝜑^′(𝜃₂𝜏₁−𝜃₁)Φ(𝜃₂𝜏₁−𝜃₁)−𝒬Φ(𝜃₂𝜏₁−𝜃₁)²−𝜏₁𝜑^′(𝜃₂𝜏₁−𝜃₁)Φ(𝜃₂𝜏₁−𝜃₁)−𝒬Φ(𝜃₂𝜏₁−𝜃₁)²−𝜏₁𝜑^′(𝜃₂𝜏₁−𝜃₁)Φ(𝜃₂𝜏₁−𝜃₁)−𝒬Φ(𝜃₂𝜏₁−𝜃₁)²𝜏²₁𝜑^′(𝜃₂𝜏₁−𝜃₁)Φ(𝜃₂𝜏₁−𝜃₁)−𝒬Φ(𝜃₂𝜏₁−𝜃₁)², (2.6) where 𝒬 denotes 𝜑²(𝜃₂𝜏₁−𝜃₁).
To show 𝐻₁ is negative semidefinite, we will verify the following two conditions: (a) 𝜕²𝑔/𝜕𝜃²₁<0 or 𝜕²𝑔/𝜕𝜃²₂<0, ∀(𝜃₁,𝜃₂)∈(−∞,∞)×(0,∞); (b) the determinant of 𝐻₁ is nonnegative, that is, |𝐻₁|≥0.
Note that (a) is equivalent to −(𝜃₂𝜏₁−𝜃₁)Φ(𝜃₂𝜏₁−𝜃₁)−𝜑(𝜃₂𝜏₁−𝜃₁)<0, ∀(𝜃₁,𝜃₂)∈(−∞,∞)×(0,∞). This is true since 𝑦[1−Φ(𝑦)]<𝜑(𝑦) holds for any 𝑦 (see, e.g., Feller [8]). Hence (a) is satisfied. The two rows of 𝐻₁ are proportional, so |𝐻₁|=0. Hence, the condition (b) is satisfied. Therefore, 𝐻₁ is negative semidefinite.
Now denote 𝑔_𝑘+1(𝜽)≡ln[1−Φ(𝜃₂𝜏_𝑘−𝜃₁)]. The Hessian matrix 𝐻_𝑘+1 of 𝑔_𝑘+1 is 𝐻_𝑘+1≡−𝜑^′(𝜃₂𝜏_𝑘−𝜃₁)[1−Φ(𝜃₂𝜏_𝑘−𝜃₁)]+𝜑²(𝜃₂𝜏_𝑘−𝜃₁)[1−Φ(𝜃₂𝜏₁−𝜃₁)]²𝜏_𝑘𝜑^′(𝜃₂𝜏_𝑘−𝜃₁)ℱ[1−Φ(𝜃₂𝜏_𝑘−𝜃₁)]²𝜏_𝑘𝜑^′(𝜃₂𝜏_𝑘−𝜃₁)ℱ[1−Φ(𝜃₂𝜏_𝑘−𝜃₁)]²−𝜏²_𝑘𝜑^′(𝜃₂𝜏_𝑘−𝜃₁)ℱ[1−Φ(𝜃₂𝜏_𝑘−𝜃₁)]², (2.7) where ℱ denotes [1−Φ(𝜃₂𝜏_𝑘−𝜃₁)]²+𝜑²(𝜃₂𝜏_𝑘−𝜃₁).
In the similar way as the above we can show that the matrix 𝐻_𝑘+1 is negative semidefinite.
Finally, let us consider ℎ(𝜃₁,𝜃₂)≡ln[Φ(𝜃₂𝜏_𝑖−𝜃₁)−Φ(𝜃₂𝜏_𝑖−1−𝜃₁)], 2≤𝑖≤𝑘. Let 𝑢=𝜃₂𝜏_𝑖−𝜃₁,𝑣=𝜃₂𝜏_𝑖−1−𝜃₁. Then ℎ(𝜃₁,𝜃₂)=ln[Φ(𝑢)−Φ(𝑣)]≡𝑔(𝑢,𝑣). The Hessian matrix 𝐻_𝑖 associated with ℎ(𝜃₁,𝜃₂) is 𝐻_𝑖=𝜕²ℎ𝜕𝜃²₁𝜕²ℎ𝜕𝜃₁𝜕𝜃₂𝜕²ℎ𝜕𝜃₁𝜕𝜃₂𝜕²ℎ𝜕𝜃²₂=𝜏−1−1_𝑖𝜏_𝑖−1𝜕²𝑔𝜕𝑢²𝜕²𝑔𝜕𝜕𝑢𝜕𝑣₂𝑔𝜕𝜕𝑢𝜕𝑣²𝑔𝜕𝑣²−1𝜏_𝑖−1𝜏_𝑖−1=𝐴^′𝐻^∗𝐴, (2.8) where 𝐻^∗=𝜕²𝑔𝜕𝑢²𝜕²𝑔𝜕𝜕𝑢𝜕𝑣₂𝑔𝜕𝜕𝑢𝜕𝑣²𝑔𝜕𝑣²,𝐴=−1𝜏_𝑖−1𝜏_𝑖−1, (2.9) and 𝐴^′ is the transpose of 𝐴. By Lemma 2.2, 𝐻^∗ is negative definite. Therefore, 𝐻_𝑖 is negative definite.
The Hessian matrix 𝐻 of the log-likelihood function ln𝐿(𝜽) can be expressed as 𝐻=𝑛₁𝐻₁+𝑛_𝑘+1𝐻_𝑘+1+∑^𝑘_𝑖=2𝑛_𝑖𝐻_𝑖. Since matrices 𝐻₁ and 𝐻_𝑘+1 are negative semidefinite, each 𝐻_𝑖(2≤𝑖≤𝑘) is negative definite, and at least one 𝑛_𝑖>0 by our assumptions, so 𝐻 must be negative definite. This completes the proof of the theorem.

Corollary 2.4. Under the conditions of Theorem 2.3, it holds that as 𝑛→∞, 𝜇_𝑛𝜎_𝑛−𝜇𝜎^𝐿−−−−→𝑁(𝟎,𝐼⁻¹(𝜇,𝜎)), (2.10) where →−^𝐿 means “converges in law,’’ and 𝜕𝐼(𝜇,𝜎)=−𝐸²ln𝐿𝜕𝜇²𝜕²ln𝐿𝜕𝜕𝜇𝜕𝜎₂ln𝐿𝜕𝜕𝜇𝜕𝜎²ln𝐿𝜕𝜎². (2.11)

Proof. For each 𝑛>𝑘, define 𝐴_𝑛≡{𝑛₁+𝑛_𝑘+1=𝑛,𝑛_𝑖−1+𝑛_𝑖=𝑛,1≤𝑖≤𝑘}. Note that 𝑃(limsup_𝑛→∞𝐴_𝑛)=0. Hence the result follows from Theorem 2.3 and the asymptotic normality of MLE (see, e.g., Lawless (2003).
The same results as in Theorem 2.3 and Corollary 2.4 also hold for the case of Type-I censored data. Let 𝑋₁,…,𝑋_𝑛 be a sample from an 𝑁(𝜇,𝜎²) population. Suppose that 𝜏 is a predetermined detection limit. Without loss of generality, we will consider left censoring, the common situation in environmental studies, that is, 𝑋_𝑗 will be observed if and only if 𝑋_𝑗≥𝜏. Even though Type-I is widely applied in literature, but according to the authors' knowledge, the existence and uniqueness of the MLEs of (𝜇,𝜎) have not been proved. This will be shown in the following theorem.

Theorem 2.5. Suppose that the number of observable 𝑋_𝑗s is at least 2, then the MLEs of (𝜇,𝜎) uniquely exist based on the Type-I censored data with 𝜏 as detection limit.

Proof. The result can be proved in the same way as Balakrishnan and Mi [9].

Remark 2.6. (a) The same result holds for the case of right censoring; (b) the results of Theorem 2.5 are true if each 𝑋_𝑗 is censored by detection limit 𝜏_𝑗(1≤𝑗≤𝑛).

2.1. Estimation of the Median and Mean

Suppose that random variable 𝑌 follows lognormal distribution 𝐿𝑁(𝜇,𝜎²). With log-transformation then 𝑋=ln𝑌 follows normal distribution 𝑁(𝜇,𝜎²). Lognormal distribution has been used to model various continuous random variables as mentioned in Section 1. Specifically, this distribution is frequently applied in environmental statistics. The lognormal random variable 𝑌 has median 𝑚≡exp{𝜇} and mean 𝜈≡𝐸(𝑌)=exp{𝜇+𝜎²/2}. The MLEs of 𝑚 and 𝜈 can easily be obtained as 𝑚=exp{𝜇} and 𝜈=exp{𝜇+𝜎²/2} due to the invariance property of MLE. We can also obtain approximate confidence intervals for 𝑚 and 𝜈 as follows.

Denote the inverse of the matrix 𝐼(𝜇,𝜎) in Corollary 2.4 to Theorem 2.3 as 𝐼⁻¹𝛽(𝜇,𝜎)=¹¹𝛽₁₂𝛽₂₁𝛽₂₂. (2.12) It is obvious that 𝜇_𝑛−𝜇→−^𝐿𝑁(0,𝛽₁₁), by large sample theory we have exp{𝜇_𝑛}−exp{𝜇}→−^𝐿𝑁(0,(exp{𝜇}²)𝛽₁₁). From these, an approximate (1−𝛼)100% confidence interval of 𝑚 can be obtained as exp{𝜇_𝑛}±𝑧_𝛼/2(exp{𝜇_𝑛}𝛽₁₁), here 𝑧_𝛼/2 is the upper 𝛼/2 percentile, and 𝛽₁₁ is obtained from substituting 𝜇_𝑛 and 𝜎 for 𝜇 and 𝜎 in the expression of 𝛽₁₁. Similarly, it holds that as 𝑛→∞𝑒𝜇_𝑛+𝜎_𝑛^2/2−𝑒^{𝜇+𝜎2𝐿/2}−−−−→𝑁(𝟎,𝜏²), (2.13) where 𝜏²=(𝑒^𝜇+𝜎2/2,𝜎𝑒^𝜇+𝜎2/2)𝐼⁻¹𝑒(𝜇,𝜎)^𝜇+𝜎2/2𝜎𝑒^𝜇+𝜎2/2. (2.14) Therefore, an approximate (1−𝛼)100% confidence interval of 𝜈=exp{𝜇+𝜎²/2} is obtained as 𝑒𝜇_𝑛+𝜎_𝑛^2/2±𝑧_𝛼/2𝜏, (2.15) where 𝜏 is obtained by substituting 𝜇 and 𝜎 by their MLEs 𝜇_𝑛 and 𝜎_𝑛.

3. Simulation Studies

In this section, we will conduct simulation studies on the MLEs and confidence intervals of 𝜇 and 𝜎 of normal distribution 𝑁(𝜇,𝜎²) based on grouped data. In addition, we will also examine point and interval estimations of the mean and median of lognormal distribution 𝐿𝑁(𝜇,𝜎²). The results obtained from grouped data will be compared with those obtained from Type-I censored data.

We create a population of size 𝑛 by drawing 𝑛 values from a normal population with 𝜇=3 and 𝜎=2. Next, for a prefixed five partition points 𝜏_𝑖,1≤𝑖≤5, we record the number of this population that fall into each interval [𝜏_𝑖−1,𝜏_𝑖). Each such samples are consider to be observed sample. The MLEs of 𝜇 and 𝜎 are then computed based on this observed sample. This process is repeated 5,000 times. Different sample size and 6 sets of partition points are considered for comparisons purpose.

We compute the MLEs of 𝜃₁=𝜇/𝜎 and 𝜃₂=1/𝜎 by solving the likelihood equations 𝜕ln𝐿𝜕𝜃₁=0,𝜕ln𝐿𝜕𝜃₂=0, (3.1) using SAS IMSL nonlinear equation solver. Then the MLEs 𝜇=𝜇_𝑛 and 𝜎=𝜎_𝑛 of 𝜇 and 𝜎 are readily obtained by the invariance of MLE. According to the large sample properties of MLEs stated in Corollary 2.4 to Theorem 2.3, we know that (𝜇_𝑛,𝜎_𝑛) is asymptotically normally distributed. Thus we can obtain approximate confidence intervals for 𝜇 and 𝜎.

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 𝜏₁ as the “detection limit’’. Figures 1, 2, 3, 4, 5, and 6 present the estimated MLEs of 𝜇 and 𝜎 under six different partition sets {𝜏₁<𝜏₂<𝜏₃<𝜏₄<𝜏₅} with 𝑛 ranges from 15 to 215. The results of median and mean of the lognormal population are listed in Tables 1, 2, 3, 4, 5, and 6.

Table 1 

Grouped data: 𝜏₁=2, 𝜏₂=2.5, 𝜏₃=3, 𝜏₄=3.5, 𝜏₅=4.

Table 2 

Grouped data: 𝜏₁=1.5, 𝜏₂=2.5, 𝜏₃=3.5, 𝜏₄=4.5, 𝜏₅=5.5.

Table 3 

Grouped data: 𝜏₁=0, 𝜏₂=1.5, 𝜏₃=3, 𝜏₄=4.5, 𝜏₅=6.

Table 4 

Grouped data: 𝜏₁=−1.5, 𝜏₂=1, 𝜏₃=3.5, 𝜏₄=6, 𝜏₅=8.5.

Table 5 

Grouped data: 𝜏₁=−2, 𝜏₂=0, 𝜏₃=2, 𝜏₄=4, 𝜏₅=6.

Table 6 

Grouped data: 𝜏₁=−3.5, 𝜏₂=−0.5, 𝜏₃=2.5, 𝜏₄=5.5, 𝜏₅=8.5.

(a)

(b)

Figure 1 

Grouped data: 𝜏1=2, 𝜏2=2.5, 𝜏3=3, 𝜏4=3.5, 𝜏5=4. Type I left censored data: 𝜏=2.

(a)

(b)

Figure 2 

Grouped data: 𝜏1=1.5, 𝜏2=2.5, 𝜏3=3.5, 𝜏4=4.5, 𝜏5=5.5. Type I left censored data: 𝜏=1.5.

(a)

(b)

Figure 3 

Grouped data: 𝜏1=0, 𝜏2=1.5, 𝜏3=3, 𝜏4=4.5, 𝜏5=6. Type I left censored data: 𝜏=0.

(a)

(b)

Figure 4 

Grouped data: 𝜏1=−1.5, 𝜏2=1, 𝜏3=3.5, 𝜏4=6, 𝜏5=8.5. Type I left censored data: 𝜏=−1.5.

(a)

(b)

Figure 5 

Grouped data: 𝜏1=−2, 𝜏2=0, 𝜏3=2, 𝜏4=4, 𝜏5=6. Type I left censored data: 𝜏=−2.

(a)

(b)

Figure 6 

Grouped data: 𝜏1=−3.5, 𝜏2=−0.5, 𝜏3=2.5, 𝜏4=5.5, 𝜏5=8.5. Type I left censored data: 𝜏=−3.5.

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 𝑛>30. The choice of 𝜏's does not seem to affect the result much except in Figure 6, where 𝜏₁=−3.5,𝜏₂=−0.5,𝜏₃=2.5,𝜏₄=5.5,𝜏₅=8.5, an interval which most samples will be observed in the middle and few on the either side. From those figures, it is not hard to see that the estimation with grouped data are uniformly better than those based on type I censoring data, especially in the estimation of 𝜎, with exception in few isolated cases. Moreover, it is interesting to observe how the 𝜇 and 𝜎 approach the true value differently with 𝜇 taking the oscillated routine and 𝜎 tends to be consistently underestimated.

4. An Application

Let us consider a sample of 47 observations from the guidance document USEPA [10, pages 6.22–6.25]. The data describe the measures of 1,2,3,4-Tetrachlorobenzene (TcCB) concentrations (in parts per billion, usually abbreviated ppb) from soil samples at a “Reference’’ site.

The normal Q-Q plot for the log-transformed TcCB data shown in the book of Millard and Neerchal (2001) indicates that the lognormal distribution appears to provide a good fit to the original data. The book gives 𝜈(𝑐)=0.60 as the MLE of the mean of the lognormal distribution, and CI(𝑐)=[0.51,0.68] as an approximate 95% confidence interval for 𝜈 based on the complete sample data with the 47 observations. The book also uses 0.5 as the detection limit, that is, any observation lower than 0.5 will be censored, which yields 19 censored observations and 28 uncensored observations. The censored data then give 𝜈(𝐼)=0.606 as the MLE of 𝜈 and CI(𝐼)=[0.51,0.73] as an approximate 95% confidence interval for 𝜈.

To apply the results in Section 2 for computing the MLEs of the parameters of this lognormal distribution, we first transform the original data to their logarithms and thus the log-transformed data constitute a sample from a normal distribution, then obtain 𝑛₁=19, 𝑛₂=5, 𝑛₃=7, 𝑛₄=6, 𝑛₅=5, 𝑛₆=5 by using the following five partition points 𝜏₁=−0.71,𝜏₂=−0.61,𝜏₃=−0.41,𝜏₄=−0.21,𝜏₅=0.11. Solving the corresponding log-likelihood equations gives 𝜇(𝑔)=−0.599, 𝜎(𝑔)=0.532, 𝑚(𝑔)=0.549, and 𝜈(𝑔)=0.633. Approximate 95% confidence intervals for 𝜇,𝜎,𝑚, and 𝜈 are given in Table 7.

Table 7 

Grouped data: 𝜏₁=−0.71,𝜏₂=−0.61,𝜏₃=−0.41,𝜏₄=−0.21,𝜏₅=0.11.

Appendix

Proof of Lemma 2.1. To prove the lemma, it is sufficient to verify the following three limits: lim_𝜃2→0+sup_{−∞<𝜃1<∞}ln𝐿(𝜃₁,𝜃₂)=−∞;lim_𝜃2→∞sup_{−∞<𝜃1<∞}ln𝐿(𝜃₁,𝜃₂)=−∞;lim_{|𝜃1|→∞}sup_𝜃2>0ln𝐿(𝜃₁,𝜃₂)=−∞. (A.1)(A.2)(A.3)
To see (A.1), from the assumption 𝑛₁+𝑛_𝑘+1<𝑛, there exists an index, say 𝑖, such that 2≤𝑖≤𝑘 and 𝑛_𝑖>0. We have ln𝐿(𝜃₁,𝜃₂)≤𝑛_𝑖ln∫^{𝜃2𝜏𝑖−𝜃1}_{𝜃2𝜏𝑖−1−𝜃1}𝜑(𝑡)𝑑𝑡≤𝑛_𝑖ln[𝜃₂(𝜏_𝑖−𝜏_𝑖−1)𝜑(0)]. So sup_{−∞<𝜃1<∞}ln𝐿(𝜃₁,𝜃₂)≤𝑛_𝑖[ln𝜑(0)+ln(𝜏_𝑖−𝜏_𝑖−1)+ln𝜃₂] and limsup_𝜃2→0+sup_{−∞<𝜃1<∞}ln𝐿(𝜃₁,𝜃₂)≤lim_𝜃2→0+𝑛_𝑖[ln𝜑(0)+ln(𝜏_𝑖−𝜏_𝑖−1)+ln𝜃₂]=−∞. Therefore, (A.1) holds.
To show (A.2), we denote 𝐼≡{1≤𝑗≤𝑘+1,𝑛_𝑗>0}. For each fixed 𝜃₂>0, it is evident that ln𝐿(𝜃₁,𝜃₂)=∑_𝑖∈𝐼𝑛_𝑖ln∫^{𝜃2𝜏𝑖−𝜃1}_{𝜃2𝜏𝑖−1−𝜃1}𝜑(𝑡)𝑑𝑡≡𝑀(𝜃₂). Thus sup_{−∞<𝜃1<∞}ln𝐿(𝜃₁,𝜃₂)=sup_{−∞<𝜃1<∞}𝑀(𝜃₂).
Note that, lim_{|𝜃1|→∞}𝑀(𝜃₁)=−∞, so there exists 𝜃^∗₁=𝜃^∗₁(𝜃₂)∈(−∞,∞), such that sup_{−∞<𝜃1<∞}ln𝐿(𝜃₁,𝜃₂)=ln𝐿(𝜃^∗₁,𝜃₂)=∑_𝑖∈𝐼𝑛_𝑖ln∫^{𝜃2𝜏𝑖−𝜃∗1}_{𝜃2𝜏𝑖−1−𝜃∗1}𝜑(𝑡)𝑑𝑡. Consider function 𝑔(𝑥)≡|𝑥|exp(𝑥²/2). For any given large number 𝐴>0, it is easy to see that there exists 𝑥₀>0 such that 𝑔(𝑥)>√2𝜋exp(𝑥²/2),∀|𝑥|>𝑥₀.
Denote 𝑐≡min_{1≤𝑗≤𝑘+1}(𝜏_𝑗−𝜏_𝑗−1)>0. For any 𝜃₂>𝑥₀/𝑐, from our assumptions there exists an index, say 𝑖, belonging to 𝐼 satisfying (a) 𝑛_𝑖>0; (b) the following two quantities 𝜃₂𝜏_𝑖−1−𝜃^∗₁=𝜃₂(𝜏_𝑖−1−𝜃^∗₁/𝜃₂) and 𝜃₂𝜏_𝑖−𝜃^∗₁=𝜃₂(𝜏_𝑖−𝜃^∗₁/𝜃₂) have the same sign; and (c) |𝜏_𝑖−1−𝜃^∗₁/𝜃₂|>𝑐, and |𝜏_𝑖−𝜃^∗₁/𝜃₂|>𝑐.
Note that, if 𝑖∈𝐼, and both 𝜃₂𝜏_𝑖−1−𝜃^∗₁>0 and 𝜃₂𝜏_𝑖−𝜃^∗₁>0, then ^{𝜃2𝜏𝑖−𝜃∗1}_{𝜃2𝜏𝑖−1−𝜃∗1}1𝜑(𝑡)𝑑𝑡<𝜃₂𝜏_𝑖−1−𝜃^∗₁^{𝜃2𝜏𝑖−𝜃∗1}_{𝜃2𝜏𝑖−1−𝜃}^∗₁=1𝑡𝜑(𝑡)𝑑𝑡𝜃₂𝜏_𝑖−1−𝜃^∗₁[𝜑(𝜃₂𝜏_𝑖−1−𝜃^∗₁)−𝜑(𝜃₂𝜏_𝑖−𝜃^∗₁<)]𝜑(𝜃₂𝜏_𝑖−1−𝜃^∗₁)𝜃₂𝜏_𝑖−1−𝜃^∗₁=1√2𝜋𝑔(𝜃₂𝜏_𝑖−1−𝜃^∗₁). (A.4) If 𝜃₂>𝑥₀/𝑐, then 𝜃₂𝜏_𝑖−1−𝜃^∗₁=𝜃₂(𝜏_𝑖−1−𝜃^∗₁/𝜃₂)>(𝑥₀/𝑐)𝑐=𝑥₀ and so 𝑔(𝜃₂𝜏_𝑖−1−𝜃^∗₁)>√2𝜋exp𝐴. Consequently, 𝑛_𝑖ln^{𝜃2𝜏𝑖−𝜃∗1}_{𝜃2𝜏𝑖−1−𝜃∗1}𝜑(𝑡)𝑑𝑡<𝑛_𝑖1ln√2𝜋exp(𝐴)=𝑛_𝑖(−ln√2𝜋−𝐴)<−𝑛_𝑖𝐴<−𝐴. (A.5)
This further implies sup_{−∞<𝜃1<∞}ln𝐿(𝜃₁,𝜃₂)=ln𝐿(𝜃^∗₁,𝜃₂)=_𝑗∈𝐼𝑛_𝑗ln^{𝜃2𝜏𝑗−𝜃∗1}_{𝜃2𝜏𝑗−1−𝜃∗1}𝜑(𝑡)𝑑𝑡<𝑛_𝑖ln^{𝜃2𝜏𝑖−𝜃∗1}_{𝜃2𝜏𝑖−1−𝜃∗1}𝜑(𝑡)𝑑𝑡<−𝐴,∀𝜃₂>𝑥₀𝑐. (A.6)
If 𝑖∈𝐼, but both 𝜏_𝑖−1−𝜃^∗₁/𝜃₂<−𝑐 and 𝜏_𝑖−𝜃^∗₁/𝜃₂<−𝑐, then similarly, it can be shown that (A.6) is true again. Therefore, we see that for any given large number 𝐴>0, it holds that sup_{−∞<𝜃1<∞}ln𝐿(𝜃₁,𝜃₂)<−𝐴,∀𝜃₂>𝑥₀𝑐. (A.7) Due to the arbitrariness of 𝐴>0, we conclude that (A.1) is true.
To verify (A.3), we let 𝜃₂𝜏_𝑘+1−𝜃₁=∞ and 𝜃₂𝜏₀−𝜃₁=−∞ for any (𝜃₁,𝜃₂)∈(−∞,∞)×(0,∞). For any fixed 𝜃₁∈(−∞,∞), we have ln𝐿(𝜃₁,𝜃₂)=_𝑗∈𝐼𝑛_𝑗ln^{𝜃2𝜏𝑗−𝜃1}_{𝜃2𝜏𝑗−1−𝜃1}𝜑(𝑡)𝑑𝑡≡𝑀(𝜃₂). (A.8) It can be easily verified that 𝑀(𝜃₂)→−∞ as 𝜃₂→0+ or 𝜃₂→∞. Thus, there exists 𝜃^∗₂≡𝜃^∗₂(𝜃₁)∈(0,∞) such that sup_𝜃2>0ln𝐿(𝜃₁,𝜃₂)=ln𝐿(𝜃₁,𝜃^∗₂).
We define function 𝑔(𝑥), 𝑥₀ for any given 𝐴>0, and 𝑐>0 as before. Consider any sequence {𝜃_1𝑚,𝑚≥1}⊂(−∞,∞) with |𝜃_1𝑚|→∞ as 𝑚→∞. Let 𝜃^∗_2𝑚≡𝜃^∗₂(𝜃_1𝑚) and {𝜃^∗_2𝑚𝑟,𝑟≥1} be any converging subsequence of {𝜃_2𝑚,𝑚≥1}, 𝜂≡lim_𝑚→∞𝜃^∗_2𝑚𝑟≤∞. Let us study two cases.
Case 1 (𝜂=∞). Notice that for any 𝑟≥1, by our assumptions there exists at least one index, say 𝑖, in 𝐼 such that (a) 𝑛_𝑖>0; (b) |𝜏_𝑖−1−𝜃_1𝑚𝑟/𝜃^∗_2𝑚𝑟|>𝑐 and |𝜏_𝑖−𝜃_1𝑚𝑟/𝜃^∗_2𝑚𝑟|>𝑐; (c) 𝜏_𝑖−1−𝜃_1𝑚𝑟/𝜃^∗_2𝑚𝑟 and 𝜏_𝑖−𝜃_1𝑚𝑟/𝜃^∗_2𝑚𝑟 have the same sign.
Since 𝜃^∗_2𝑚→∞ as 𝑚→∞, there exists 𝑟₀ sufficiently large such that 𝜃^∗_2𝑚𝑟>𝑥₀/𝑐,∀𝑟≥𝑟₀. Thus, |𝜃^∗_2𝑚𝑟𝜏_𝑖−1−𝜃_1𝑚𝑟|=𝜃^∗_2𝑚𝑟|𝜏_𝑖−1−𝜃_1𝑚𝑟𝜃^∗_2𝑚𝑟|>𝑥₀,∀𝑟≥𝑟₀,|𝜃^∗_2𝑚𝑟𝜏_𝑖−𝜃_1𝑚𝑟|=𝜃^∗_2𝑚𝑟|𝜏_𝑖−𝜃_1𝑚𝑟𝜃^∗_2𝑚𝑟|>𝑥₀,∀𝑟≥𝑟₀. (A.9) From these, as what we did before we obtain ln𝐿(𝜃_1𝑚𝑟,𝜃^∗_2𝑚𝑟)<𝑛_𝑖ln^{𝜃∗𝑟2𝑚𝜏𝑖−𝜃𝑟1𝑚}_{𝜃∗𝑟2𝑚𝜏𝑖−1−𝜃𝑟1𝑚}𝜑(𝑡)𝑑𝑡<−𝐴,∀𝑟≥𝑟₀, (A.10) this implies lim_𝑟→∞ln𝐿(𝜃_1𝑚𝑟,𝜃^∗_2𝑚𝑟)=−∞.Case 2 (0≤𝜂<∞). In this case, the inequality lim_𝑟→∞ln𝐿(𝜃_1𝑚𝑟,𝜃^∗_2𝑚𝑟)=−∞ can be proved in the same way as Case 1.From the results in the above two cases, we conclude that lim_𝑚→∞ln𝐿(𝜃_1𝑚,𝜃^∗_2𝑚)=−∞. Since {𝜃_1𝑚,𝑚≥1} is an arbitrary sequence satisfying |𝜃_1𝑚|→∞, so finally (A.3) is true.

Proof of Lemma 2.2. For any given 𝑣<𝑢, we have 𝑔(𝑢,𝑣)≡ln(Φ(𝑢)−Φ(𝑣)). The Hessian matrix of 𝑔(𝑢,𝑣) is 𝐻^∗=𝜑^′(𝑢)(Φ(𝑢)−Φ(𝑣))−𝜑²(𝑢)(Φ(𝑢)−Φ(𝑣))²𝜑(𝑢)𝜑(𝑣)(Φ(𝑢)−Φ(𝑣))²𝜑(𝑢)𝜑(𝑣)(Φ(𝑢)−Φ(𝑣))²−𝜑^′(𝑣)(Φ(𝑢)−Φ(𝑣))+𝜑²(𝑣)(Φ(𝑢)−Φ(𝑣))². (A.11) In order to prove 𝐻^∗ is negative definite, the following two conditions must be satisfied: (i) 𝜕²𝑔/𝜕𝑢²<0 or 𝜕²𝑔/𝜕𝑣²<0; (ii) the determinant of the Hessian matrix 𝐻^∗ is positive.
The inequality 𝜕²𝑔/𝜕𝑣²<0 is equivalent to 𝜑(𝑣)>𝑣(Φ(𝑢)−Φ(𝑣)). This inequality follows from 𝑦[1−Φ(𝑦)]<𝜙(𝑦),∀𝑦. Thus the desired inequality is true.
From the expression of 𝐻^∗, it follows that (Φ(𝑢)−Φ(𝑣))²|𝐻^∗|=−𝜑^′(𝑢)𝜑^′(𝑣)(Φ(𝑢)−Φ(𝑣))²−𝜑^′(𝑢)𝜑²(𝑣)(Φ(𝑢)−Φ(𝑣))+𝜑²(𝑢)𝜑^′(𝑣)(Φ(𝑢)−Φ(𝑣)). (A.12) The inequality |𝐻^∗|>0 is equivalent to 𝑢𝜑(𝑣)−𝑣𝜑(𝑢)−𝑢𝑣(Φ(𝑢)−Φ(𝑣))>0. We discuss three cases.
Case 1 (𝑣<𝑢≤0). We have −𝑢(Φ(𝑢)−Φ(𝑣))=∫^𝑢_𝑣−𝑢𝜑(𝑡)𝑑𝑡<∫^𝑢_𝑣−𝑡𝜑(𝑡)𝑑𝑡=𝜑(𝑢)−𝜑(𝑣). From this, we see that 𝑢𝜑(𝑣)−𝑣𝜑(𝑢)−𝑢𝑣(Φ(𝑢)−Φ(𝑣))>𝑢𝜑(𝑣)−𝑣𝜑(𝑢)+𝑣[𝜑(𝑢)−𝜑(𝑣)]=(𝑢−𝑣)𝜑(𝑣)>0. (A.13)Case 2 (𝑣<0<𝑢). It is obvious that 𝑢𝜑(𝑣)−𝑣𝜑(𝑢)−𝑢𝑣(Φ(𝑢)−Φ(𝑣))>0.Case 3 (0<𝑣<𝑢). It holds that 𝑣(Φ(𝑢)−Φ(𝑣))=^𝑢_𝑣𝑣𝜑(𝑡)𝑑𝑡<^𝑢_𝑣𝑡𝜑(𝑡)𝑑𝑡=^𝑢_𝑣−𝜑^′(𝑡)𝑑𝑡=𝜑(𝑣)−𝜑(𝑢). (A.14) From this, we see that −𝑢𝑣(Φ(𝑢)−Φ(𝑣))>−𝑢(𝜑(𝑣)−𝜑(𝑢))=−𝑢𝜑(𝑣)+𝑢𝜑(𝑢) since 𝑢>0. It means that 𝑢𝜑(𝑣)−𝑢𝜑(𝑢)−𝑢𝑣(Φ(𝑢)−Φ(𝑣))>0. This further implies that 𝑢𝜑(𝑣)−𝑣𝜑(𝑢)−𝑢𝑣(Φ(𝑢)−Φ(𝑣))>0 since 𝑢>𝑣>0. Hence, in all the three cases, we obtain |𝐻^∗|>0.
From all the above, we conclude that both conditions (i) and (ii) are satisfied and thus the Hessian matrix 𝐻^∗(𝑢,𝑣) is negative definite.

Notations

S.S.:	Sample size
𝜇(𝑔), 𝜎(𝑔):	MLEs of 𝜇, 𝜎 with grouped data
𝜇(𝐼), 𝜎(𝐼):	MLEs of 𝜇, 𝜎 with type I left censored data
𝑚:	MLE of median 𝑚=exp{𝜇} of 𝐿𝑁(𝜇,𝜎2) distribution with grouped data
𝑣:	MLE of mean 𝑣=exp{𝜇+𝜎2/2} of 𝐿𝑁(𝜇,𝜎2) distribution with grouped data
Average:	The average of estimates from 5000 simulations
A.W.:	The average width of 5000 approximate 95% confidence intervals
C.R.:	The average coverage rate of 5000 approximate 95% confidence intervals.