JPSJournal of Probability and Statistics1687-95381687-952XHindawi Publishing Corporation31057510.1155/2009/310575310575Research ArticleOn the Existence and Uniqueness of the Maximum Likelihood Estimators of Normal and Lognormal Population Parameters with Grouped DataXiaJin1MiJie2ZhouYanYan3ThavaneswaranA.1Center on Aging and HealthJohn Hopkins UniversityBaltimore, MD 21287USAjhu.edu2Department of StatisticsFlorida International UniversityMiami, FL 33139USAfiu.edu3Department of Statistics & biostatisticsCalifornia State UniversityHayward, CA 94542USAcalstate.edu200928072009200911032009160620092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.  and Mould et al.  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 , reliability analysis , size distributions in economics and actuarial sciences , cell growth , 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,,Xn is drawn from a normal population, the values of Xjs are unknown; however, according to k preestablished partition points τ1<τ2<<τk, we know ni, the number of Xjs that fall into the interval [τi-1,τi), 1ik+1 where τ0- and τk+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)+nk+1ln[1-Φ(θ2τk-θ1)]+i=2kniln[Φ(θ2τi-θ1)-Φ(θ2τi-1-θ1)], 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+nk+1<n, nj-1+nj<n, 2jk+1. For any given η>0, there exists a compact subset KK(η)(-,)×(0,) such that {(θ1,θ2):lnL(θ1,θ2)-η}K.

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*=(2gu22guv2guv2gv2), is negative definite.

Theorem 2.3.

Suppose that the observed n1,,nk+1 satisfy n1+nk+1<n and nj-1+nj<n,2jk+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. , in order to show the existence and uniqueness of the MLEs of (θ1,θ2), it is sufficient to verify the following two conditions.

For any given η>0, (2.2) holds.

The Hessian matrix of lnL, H(θ1,θ2)=(2lnLθiθj), 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,).

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(2g1θ122g1θ1θ22g1θ1θ22g1θ22)=(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)2τ12φ(θ2τ1-θ1)Φ(θ2τ1-θ1)-𝒬Φ(θ2τ1-θ1)2), where 𝒬 denotes φ2(θ2τ1-θ1).

To show H1 is negative semidefinite, we will verify the following two conditions: (a) 2g/θ12<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 ). 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 gk+1(θ)ln[1-Φ(θ2τk-θ1)]. The Hessian matrix Hk+1 of gk+1 is Hk+1(-φ(θ2τk-θ1)[1-Φ(θ2τk-θ1)]+φ2(θ2τk-θ1)[1-Φ(θ2τ1-θ1)]2τkφ(θ2τk-θ1)[1-Φ(θ2τk-θ1)]2τkφ(θ2τk-θ1)[1-Φ(θ2τk-θ1)]2-τk2φ(θ2τk-θ1)[1-Φ(θ2τk-θ1)]2), where denotes [1-Φ(θ2τk-θ1)]2+φ2(θ2τk-θ1).

In the similar way as the above we can show that the matrix Hk+1 is negative semidefinite.

Finally, let us consider h(θ1,θ2)ln[Φ(θ2τi-θ1)-Φ(θ2τi-1-θ1)], 2ik. Let u=θ2τi-θ1,v=θ2τi-1-θ1. Then h(θ1,θ2)=ln[Φ(u)-Φ(v)]g(u,v). The Hessian matrix Hi associated with h(θ1,θ2) is Hi=(2hθ122hθ1θ22hθ1θ22hθ22)=(-1-1τiτi-1)(2gu22guv2guv2gv2)(-1τi-1τi-1)=AH*A, where H*=(2gu22guv2guv2gv2),A=(-1τi-1τi-1), and A is the transpose of A. By Lemma 2.2, H* is negative definite. Therefore, Hi is negative definite.

The Hessian matrix H of the log-likelihood function lnL(θ) can be expressed as H=n1H1+nk+1Hk+1+i=2kniHi. Since matrices H1 and Hk+1 are negative semidefinite, each Hi(2ik) is negative definite, and at least one ni>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, (μ̂nσ̂n)-(μσ)    L    N(0,I-1(μ,σ)), where L means “converges in law,’’ and I(μ,σ)=-E(2lnLμ22lnLμσ2lnLμσ2lnLσ2).

Proof.

For each n>k, define An{n1+nk+1=n,ni-1+ni=n,1ik}. Note that P(lim supnAn)=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,,Xn 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, Xj will be observed if and only if Xjτ. 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 Xjs 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 .

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 Xj is censored by detection limit τj(1jn).

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 mexp{μ} and mean νE(Y)=exp{μ+σ2/2}. The MLEs of m and ν can easily be obtained as m̂=exp{μ̂} and ν̂=exp{μ̂+σ̂2/2} 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-1(μ,σ)=(β11β12β21β22). It is obvious that μ̂n-μLN(0,β11), by large sample theory we have exp{μ̂n}-exp{μ}LN(0,(exp{μ}2)β11). From these, an approximate (1-α)100% confidence interval of m can be obtained as exp{μ̂n}±zα/2(exp{μ̂n}β̂11), here zα/2 is the upper α/2 percentile, and β̂11 is obtained from substituting μ̂n and σ̂ for μ and σ in the expression of β11. Similarly, it holds that as neμ̂n+σ̂n2/2-eμ+σ2/2    L    N(0,τ2), where τ2=(eμ+σ2/2,σeμ+σ2/2)I-1(μ,σ)(eμ+σ2/2σeμ+σ2/2). Therefore, an approximate (1-α)100% confidence interval of ν=exp{μ+σ2/2} is obtained as eμ̂n+σ̂n2/2±zα/2τ̂, where τ̂ is obtained by substituting μ and σ by their MLEs μ̂n and σ̂n.

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 τi,1i5, we record the number of this population that fall into each interval [τi-1,τi). 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 lnLθ1=0,lnLθ2=0, using SAS IMSL nonlinear equation solver. Then the MLEs μ̂=μ̂n and σ̂=σ̂n 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 (μ̂n,σ̂n) 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.

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

S.Sm̂ν̂
n Average A.W. C.R. Average A.W. C.R.

30 21.992 33.374 90.9% 245.490 1517.67470.7%
35 21.665 31.882 91.7% 229.715 2810.49078.8%
40 21.441 29.207 92.9% 231.326 2356.41281.4%
50 21.211 25.697 93.4% 222.684 1632.30182.9%
100 20.614 17.983 94.5% 199.014 647.06585.6%

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

S.Sm̂ν̂
n 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: τ1=0, τ2=1.5, τ3=3, τ4=4.5, τ5=6.

S.Sm̂ν̂
n Average A.W. C.R. Average A.W. C.R.

30 21.657 31.826 92.7% 203.855 845.74282.2%
35 21.397 29.168 93.4% 189.163 679.64683.8%
40 21.201 26.989 93.1% 181.097 558.17684.6%
50 20.974 23.931 93.6% 172.924 429.35385.1%
100 20.495 16.577 94.3% 159.906 274.16689.2%

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

S.Sm̂ν̂
n 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: τ1=-2, τ2=0, τ3=2, τ4=4, τ5=6.

S.Sm̂ν̂
n Average A.W. C.R. Average A.W. C.R.

30 21.688 32.743 93.1% 183.834 659.78979.8%
35 21.496 29.564 92.3% 180.633 569.35682.7%
40 21.141 27.514 92.5% 175.077 554.14284.1%
50 21.093 24.082 93.1% 170.522 360.50882.7%
100 20.562 16.932 94.3% 158.738 266.68789.1%

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

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

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

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

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

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 μ̂ and σ̂ 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 ν̂(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 ν̂(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 μ̂(g)=-0.599, σ̂(g)=0.532, m̂(g)=0.549, and ν̂(g)=0.633. Approximate 95% confidence intervals for μ,σ,m, and ν are given in Table 7.

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

n95% CI for μ95% CI for σ95% CI for m95% CI for ν
47(-0.771, -0.426)(0.354, 0.709)(0.455, 0.645)(0.525, 0.741)
AppendixProof of Lemma <xref ref-type="statement" rid="lem1">2.1</xref>.

To prove the lemma, it is sufficient to verify the following three limits: limθ20+sup-<θ1<lnL(θ1,θ2)=-;limθ2sup-<θ1<lnL(θ1,θ2)=-;lim|θ1|supθ2>0lnL(θ1,θ2)=-.

To see (A.1), from the assumption n1+nk+1<n, there exists an index, say i, such that 2ik and ni>0. We have lnL(θ1,θ2)nilnθ2τi-1-θ1θ2τi-θ1φ(t)dtniln[θ2(τi-τi-1)φ(0)]. So sup-<θ1<lnL(θ1,θ2)ni[lnφ(0)+ln(τi-τi-1)+lnθ2] and lim supθ20+sup-<θ1<lnL(θ1,θ2)limθ20+ni[lnφ(0)+ln(τi-τi-1)+lnθ2]=-. Therefore, (A.1) holds.

To show (A.2), we denote I{1jk+1,nj>0}. For each fixed θ2>0, it is evident that lnL(θ1,θ2)=iInilnθ2τi-1-θ1θ2τi-θ1φ(t)dtM(θ2). Thus sup-<θ1<lnL(θ1,θ2)=sup-<θ1<M(θ2).

Note that, lim|θ1|M(θ1)=-, so there exists θ1*=θ1*(θ2)(-,), such that sup-<θ1<lnL(θ1,θ2)=lnL(θ1*,θ2)=iInilnθ2τi-1-θ1*θ2τi-θ1*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 cmin1jk+1(τj-τj-1)>0. For any θ2>x0/c, from our assumptions there exists an index, say i, belonging to I satisfying (a) ni>0; (b) the following two quantities θ2τi-1-θ1*=θ2(τi-1-θ1*/θ2) and θ2τi-θ1*=θ2(τi-θ1*/θ2) have the same sign; and (c) |τi-1-θ1*/θ2|>c, and |τi-θ1*/θ2|>c.

Note that, if iI, and both θ2τi-1-θ1*>0 and θ2τi-θ1*>0, then θ2τi-1-θ1*θ2τi-θ1*φ(t)dt<1θ2τi-1-θ1*θ2τi-1-θ1*θ2τi-θ1*tφ(t)dt=1θ2τi-1-θ1*[φ(θ2τi-1-θ1*)-φ(θ2τi-θ1*)]<φ(θ2τi-1-θ1*)θ2τi-1-θ1*=12π    g(θ2τi-1-θ1*). If θ2>x0/c, then θ2τi-1-θ1*=θ2(τi-1-θ1*/θ2)>(x0/c)c=x0 and so g(θ2τi-1-θ1*)>2πexpA. Consequently, nilnθ2τi-1-θ1*θ2τi-θ1*φ(t)dt<niln12πexp(A)=ni(-ln2π-A)<-niA<-A.

This further implies sup-<θ1<lnL(θ1,θ2)=lnL(θ1*,θ2)=jInjlnθ2τj-1-θ1*θ2τj-θ1*φ(t)dt<nilnθ2τi-1-θ1*θ2τi-θ1*φ(t)dt<-A,θ2>x0c.

If iI, but both τi-1-θ1*/θ2<-c and τi-θ1*/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. Due to the arbitrariness of A>0, we conclude that (A.1) is true.

To verify (A.3), we let θ2τk+1-θ1= and θ2τ0-θ1=- for any (θ1,θ2)(-,)×(0,). For any fixed θ1(-,), we have lnL(θ1,θ2)=jInjlnθ2τj-1-θ1θ2τj-θ1φ(t)dtM(θ2). It can be easily verified that M(θ2)- as θ20+ or θ2. Thus, there exists θ2*θ2*(θ1)(0,) such that supθ2>0lnL(θ1,θ2)=lnL(θ1,θ2*).

We define function g(x), x0 for any given A>0, and c>0 as before. Consider any sequence {θ1m,m1}(-,) with |θ1m| as m. Let θ2m*θ2*(θ1m) and {θ2mr*,  r1} be any converging subsequence of {θ2m,  m1}, ηlimmθ2mr*. Let us study two cases.

Case 1 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M391"><mml:mi>η</mml:mi><mml:mo>=</mml:mo><mml:mi>∞</mml:mi></mml:math></inline-formula>).

Notice that for any r1, by our assumptions there exists at least one index, say i, in I such that (a) ni>0; (b) |τi-1-θ1mr/θ2mr*|>c and |τi-θ1mr/θ2mr*|>c; (c) τi-1-θ1mr/θ2mr* and τi-θ1mr/θ2mr* have the same sign.

Since θ2m* as m, there exists r0 sufficiently large such that θ2mr*>x0/c,  rr0. Thus, |θ2mr*τi-1-θ1mr|=θ2mr*|τi-1-θ1mrθ2mr*|>x0,rr0,|θ2mr*τi-θ1mr|=θ2mr*|τi-θ1mrθ2mr*|>x0,rr0. From these, as what we did before we obtain lnL(θ1mr,θ2mr*)<nilnθ2mr*τi-1-θ1mrθ2mr*τi-θ1mrφ(t)dt<-A,rr0, this implies limrlnL(θ1mr,θ2mr*)=-.

Case 2 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M407"><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>η</mml:mi><mml:mo><</mml:mo><mml:mi>∞</mml:mi></mml:math></inline-formula>).

In this case, the inequality limrlnL(θ1mr,θ2mr*)=- can be proved in the same way as Case 1.

From the results in the above two cases, we conclude that limmlnL(θ1m,θ2m*)=-. Since {θ1m,m1} is an arbitrary sequence satisfying |θ1m|, so finally (A.3) is true.

Proof of Lemma <xref ref-type="statement" rid="lem2">2.2</xref>.

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). 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)). The inequality |H*|>0 is equivalent to uφ(v)-vφ(u)-uv(Φ(u)-Φ(v))>0. We discuss three cases.

Case 1 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M427"><mml:mi>v</mml:mi><mml:mo><</mml:mo><mml:mi>u</mml:mi><mml:mo>≤</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>).

We have -u(Φ(u)-Φ(v))=vu-uφ(t)dt<vu-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.

Case 2 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M430"><mml:mi>v</mml:mi><mml:mo><</mml:mo><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi>u</mml:mi></mml:math></inline-formula>).

It is obvious that uφ(v)-vφ(u)-uv(Φ(u)-Φ(v))>0.

Case 3 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M432"><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi>v</mml:mi><mml:mo><</mml:mo><mml:mi>u</mml:mi></mml:math></inline-formula>).

It holds that v(Φ(u)-Φ(v))=vuvφ(t)dt<vutφ(t)dt=vu-φ(t)dt=φ(v)-φ(u). 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.

NotationsS.S.:

Sample size

μ̂(g), σ̂(g):

MLEs of μ, σ with grouped data

μ̂(I), σ̂(I):

MLEs of μ, σ with type I left censored data

m̂:

MLE of median m=exp{μ} of LN(μ,σ2) distribution with grouped data

v̂:

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.

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