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

Authors: Jin Xia; Jie Mi; YanYan Zhou
Journal: Journal of Probability and Statistics
 (2009)

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## 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.


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## Body

## 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 distributionLN(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 Section2, 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 parametersμ 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),  1≤i≤k+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(2.1)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.
Assumen1+nk+1<n,  nj-1+nj<n, 2≤j≤k+1. For any given η>0, there exists a compact subset K≡K(η)⊂(-∞,∞)×(0,∞) such that
(2.2){(θ1,θ2):lnL(θ1,θ2)≥-η}⊂K.Basically, Lemma2.1 means that the log-likelihood function lnL(θ1,θ2) will not achieve its maximum value at the boundary of its domain.Lemma 2.2.
Letg(u,v)≡ln(Φ(u)-Φ(v)) for v<u. Then the Hessian matrix H* of g(u,v),
(2.3)H*=(∂2g∂u2∂2g∂u∂v∂2g∂u∂v∂2g∂v2),
is negative definite.Theorem 2.3.
Suppose that the observedn1,…,nk+1 satisfy n1+nk+1<n and nj-1+nj<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 parametersθ1 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 givenη>0, (2.2) holds.(ii)
The Hessian matrix oflnL,
(2.4)H(θ1,θ2)=(∂2lnL∂θi∂θj),
is negative definite at every point (θ1,θ2)∈(-∞,∞)×(0,∞).
Condition (i) is certainly satisfied by Lemma2.1. Therefore, to prove the theorem, we need only to show (ii), that is, the log-likelihood function lnL is negative definite function of
(2.5)θ=(θ1,θ2)∈(-∞,∞)×(0,∞).
To this end we should consider each of the three terms in the expression (2.1) of lnL(θ).
Letg1(θ)≡lnΦ(θ2τ1-θ1). It is evident that the Hessian matrix of g1 is
(2.6)H1≡(∂2g1∂θ12∂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)2τ12φ′(θ2τ1-θ1)Φ(θ2τ1-θ1)-𝒬Φ(θ2τ1-θ1)2),
where 𝒬 denotes φ2(θ2τ1-θ1).
To showH1 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 [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 denotegk+1(θ)≡ln[1-Φ(θ2τk-θ1)]. The Hessian matrix Hk+1 of gk+1 is
(2.7)Hk+1≡(-φ′(θ2τk-θ1)[1-Φ(θ2τk-θ1)]+φ2(θ2τk-θ1)[1-Φ(θ2τ1-θ1)]2τkφ′(θ2τk-θ1)ℱ[1-Φ(θ2τk-θ1)]2τkφ′(θ2τk-θ1)ℱ[1-Φ(θ2τk-θ1)]2-τk2φ′(θ2τk-θ1)ℱ[1-Φ(θ2τk-θ1)]2),
where ℱ denotes [1-Φ(θ2τk-θ1)]2+φ2(θ2τk-θ1).
In the similar way as the above we can show that the matrixHk+1 is negative semidefinite.
Finally, let us considerh(θ1,θ2)≡ln[Φ(θ2τi-θ1)-Φ(θ2τi-1-θ1)], 2≤i≤k. 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
(2.8)Hi=(∂2h∂θ12∂2h∂θ1∂θ2∂2h∂θ1∂θ2∂2h∂θ22)=(-1-1τiτi-1)(∂2g∂u2∂2g∂u∂v∂2g∂u∂v∂2g∂v2)(-1τi-1τi-1)=A′H*A,
where
(2.9)H*=(∂2g∂u2∂2g∂u∂v∂2g∂u∂v∂2g∂v2),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 matrixH 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(2≤i≤k) 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 Theorem2.3, it holds that as n→∞,
(2.10)(μ̂nσ̂n)-(μσ)→LN(0,I-1(μ,σ)),
where →L means “converges in law,’’ and
(2.11)I(μ,σ)=-E(∂2lnL∂μ2∂2lnL∂μ∂σ∂2lnL∂μ∂σ∂2lnL∂σ2).Proof.
For eachn>k, define An≡{n1+nk+1=n,ni-1+ni=n,1≤i≤k}. Note that P(lim supn→∞An)=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 Theorem2.3 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 Xj≥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 observableXjs 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 Theorem2.5 are true if each Xj is censored by detection limit τj(1≤j≤n).


### 2.1. Estimation of the Median and Mean

Suppose that random variableY 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 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 matrixI(μ,σ) in Corollary 2.4 to Theorem 2.3 as
(2.12)I-1(μ,σ)=(β11β12β21β22).
It is obvious that μ̂n-μ→LN(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 n→∞(2.13)eμ̂n+σ̂n2/2-eμ+σ2/2→LN(0,τ2),
where
(2.14)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
(2.15)eμ̂n+σ̂n2/2±zα/2τ̂,
where τ̂ is obtained by substituting μ and σ by their MLEs μ̂n and σ̂n.


## 2.1. Estimation of the Median and Mean

Suppose that random variableY 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 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 matrixI(μ,σ) in Corollary 2.4 to Theorem 2.3 as
(2.12)I-1(μ,σ)=(β11β12β21β22).
It is obvious that μ̂n-μ→LN(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 n→∞(2.13)eμ̂n+σ̂n2/2-eμ+σ2/2→LN(0,τ2),
where
(2.14)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
(2.15)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 ofμ 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 sizen by drawing n values from a normal population with μ=3 and σ=2. Next, for a prefixed five partition points τi,1≤i≤5, 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 ofθ1=μ/σ and θ2=1/σ by solving the likelihood equations
(3.1)∂lnL∂θ1=0,∂lnL∂θ2=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 useτ1 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.
S.Sm̂ν̂nAverageA.W.C.R.AverageA.W.C.R.3021.99233.37490.9%245.4901517.67470.7%3521.66531.88291.7%229.7152810.49078.8%4021.44129.20792.9%231.3262356.41281.4%5021.21125.69793.4%222.6841632.30182.9%10020.61417.98394.5%199.014647.06585.6%Table 2
Grouped data:τ1=1.5, τ2=2.5, τ3=3.5, τ4=4.5, τ5=5.5.
S.Sm̂ν̂nAverageA.W.C.R.AverageA.W.C.R.3021.66034.09192.4%245.5691835.28485.4%3521.32030.02792.1%211.600794.54781.9%4021.15927.38693.4%205.518763.31084.1%5020.88424.58894.4%190.732525.01983.3%10020.46516.79094.3%167.344343.43590.0%Table 3
Grouped data:τ1=0, τ2=1.5, τ3=3, τ4=4.5, τ5=6.
S.Sm̂ν̂nAverageA.W.C.R.AverageA.W.C.R.3021.65731.82692.7%203.855845.74282.2%3521.39729.16893.4%189.163679.64683.8%4021.20126.98993.1%181.097558.17684.6%5020.97423.93193.6%172.924429.35385.1%10020.49516.57794.3%159.906274.16689.2%Table 4
Grouped data:τ1=-1.5, τ2=1, τ3=3.5, τ4=6, τ5=8.5.
S.Sm̂ν̂nAverageA.W.C.R.AverageA.W.C.R.3021.75232.37492.4%176.952610.02080.1%3521.38129.63592.5%169.389512.87781.2%4021.24728.55292.6%168.496552.49685.0%5021.17924.73893.8%166.919373.01283.3%10020.50717.13294.3%156.027264.16588.8%Table 5
Grouped data:τ1=-2, τ2=0, τ3=2, τ4=4, τ5=6.
S.Sm̂ν̂nAverageA.W.C.R.AverageA.W.C.R.3021.68832.74393.1%183.834659.78979.8%3521.49629.56492.3%180.633569.35682.7%4021.14127.51492.5%175.077554.14284.1%5021.09324.08293.1%170.522360.50882.7%10020.56216.93294.3%158.738266.68789.1%Table 6
Grouped data:τ1=-3.5, τ2=-0.5, τ3=2.5, τ4=5.5, τ5=8.5.
S.Sm̂ν̂nAverageA.W.C.R.AverageA.W.C.R.3021.64132.12290.8%170.705505.26777.8%3521.38630.49892.8%167.457517.09180.3%4021.25028.96093.5%164.214555.64683.9%5021.03526.62395.2%161.572583.24988.7%10020.46917.48194.1%154.492271.90088.7%Grouped data:τ1=2, τ2=2.5, τ3=3, τ4=3.5, τ5=4. Type I left censored data: τ=2.
(a)(b)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)Grouped data:τ1=0, τ2=1.5, τ3=3, τ4=4.5, τ5=6. Type I left censored data: τ=0.
(a)(b)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)Grouped data:τ1=-2, τ2=0, τ3=2, τ4=4, τ5=6. Type I left censored data: τ=-2.
(a)(b)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.
(a)(b)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% whenn>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 normalQ-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 Section2 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.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 ν47(-0.771, -0.426)(0.354, 0.709)(0.455, 0.645)(0.525, 0.741)



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*Source: 310575-2009-09-14.xml*