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On the maximum likelihood estimation for a normal distribution under random censoring

Namhyun Kim1,a

aDepartment of Science, Hongik University, Korea
Correspondence to: 1Department of Science, Hongik University, 94 Wausan-Ro, Mapo-Gu, Seoul 04066, Korea. E-mail: nhkim@hongik.ac.kr
Received June 28, 2018; Revised August 16, 2018; Accepted September 26, 2018.
Abstract

In this paper, we study statistical inferences on the maximum likelihood estimation of a normal distribution when data are randomly censored. Likelihood equations are derived assuming that the censoring distribution does not involve any parameters of interest. The maximum likelihood estimators (MLEs) of the censored normal distribution do not have an explicit form, and it should be solved in an iterative way. We consider a simple method to derive an explicit form of the approximate MLEs with no iterations by expanding the nonlinear parts of the likelihood equations in Taylor series around some suitable points. The points are closely related to Kaplan-Meier estimators. By using the same method, the observed Fisher information is also approximated to obtain asymptotic variances of the estimators. An illustrative example is presented, and a simulation study is conducted to compare the performances of the estimators. In addition to their explicit form, the approximate MLEs are as efficient as the MLEs in terms of variances.

Keywords : Kaplan-Meier estimators, Koziol-Green model, maximum likelihood estimators, normal distribution, random censoring
1. Introduction

In statistical analysis of life time data, some well-known common distributions are exponential, Weibull, lognormal, and gamma distribution. The lognormal distribution is especially useful when a hazard rate is initially increasing and then decreasing. We also need inferences for a normal distribution since the logarithm of a lognormal variable follows a normal. As for censoring types, the most common and simplest censoring schemes are type I or type II censoring. For numerous censoring types, see Tableman and Kim (2004), and Lee and Wang (2003).

Gupta (1952), Cohen (1959, 1961), and Kim (2014b) studied the estimation of a type II censored normal distribution. Balakrishinan et al. (2003) dealt with the estimation of a normal distribution on a progressively type II censoring scheme, which is a generalization of traditional type II censoring. Maximum likelihood estimators (MLEs) based on type II or progressive type II censored data from a normal distribution do not have explicit forms, and the situation remains as it is for many distributions when data are censored. Kim (2014b) and Balakrishinan et al. (2003) derived approximate MLEs under type II censoring and progressively type II censoring of a normal, respectively.

Random censoring occurs frequently in survival studies. In this paper, we consider the parameter estimation of a normal distribution for randomly censored data by applying the same approximate method used in Kim (2014b) and Balakrishinan et al. (2003). As it is mentioned, they dealt with type II or progressively type II censoring, not random censoring. The Kaplan and Meier (1958) estimator is used for the plotting position of randomly censored data.

The approximation method was first developed by Balakrishnan (1989) to find the approximate MLE of the scalar parameter in the Rayleigh distribution with left and right type II censoring. The method approximates the nonlinear part of the likelihood equations; subsequently, many researchers used it for other distributions under several censoring schemes that are most often progressively type II censoring. Balakrishinan et al. (2003) did it for a normal distribution. Balakrishnan and Kannan (2001), and Balakrishnan et al. (2004) studied the estimation for the logistic distribution and the extreme value distribution, respectively. Asgharzadeh (2006, 2009) dealt with the problem for generalized logistic distribution and generalized exponential distribution, respectively. Balakrishnan and Asgharzadeh (2005), and Kang et al. (2008) used the method for the half logistic distribution. Seo and Kang (2007), and Kim and Han (2009) discussed the procedure for the Rayleigh distribution. Sultan et al. (2014) used it for inverse Weibull distribution. As for random censoring, Kim (2014a, 2016) applied the approximate method to generalized exponential distributions and Weibull distribution, respectively.

In Section 2, we derive MLEs and approximate MLEs for a normal under random censoring. In Section 3, we provide expressions for observed Fisher information to obtain the approximate variances of the estimators. Section 4 presents simulation results that compare MLEs and approximate MLEs. Section 5 ends the paper with some concluding remarks.

2. MLEs and the approximate MLEs

Let T1, . . . , Tn be lifetimes with distribution function F and probability density function (pdf) f , and C1, . . . ,Cn be random censoring times drawn independently of the T1, . . . , Tn from distribution function G and pdf g. The Ti’s are censored on the right by Ci. On each of n individuals, we observe n random pairs (Xi, δi), i = 1, . . . , n, where

$Xi=min(Ti,Ci) and δi={1,if Ti≤Ci,0,if Ti>Ci.$

The observed random pairs (Xi, δi) could be written as (X(i), δ(i)) where X(1) ≤ · · · ≤ X(n) are the ordered observations of X1, . . . , Xn, and δ(i) is the δ corresponding to X(i). Then the likelihood function based on the ordered (X(i), δ(i)) becomes

$L=n!∏j=1n(f(x(j))G¯(x(j)))δ(j)(g(x(j))F¯(x(j)))1-δ(j)=n!∏j=1n(f(x(j)))δ(j)(F¯(x(j)))1-δ(j)∏j=1n(g(x(j)))1-δ(j)(G¯(x(j)))δ(j)$

with = 1 – F, = 1 – G. If we assume the distribution of Ci’s does not involve any parameters of interest, the last factor in (2.1) plays no role in the maximization. Hence we have

$L∝∏j=1n(f(x(j)))δ(j)(F¯(x(j)))1-δ(j).$

Usually lifetimes are positive. When we assume the lifetimes follow a lognormal distribution, the logarithm of the lifetimes should follow a normal distribution. In this case we need the inference for a normal. Let Ti’s follow the normal distribution N(μ,σ2) with the density

$f(x)=12πσexp(-(x-μ)22σ2), -∞

Here, the lifetime itself should be eTi. Then the log-likelihood function l = ln L gives

$l∝∑j=1nδ(j)(-log σ-12ξj2)+∑j=1n(1-δ(j)) log Φ¯(ξj),$

where ξjξj(μ,σ) = (x(j)μ), $ϕ(t)=(1/2π)e-(1/2)t2,Φ(ξj)=∫-∞ξjϕ(t)dt$, and $Φ-$(ξj) = 1–Φ(ξj).

Using ∂ξj/∂μ = −1, ∂ξj/∂σ = −ξj,

$∂∂μlog Φ¯(ξj)=1σQ(ξj),∂∂σlog Φ¯(ξj)=ξjσQ(ξj),$

with Q(ξj) = ϕ(ξj)/$Φ-$(ξj), we have the likelihood equations

$∂l∂μ=1σ∑j=1nδ(j)ξj+1σ∑j=1n(1-δ(j))Q(ξj)=0,$$∂l∂σ=-nuσ+1σ∑j=1nδ(j)ξj2+1σ∑j=1n(1-δ(j))ξjQ(ξj)=0,$

where $nu=∑j=1nδ(j)$ is the number of uncensored data. We let $μ^$, $σ^$ be the solutions of the above equations.

The likelihood equations (2.2) and (2.3) do not have an explicit solution except for all δ( j) = 1, a complete sample. Therefore we consider approximate MLEs that give an explicit form and do not need any iterations for computations. As mentioned in Section 1, the approximation method was first developed by Balakrishnan (1989); subsequently, researchers have used it for other distributions under several censoring schemes.

The process can be done by approximating Q(ξi) ≅ ai + biξi. The idea is to expand Q(ξi) in the Taylor series by keeping the first two terms around a suitable point ξi0. We will define ai, bi later in this section. By replacing Q(ξi) ≅ ai + biξi in (2.2), and (2.3), we have the approximate likelihood equations

$∂l∂μ≅1σ∑j=1nδ(j)ξj+1σ∑j=1n(1-δ(j)) (aj+bjξj)=0,$$∂l∂σ≅-nuσ+1σ∑j=1nδ(j)ξj2+1σ∑j=1n(1-δ(j))ξj(aj+bjξj)=0.$

Substituting ξj = (x( j)μ) and multiplying σ2 or σ3, we have

$∑j=1nδ(j)(x(j)-μ)+∑j=1n(1-δ(j)) (ajσ+bj(x(j)-μ))=0,$$-nuσ2+σ∑j=1n(1-δ(j))aj(x(j)-μ)+∑j=1nδ(j)(x(j)-μ)2+∑j=1n(1-δ(j))bj(x(j)-μ)2=0.$

From (2.6), we have the solution $μ~$,

$μ˜=D+Eσ$

with

$D=∑j=1nδ(j)x(j)+∑j=1n(1-δ(j))bjx(j)nu+∑j=1n(1-δ(j))bj, E=∑j=1n(1-δ(j))ajnu+∑j=1n(1-δ(j))bj.$

Substituting μ = D + into (2.7), we have

$Aσ2-Bσ-C=0$

with

$A=nu-nuE2+∑j=1n(1-δ(j)) (ajE-bjE2),B=-2E∑j=1nδ(j)(x(j)-D)-2E∑j=1n(1-δ(j))bj(x(j)-D)+∑j=1n(1-δ(j))aj(x(j)-D),C=∑j=1nδ(j)(x(j)-D)2+∑j=1n(1-δ(j))bj(x(j)-D)2.$

Using (2.9), we can easily see that A = nu, and the first two terms in B vanish. Hence the quadratic equation (2.10) in σ becomes

$nuσ2-Bσ-C=0$

with

$B=∑j=1n(1-δ(j))aj(x(j)-D)$

and C in (2.11). The solution $σ~$ of the quadratic equation (2.12) is

$σ˜=B+B2+4nuC2nu.$

Since we can show bi ≥ 0, it follows C ≥ 0 (see Remark 1 below). Therefore the other root of the equation (2.12) can not be a solution of σ. Note that the approximate MLEs in (2.8) and (2.13) lead to the MLEs $μ^$ = χ̄, $σ^2=∑j=1n(xj-x¯)2/n$ of the normal distribution for a complete sample.

Now let us think about the explicit form of ai and bi in Q(ξi) ≅ ai + biξi. Let

$ξi0=Φ-1(pi),$

where Φ−1 is the inverse of Φ, and pi is a plotting position or a quantile probability. By expanding Q(ξi) around ξi0 in the Taylor series keeping only the first two terms, Q(ξi) can be approximated by

$Q(ξi)≅Q(ξi0)+Q′(ξi0)(ξi-ξi0)≡ai+biξi$

with

$ai=Q(ξi0)-Q′(ξi0)ξi0,$$bi=Q′(ξi0)=Q(ξi0)(Q(ξi0)-ξi0).$

### Remark 1

The second equality in (2.16) follows easily using

$ϕ′(ξi0)=-ξi0ϕ(ξi0) and Φ¯′(ξi0)=-ϕ(ξi0).$

From

$bi(Φ¯(ξi0))2=ϕ(ξi0) (ϕ(ξi0)-ξi0Φ¯(ξi0)),ϕ(ξi0)-ξi0Φ¯(ξi0)≥ϕ(ξi0)-∫ξi0∞zϕ(z)dz≥ϕ(ξi0)+∫ξi0∞ϕ′(z)dz≥0,$

bi in (2.16) should be nonnegative, and C in (2.11) is also nonnegative. See also Kim (2014b), and Balakrishnan et al. (2003).

### Remark 2

Let Zi, i = 1, . . . , n, be a sample from a normal distribution with μ = 0, σ = 1, and let $μ^$z, $σ^$z be their MLEs. Then we can easily show

$μ^=σμ^z+μ, σ^=σσ^z$

from the likelihood equations (2.2), (2.3). The same relations hold for the approximate MLEs by the equations (2.8), (2.13). Hence we can assume the true value of μ = 0 and σ = 1 without loss of generality when we simulate the performance of the estimators.

For randomly censored data, the Kaplan-Meier estimator $piKM$ is usually used for the plotting position pi in (2.14),

$piKM=1-∏j≤i(n-jn-j+1)δ(j).$

The estimator has been studied in Kaplan and Meier (1958), Efron (1967), Breslow and Crowley (1974), and Meier (1975). Michael and Schucany (1986) suggested the modified Kaplan-Meier estimator $pi,cMS$,

$pi,cMS=1-n-c+1n-2c+1∏j≤i(n-j-c+1n-j-c+2)δ(j), 0≤c≤1,i=1,…,n,$

that reduces to pi,c = (ic)/(n – 2c + 1) for a complete sample. However $pi,cMS$ in (2.17) could have a negative value if the smallest data value is censored. Therefore we modify $pi,cMS$ again as

$pi,c=1-∏j≤i(n-j-c+1n-j-c+2)δ(j)(n-c+1n-2c+1)δ(1)(n-cn-2c+1)1-δ(1), 0≤c≤1,i=1,…,n.$

The last term in pi,c is to avoid 0 value. Note that we assumed the data x(1), . . . , x(n) are ordered observations and δ(1) corresponds to the indicator of the minimum. The particular choice of c is of little consequence, and popular values for c is c = 0 or c = 0.5. We often use the Blom (1958)’s position c = 3/8 = 0.375 for a normal distribution, because it approximates the expected value of the order statistics from the standard normal distribution.

Now we have defined ai, bi, ξi0, pi,c in (2.15), (2.16), (2.14), (2.18), respectively, and the approximate MLEs $μ~$, $σ~$ of μ, σ in (2.8), (2.13) have been completely defined.

3. Asymptotic variances and covariance

In this section, the observed Fisher information is computed to give the asymptotic variances and covariance of the MLEs $μ^$, $σ^$ or the approximate MLEs $μ~$, $σ~$ . From the likelihood equation (2.2), and

$∂ϕ(ξi)∂μ=ξiσϕ(ξi), ∂Φ¯(ξi)∂μ=1σϕ(ξi), ∂Q(ξi)∂μ=1σ(ξiQ(ξi)-(Q(ξi))2)≡1σJ(ξi),$

we obtain

$-∂2l∂μ2=1σ2v11 with v11=nu-∑j=1n(1-δ(j))J(ξj),$$-∂2l∂σtialμ=1σ2v12 with v12=2∑j=1nδ(j)ξj+∑j=1n(1-δ(j))Q(ξj)-∑j=1n(1-δ(j))ξjJ(ξj).$

From (2.3) and

$∂ϕ(ξi)∂σ=ξi2σϕ(ξi), ∂Φ¯(ξi)∂σ=ξiσϕ(ξi), ∂Q(ξi)∂σ=1σξiJ(ξi),$

we get

$-∂2l∂σ2=1σ2v22 with v22=-nu+3∑j=1nδ(j)ξj2+2∑j=1n(1-δ(j))ξjQ(ξj)-∑j=1n(1-δ(j))ξj2J(ξj).$

From (3.1), (3.2), and (3.3), the observed Fisher information matrix becomes

$I=1σ2[v11v12v12v22].$

By inverting (3.4), we obtain the asymptotic variance-covariance matrix of the estimates $μ^$ and $σ^$ as

$I-1=σ2[v11v12v12v22]-1≅[Var^(μ^)Cov^(μ^,σ^)Cov^(μ^,σ^)Var^(σ^)].$

From the approximate likelihood equations (2.4) and (2.5), we get

$-∂2l∂μ2≅1σ2v˜11 with v˜11=nu+∑j=1n(1-δ(j))bj,-∂2l∂σ∂μ≅1σ2v˜12 with v˜12=2∑j=1nδ(j)ξj+∑j=1n(1-δ(j)) (aj+bjξj)+∑j=1n(1-δ(j))bjξj,-∂2l∂σ2≅1σ2v˜22 with v˜22=-nu+3∑j=1nδ(j)ξj2+2∑j=1n(1-δ(j))ξj(aj+bjξj)+∑j=1n(1-δ(j))bjξj2,$

and the approximate observed Fisher information matrix $I~$,

$I˜=1σ2[v˜11v˜12v˜12v˜22],I˜-1=σ2[v˜11v˜12v˜12v˜22]-1≅[Var˜(μ˜)Cov˜(μ˜,σ˜)Cov˜(μ˜,σ˜)Var˜(σ˜)].$
4. Simulation study and example

### 4.1. Simulation results

We compare the performance of the MLEs with the approximate MLEs through a simulation study. To control the ratio of the censored data, we use two different random censoring models. One is the Koziol and Green (1976) censorship model, that is

$1-G=(1-F)β1 for some β1>0,$

where G is the distribution function of the censoring times C1, . . . ,Cn, and β1 is called a censoring parameter. Csörgő and Horváth (1981), Chen et al. (1982), and Kim (2014a) discussed the motivation and characterization of this model. According to Chen et al. (1982), the model could arise in a series system with two components. In this system, it can only work if both components are functioning. Let Ti be the life time of the first component with the distribution function F, and Ci be the second with G. Then Xi = min(Ti,Ci) should be the lifetime of the system. If the second component is also a series system of β1 independent and identically distributed subcomponents with F, then Ci should be the minimum of β1’s life times, and the distribution function G of Ci should be (4.1). Under the Koziol-Green model, the expected ratio of the censored data could be

$P(Ti>Ci)=∫-∞∞(1-F(x))dG(x)=∫01β1(1-x)β1dx=β1β1+1≡γ,$

and we call γ the censoring ratio of the model.

When the second component is parallel with β2 subcomponents, it works if at least one component is functioning. In this case the censoring distribution G becomes

$G=Fβ2 for some β2>0.$

We call it P model. The model is closely related to generalized exponential distribution. See Gupta and Kundu (2007), and Kim (2014a). In this case, the expected censoring ratio is

$P(Ti>Ci)=∫-∞∞(1-F(x))dG(x)=1β2+1≡γ.$

The model is also considered in Kim (2011). The two models are the same when β1 = β2 = 1. Tables 14 present the simulation results for the sample sizes n = 20, 30, 40, 50, and the censoring ratio γ = 3/5, 1/2, 1/3, 1/4 with N = 2000 repetitions for each of the censoring model in (4.1) and (4.2). The data are generated by S-plus package. They are generated with the true value of μ = 0, σ = 1 without loss of generality by Remark 2 in Section 2. Note that the MLEs of μ and σ are derived under the assumption that the distribution of the censoring time does not involve any parameters of interest, and the two models in (4.1) and (4.2) do not satisfy the assumption. However we usually do not know the true censoring model in a real situation, and the ratio of the simulated censored data could hardly be controlled without a censoring model.

Table 1 gives the averages of the MLEs of μ and σ, their variances and covariance by the simulation, and the asymptotic variances and covariance from the inverse of the observed Fisher information given in (3.5) under the Koziol-Green model in (4.1). The MLEs are computed by the S-plus function survReg. Table 2 gives the same statistics for the approximate MLEs under the same model. The last three columns are from the inverse of the approximated observed Fisher information given in (3.6). It also provides the average differences of the two estimates. In Tables 3 and 4, the same statistics are given under the P model in (4.2). For true values of σ not equal to 1, the variances and covariances of μ and σ should be multiplied by σ2 by (3.5) and (3.6).

From Tables 14, we observe the following. The average differences between the MLEs and the approximate MLEs are negligible. The differences tend to decrease as the sample size increases or if the censoring ratio becomes smaller. The variances of the estimators, not the mean squared errors, are provided since the biases of the parameters look very small. All the variances and covariances of the MLEs and the approximate MLEs are almost identical. The variances and covariances from the simulation and from the observed Fisher information are also similar. Apparently the approximate MLEs are as efficient as the MLEs. The variances and covariances also reduce considerably as the sample size becomes bigger or the censoring ratio becomes smaller. All the same phenomena happen in both the Koziol-Green model and P model. It seems that the censoring models do not have any clear influence on the estimation process, since we assumed that the censoring distribution does not have any information for the parameters. Kim (2016) studied the influence of the assumption for the censoring model on the estimation for the parameters of a Weibull distribution.

### 4.2. Example

As an illustrative example, we consider the tumor-free time data of the 30 rats fed with saturated diets. The data set is to investigate the relationship between diet and the development of tumors. The study divided 90 rats into three groups and fed them low fat, saturated fat, and unsaturated fat diets. The data were originally reported by King et al. (1979) and discussed in Lee and Wang (2003).

The data are sorted in order and given below. The other two groups are not shown.

$43465658687579818686899698105107110117124126133142142165170+200+200+200+200+200+200+$

Lee and Wang (2003, Chapter 8, 9) checked the Cox-Snell residual plot for the fitted lognormal model, and showed the goodness-of-fit tests based on the asymptotic likelihood inference and the BIC, AIC. As a result, the lognormal distribution would be selected rather than the exponential or the Weibull distribution by the procedures.

When we compute the MLEs of the parameters, we have

$μ^=4.764583, σ^=0.5605291,μ˜=4.762847, σ˜=0.5593185.$

The variance-covariance matrices in (3.5) and (3.6) are

$[Var^(μ^)Cov^(μ^,σ^)Cov^(μ^,σ^)Var^(σ^)]=[0.011270.0014010.0014010.007777],[Var˜(μ˜)Cov˜(μ˜,σ˜)Cov˜(μ˜,σ˜)Var˜(σ˜)]=[0.011400.0015000.0015000.007818].$

We can see that the results are very close each other.

5. Concluding remarks

In this study, the approximate MLEs for a normal distribution under random censoring are proposed by linearizing the nonlinear functions in the likelihood equations. As results, they give an explicit form and need no iterations contrary to the MLEs. In addition, the approximate MLEs are as efficient as the MLEs in terms of biases and variances from the simulation study. As for censoring models, we considered the Koziol-Green model and P model. Apparently, the censoring models do not have clear influence on the estimation process since we assumed that the censoring distribution does not involve any parameters of interest and ignore the information contained in the censoring model.

This paper assumes no covariates, and the problem with covariates remains a good research topic for studying.

TABLES

### Table 1

Averages of the MLEs $μ^$, $σ^$, variances of $μ^$, $σ^$ and covariances under the Koziol-Green model

γ (β1) n $μ^$ $σ^$ Var($μ^$) Var($σ^$) Cov($μ^$,$σ^$) $Var^(μ^)$ $Var^(σ^)$ $Cov^(μ^,σ^)$
$35(32)$ 20 0.016 0.962 0.122 0.077 0.053 0.126 0.079 0.057
30 0.009 0.976 0.072 0.045 0.030 0.074 0.048 0.032
40 0.004 0.974 0.051 0.033 0.021 0.052 0.034 0.022
50 −0.001 0.982 0.039 0.026 0.016 0.041 0.027 0.017

$12(1)$ 20 0.003 0.957 0.082 0.052 0.026 0.084 0.055 0.029
30 0.010 0.976 0.056 0.035 0.017 0.055 0.036 0.018
40 0.008 0.989 0.042 0.026 0.012 0.041 0.027 0.013
50 0.001 0.987 0.032 0.021 0.009 0.032 0.021 0.010

$13(12)$ 20 −0.003 0.955 0.063 0.039 0.011 0.061 0.038 0.012
30 0.003 0.974 0.042 0.027 0.008 0.041 0.026 0.008
40 −0.004 0.981 0.032 0.019 0.005 0.031 0.019 0.006
50 −0.003 0.984 0.027 0.016 0.005 0.025 0.015 0.004

$14(13)$ 20 −0.001 0.960 0.056 0.035 0.008 0.057 0.034 0.008
30 −0.001 0.974 0.039 0.021 0.004 0.038 0.022 0.005
40 0.000 0.981 0.028 0.017 0.003 0.028 0.017 0.003
50 −0.005 0.986 0.023 0.014 0.003 0.023 0.013 0.003

MLEs = maximum likelihood estimators.

### Table 2

Averages of the approximate MLEs $μ~$, $σ~$, variances of $μ~$, $σ~$ and covariances under the Koziol-Green model

γ (β1) n $μ~$ $σ~$ $μ^$$μ~$ $σ^$$σ~$ Var($μ~$) Var($σ~$) Cov($μ~$,$σ~$) $Var˜(μ˜)$ $Var˜(σ˜)$ $Cov˜(μ˜,σ˜)$
$35(32)$ 20 0.009 0.970 0.0069 −0.0075 0.120 0.073 0.052 0.129 0.079 0.059
30 0.004 0.979 0.0051 −0.0034 0.071 0.043 0.029 0.075 0.048 0.033
40 0.000 0.976 0.0036 −0.0021 0.051 0.032 0.020 0.053 0.034 0.022
50 −0.004 0.983 0.0030 −0.0013 0.039 0.026 0.015 0.041 0.027 0.017

$12(1)$ 20 −0.003 0.961 0.0056 −0.0039 0.081 0.051 0.026 0.086 0.055 0.030
30 0.006 0.978 0.0037 −0.0019 0.055 0.034 0.016 0.056 0.036 0.018
40 0.006 0.990 0.0027 −0.0011 0.041 0.026 0.012 0.042 0.027 0.013
50 −0.001 0.988 0.0022 −0.0007 0.032 0.021 0.009 0.032 0.021 0.010

$13(12)$ 20 −0.006 0.957 0.0031 −0.0015 0.062 0.039 0.012 0.062 0.038 0.012
30 0.002 0.975 0.0019 −0.0009 0.042 0.027 0.008 0.042 0.026 0.008
40 −0.005 0.981 0.0015 −0.0004 0.032 0.019 0.005 0.031 0.019 0.006
50 −0.004 0.985 0.0010 −0.0005 0.027 0.016 0.005 0.025 0.015 0.004

$14(13)$ 20 −0.003 0.961 0.0022 −0.0009 0.055 0.035 0.008 0.057 0.034 0.008
30 −0.003 0.974 0.0013 −0.0004 0.039 0.021 0.004 0.038 0.022 0.005
40 −0.001 0.981 0.0009 −0.0004 0.028 0.017 0.003 0.029 0.017 0.003
50 −0.006 0.986 0.0008 −0.0002 0.023 0.014 0.003 0.023 0.013 0.003

MLEs = maximum likelihood estimators.

### Table 3

Averages of the MLEs $μ^$, $σ^$, variances of $μ^$, $σ^$ and covariances under the P model

γ (β2) n $μ^$ $σ^$ $Var^(μ^)$ $Var^(σ^)$ $Cov^(μ^,σ^)$ $Var^(μ^)$ $Var^(σ^)$ $Cov^(μ^,σ^)$
$35(23)$ 20 0.015 0.964 0.107 0.070 0.040 0.120 0.073 0.050
30 0.008 0.975 0.075 0.047 0.030 0.074 0.046 0.029
40 0.005 0.979 0.049 0.033 0.019 0.052 0.032 0.020
50 −0.001 0.983 0.041 0.026 0.016 0.041 0.026 0.015

$12(1)$ 20 0.020 0.966 0.088 0.053 0.027 0.088 0.058 0.031
30 0.012 0.971 0.055 0.035 0.017 0.055 0.036 0.018
40 0.003 0.978 0.042 0.026 0.013 0.040 0.026 0.013
50 0.002 0.986 0.034 0.022 0.011 0.032 0.021 0.010

$13(2)$ 20 0.010 0.966 0.064 0.041 0.011 0.062 0.042 0.014
30 0.017 0.985 0.043 0.028 0.009 0.042 0.028 0.009
40 0.002 0.984 0.030 0.021 0.006 0.031 0.020 0.006
50 0.004 0.982 0.023 0.016 0.005 0.024 0.016 0.005

$14(3)$ 20 0.001 0.965 0.056 0.037 0.008 0.056 0.036 0.009
30 0.008 0.983 0.038 0.025 0.006 0.038 0.024 0.006
40 0.006 0.988 0.026 0.019 0.003 0.028 0.018 0.004
50 0.001 0.985 0.022 0.014 0.003 0.022 0.014 0.003

MLEs = maximum likelihood estimators.

### Table 4

Averages of the approximate MLEs $μ~$, $σ~$, variances of $μ~$, $σ~$ and covariances under the P model

γ (β2) n $μ~$ $σ~$ $μ^$$μ~$ $σ^$$σ~$ Var($μ~$) Var($σ~$) Cov($μ~$, $σ~$) $Var˜(μ˜)$ $Var˜(σ˜)$ $Cov˜(μ˜,σ˜)$
$35(23)$ 20 0.010 0.987 0.0044 −0.0229 0.108 0.063 0.042 0.126 0.074 0.053
30 0.004 0.987 0.0041 −0.0122 0.075 0.044 0.030 0.076 0.046 0.030
40 0.002 0.987 0.0030 −0.0082 0.049 0.031 0.019 0.053 0.032 0.020
50 −0.003 0.989 0.0025 −0.0059 0.041 0.025 0.016 0.041 0.025 0.015

$12(1)$ 20 0.014 0.970 0.0058 −0.0040 0.087 0.051 0.027 0.090 0.058 0.032
30 0.008 0.974 0.0037 −0.0021 0.054 0.034 0.017 0.055 0.036 0.018
40 0.001 0.979 0.0026 −0.0011 0.042 0.026 0.013 0.041 0.026 0.013
50 0.000 0.987 0.0022 −0.0007 0.034 0.022 0.011 0.033 0.021 0.010

$13(2)$ 20 0.007 0.965 0.0029 0.0006 0.063 0.040 0.011 0.063 0.042 0.014
30 0.015 0.985 0.0018 0.0005 0.043 0.027 0.009 0.042 0.028 0.009
40 0.001 0.983 0.0012 0.0002 0.030 0.021 0.006 0.031 0.020 0.006
50 0.003 0.981 0.0010 0.0003 0.023 0.016 0.005 0.024 0.016 0.005

$14(3)$ 20 −0.001 0.965 0.0018 0.0006 0.056 0.036 0.008 0.056 0.036 0.009
30 0.007 0.982 0.0011 0.0005 0.038 0.024 0.006 0.038 0.024 0.006
40 0.005 0.988 0.0008 0.0003 0.026 0.019 0.003 0.028 0.018 0.004
50 0.000 0.985 0.0006 0.0002 0.022 0.014 0.003 0.022 0.014 0.003

MLEs = maximum likelihood estimators.

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