Selçuk J. Appl. Math. Selçuk Journal of Special Issue. pp. 93-108, 2011 Applied Mathematics
Comparisons of Methods of Estimation for the Half-Logistic Distrib-ution
A. Asgharzadeh, R. Rezaie, M. Abdi
Department of Statistics, School of Mathematical Sciences, University of Mazandaran, Post code 47416-1467, Babolsar, Iran
e-mail: a.asgharzadeh@ um z.ac.ir
Abstract. In this paper, we consider several methods for estimating the lo-cation and scale parameters of the half-logistic distribution. The estimators considered are the maximum likelihood estimators, the approximate maximum likelihood estimators, method of moment estimators, estimators based on per-centiles, least squares estimators, weighted least squares estimators and the es-timators based on the linear combinations of order statistics. These eses-timators are compared via Monte Carlo simulations in terms of their biases and mean square errors.
Key words: Half-logistic distribution; Maximum likelihood estimator; Monte Carlo simulation; Method of moment estimators; Least squares estimators; Weighted least squares estimators; Percentiles estimators; L-estimators. 2000 Mathematics Subject Classification: 62F10,62G30.
1. Introduction
Consider the half-logistic distribution with probability density function (pdf) (1.1) g(y; μ, σ) = 2e−(y−μ)/σ
σ{1 + e−(y−μ)/σ}2, y > μ , σ > 0,
and cumulative distribution function (cdf) (1.2) G(y; μ, σ) = 1 − e
−(y−μ)/σ
1 + e−(y−μ)/σ, y > μ , σ > 0.
Here μ and σ are the location and scale parameters, respectively. We denote the half-logistic distribution with the location parameter μ and the scale parameter σ as HL(μ, σ). If Y follows HL(μ, σ), then the random variable X = (Y − μ)/σ has the standard half logistic distribution with pdf and cdf as
(1.3) f (x) = 2e−x
and
(1.4) F (x) = 1 − e−x
1 + e−x, x > 0,
respectively. It can be shown that (see Balakrishnan and Wong, 1991)
(1.5) E(X) = ln 4 and E(X2) = π
2
3 .
The half-logistic density function g(y; μ, σ) in (1.1) is a monotonic decreasing function in y and resembles an exponential density function. Also, its hazard function
h(y; μ, σ) = g(y; μ, σ) 1 − G(y; μ, σ) =
1
σ(1 + e−(y−μ)/σ)
is a monotonically increasing function of y and tends to σ1 as y → ∞, and so the distribution has an increasing failure rate. Hence, as mentioned by Polovko (1968), the half-logistic distribution can be used as a failure-time model in life-testing studies.
The increasing failure rate property of the half logistic distribution makes it suitable as a failure time model for a component which possibly has no manu-facturing defects but ages rapidly. Use of this distribution as a possible life-time model has been suggested by Balakrishnan (1985) who has derived several re-currence relations for the single and the product moments of order statistics. Later, Balakrishnan and Puthenpura (1986) considered data representing fail-ure times for a specific type of electrical insulation subjected to continuously increasing voltage stress. They demonstrated that for such applications, the half-logistic distribution was an appropriate failure-time model. For more de-tails and applications, see Balakrishnan and Puthenpura (1986), Balakrishnan and Wong (1991), Balakrishnan and Chan (1992), Adatia (2000) and Balakr-ishnan and Asgharzadeh (2005) who have discussed some inferential procedures for the half-logistic distribution. In this paper the half-logistic distribution has been considered due to its possible applications in applied statistics.
The objective of this paper is to compare several methods for estimating the location and scale parameters of the half-logistic distribution. We compare the maximum likelihood estimates (MLEs), the approximate maximum likelihood estimates (AMLEs), method of moment estimates (MMEs), estimates based on percentiles (PCEs), least squares estimates (LSEs), weighted least squares estimates (WLSEs) and the estimates based on the linear combinations of order statistics (LMEs) with respect to their biases and mean square errors (MSEs) via the Monte Carlo simulation.
The paper is organized as follows. In Section 2, we discuss the MLEs of the location and scale parameters the half-logistic distribution. Section 3 provides an explicit estimator for the scale parameter by appropriately approximating the likelihood function. In Sections 4 to 7 we discuss other methods. Simulation results are provided in Section 8.
2. Maximum Likelihood Estimators
In this section, the maximum likelihood estimates of HL(μ, σ) are considered. We consider two different cases. First consider estimation of μ and σ when both are unknown. If y1, · · · , yn is a random sample from HL(μ, σ), then the
likelihood function, L(μ, σ), for μ < y1:n= min(y1, · · · , yn) is
(2.1) L(μ, σ) = n Y i=1 g(yi, μ, σ) = 2nσ−ne− Sn i=1(yi−μ)/σ n Y i=1 [1 + e−(yi−μ)/σ]−2.
The log-likelihood function is
(2.2) ln L(μ, σ) = n ln 2 − n ln σ − 1 σ n X i=1 (yi− μ) − 2 n X i=1 ln[1 + e−(yi−μ)/σ].
The log-likelihood function is a monotonically increasing function of μ, thus the MLE of μ (say ˆμM LE) is (2.3) μˆM LE= y1:n. Putting ˆμM LEin (2.2), we obtain (2.4) ln L(y1:n, σ) = n ln 2−n ln σ − 1 σ n X i=1 (yi−y1:n)−2 n X i=1
ln[1+e−(yi−y1:n)/σ].
It is easy to show that the MLE of σ ( say ˆσM LE) which maximizes the
log-likelihood function (2.4) with respect to σ, can be obtained from the fixed point solution of (2.5) h(σ) = σ, where (2.6) h(σ) = ¯y − y1:n− 2 n n X i=1 (yi− y1:n)e−(yi−y1:n)/σ 1 + e−(yi−y1:n)/σ .
The simple iterative procedure
h(σ(j)) = σ(j+1), j = 1, 2, 3, ...
can be used to obtain the MLE, where σ(j)is jth iterative value (j = 1, 2, 3, ...)
and σ(1) is an initiate value.
Now consider the MLE of σ when the location parameter μ is known. Without loss of generality we can take μ = 0. For known μ, the MLE of σ, say ˆσM LELOK,
can be obtained by maximizing (2.7) U (σ) = −n ln σ −1 σ n X i=1 yi− 2 n X i=1 ln(1 + e−yi/σ),
with respect to σ. Now ˆσM LELOK, which maximizes U (σ) can be obtained as
the fixed point solution of
(2.8) ν(σ) = σ, where (2.9) ν(σ) = ¯y − 2 n n X i=1 yie−yi/σ 1 + e−yi/σ. 3. Approximate MLE
Suppose y1, ..., yn is a random sample of size n from HL(μ, σ) and y1:n≤ ... ≤
yn:n denote the ordered statistics of the sample. As mentioned before, the
maximum likelihood method does not provide an explicit estimator for the scale parameter σ. Balakrishnan and Wong (1991) proposed a simple method of deriving an explicit estimator by approximating the likelihood function. By approximating the likelihood function, the approximate MLE of σ (for more details see Balakrishnan and Wong, 1991) is given by
(3.1) σˆAM LE= B +√B2+ 8AC 4A , where A = n, B = 2 n X i=2 αi(yi:n− y1:n), C = 2 n X i=2 βi(yi:n− y1:n)2, αi= pi− 1 2qi(1 + pi) ln( 1 + pi qi ), βi= qi(1 + pi) 2 , pi= i n + 1, qi= 1 − pi.
If the location parameter μ is known, we put the real value of μ instead of y1:n
in (3.1).
4. Method of Moment Estimators
If Y follows HL(μ, σ), then the random variable X = (Y −μ)/σ has the standard half logistic distribution HL(0, 1). From (1.5), we have
E(Y ) = μ + σE(X) = μ + σ ln 4 and
E(Y2) = E(μ + σX)2= μ2+ 2μσ ln 4 +σ
2π2
First let us consider the case, when both the parameters are unknown. To obtain the MMEs of the unknown parameters μ and σ, we need to equate the sample moments with the population moments and solve the following equations
(4.1) μ + σ ln 4 = ¯y
(4.2) μ2+ 2μσ ln 4 + σ
2π2
3 = y
2,
where y = (1/n)Pni=1yi and y2 = (1/n)Pni=1y2i. We obtain the MMEs as
follows (4.3) μˆM M E= ¯y − ln 4 s y2− ¯y2 π2 3 − (ln 4)2 , and (4.4) σˆM M E = s y2− ¯y2 π2 3 − (ln 4)2 .
If the location parameter μ is known, then the MME of σ is
(4.5) ˆσM M ELOK=
¯ y − μ
ln 4 . 5. Estimators Based on Percentiles
This method is used when the data come from a distribution function which has a closed form. In this case the unknown parameters can be estimated by equating the sample percentile points with the population percentile points and it is known as the percentile method. This method was first introduced in Kao (1958, 1959) for estimating Weibull parameters. Many authors have used this method of estimation including Mann, Schafer and Singpurwalla (1974), Johnson, Kotz and Balakrishnan (1994), Gupta and Kundu (2001), Kundu and Raqab (2005), Kantar and Senoglu (2008) and Alkasabeh and Raqab (2009). In this section, we apply this method of estimation for the half-logistic distri-bution because of the nature of its distridistri-bution function. First let us consider the case when both the parameters are unknown. Since
F (y; μ, σ) = 1 − e −(y−μ)/σ 1 + e−(y−μ)/σ, therefore (5.1) σ ln[1 + F (y; μ, σ) 1 − F (y; μ, σ)] + μ = y.
If pi denotes some estimate of F (yi:n; μ, σ) where yi:nis the i th order statistic,
then the PCEs of μ and σ can be obtained by minimizing (5.2) n X i=1 [yi:n− μ − σ ln( 1 + pi 1 − pi )]2,
with respect to μ and σ. By differentiating (5.2) with respect to μ and σ, respectively, and equating them to zero, we obtain the PCEs of μ and σ as
(5.3) bμP CE = y −σb n n X i=1 ln(1 + pi 1 − pi ), and (5.4) σbP CE = Pn i=1[ln( 1+pi 1−pi) − 1 n Pn i=1ln( 1+pi 1−pi)](yi:n− y) Pn i=1[ln( 1+pi 1−pi) − 1 n Pn i=1ln( 1+pi 1−pi)] 2 .
We use pi= i/(n+1) as an estimator of F (yi:n) which is the most used estimator
of F (yi:n).
If the location parameter μ is known, without loss of generality we can assume μ = 0. Similarly as before the percentile estimator of σ for μ = 0 can be obtained by minimizing (5.2) with respect to σ only. Solving
∂ ∂σ n X i=1 [yi:n− σ ln( 1 + pi 1 − pi )]2= 0,
the PCE of σ, say ˆσP CELOK, becomes
(5.5) ˆσP CELOK= Pn i=1yi:nln( 1+pi 1−pi) Pn i=1[ln( 1+pi 1−pi)] 2 .
6. Least Squares and Weighted Least Squares Estimators
The method of least squares goes back to Gauss, who first developed this tech-nique of estimation. Generally it is used for estimation of parameters in a linear model. The least squares estimates and weighted least squares estimates were used by Swain, Venkatraman and Wilson (1988) to estimate the parameters of beta distributions. See also Gupta and Kundu ( 2001), Kundu and Raqab (2005), Kantar and Senoglu (2008) and Alkasabeh and Raqab (2009).
Let Y1:n ≤ ... ≤ Yn:n be order statistics from a random sample of size n from
a distribution function G(.). From the fact that G(Yj:n) behaves like j th order
statistic of a sample of size n from U (0, 1), we have E[G(Yj:n)] =
j
n + 1, V ar[G(Yj:n)] =
j(n − j + 1) (n + 1)2(n + 2),
and
Cov[G(Yi:n), G(Yj:n)] = i(n − j + 1)
(n + 1)2(n + 2), i < j.
The least squares estimates can be obtained by minimizing (6.1) n X j=1 [G(Yj:n) − j n + 1] 2
with respect to the unknown parameters. Therefore, in case of HL distribution, the least squares estimates of μ and σ, say ˆμLSE and ˆσLSE, respectively, can
be obtained by minimizing (6.2) n X j=1 [1 − e−(y j:n−μ)/σ 1 + e−(yj:n−μ)/σ − j n + 1] 2,
with respect to μ and σ.
The weighted least squares estimates of the unknown parameters can be ob-tained by minimizing (6.3) n X j=1 wj[G(Yj:n) − j n + 1] 2,
with respect to the unknown parameters, where wj = 1 V ar[G(Xj:n)] = (n + 1) 2(n + 2) j(n − j + 1) .
Therefore, in case of HL distribution the weighted least squares of μ and σ, say ˆ
μW LSE and ˆσW LSE, respectively, can be obtained by minimizing
(6.4) n X j=1 wj[1 − e −(yj:n−μ)/σ 1 + e−(yj:n−μ)/σ − j n + 1] 2,
with respect to μ and σ only. 7. L-Moment Estimators
The L-moment estimates (LMEs) are analogous to the conventional moment estimates but obtained based on the linear combinations of order statistics, i.e., by L-statistics, see for example David (1981) or Hosking (1990) for more details. The LMEs have certain advantages over the conventional moment estimates of being more robust to the presence of outliers in the data. It is observed that LMEs are less subject to bias in estimation and sometimes more accurate in small samples than even the MLEs.
First we discuss the case how to obtain the LMEs when both the parameters of a HL distribution are unknown. If y1:n≤ ... ≤ yn:n denote the ordered sample
then using the same notation as Hosking (1990), we obtain the first and second sample L-moments as (7.1) l1= 1 n n X i=1 yi:n and (7.2) l2= 2 n(n − 1) n X i=1 (i − 1)yi:n− l1.
Similarly, the first two population L-moments of the HL distribution are found to be (7.3) L1= E(Y1:1) = σ ln 4 + μ, and (7.4) L2= 12[E(Y2:2) − E(Y1:2)] = 12[(2σ + μ) − (2σ ln 4 − 2σ + μ)] = (2 − ln 4)σ.
Note that (7.3) and (7.4) follow from the single moments E(Yr:n) (1 ≤ r ≤ n)
from the half-logistic distribution (see Balakrishnan and Puthenpura, 1986). Now to obtain the LMEs of the unknown parameters μ and σ, we need to equate the sample L-moments with the population L- moments. Therefore, the LMEs can be obtained by solving the following equations:
(7.5) l1= μ + σ ln 4, and (7.6) 2 n(n − 1) n X i=1 (i − 1)yi:n− l1= (2 − ln 4)σ.
We obtain the LME of σ, say ˆσLM E, as
(7.7) σˆLM E= 1 2 − ln4[ 2 n(n − 1) n X i=1 (i − 1)yi:n− l1].
Once ˆσLM E is obtained, the LME of μ, say ˆμLM E, can be obtained from (7.5)
as
(7.8) μˆLM E= l1− (ln 4)ˆσLM E.
If the location parameter μ is known, the LME of σ, say ˆσLM ELOK, becomes
(7.9) σˆLM ELOK =
l1− μ
It is noted here that if μ is known, then the LME of σ is the same as the method of moment estimator given in Section 4.
8. Simulation Results
A simulation study was carried out to compare the performance of the different estimates proposed in the previous sections. We compare the different estimates in the sense of bias and mean square error (MSE). We generated random samples from the half logistic HL(μ, σ) distribution for the sample sizes 20, 40, 60, 80 and 100. We then computed the average biases of the various estimates and their MSEs over 10000 replications. Note that the generation of the HL(μ, σ) is very simple. If U is uniform (0, 1), then X = σ ln(1+U1−U) + μ is HL(μ, σ). 8.1. Estimation of σ when μ is known
If the location parameter μ is known, without loss of generality we can take μ = 0. In this case, the MLE of σ can be obtained by maximizing (2.7) or equivalently solving for the fixed point solution of (2.8). The AMLE, MME, PCE and LME of σ can be obtained directly from (3.1), (4.5), (5.5) and (7.9) respectively. The LSE and WLSE of σ can be obtained by minimizing (6.2) and (6.4), respectively, with respect to σ only. Eqs. (6.2) and (6.4) can be minimized by using the functions "optim" or "nlm" from the statistical software "SPLUS". Since σ is the scale parameter, we take σ = 1 without loss of generality. Table 1 provides the average biases of ˆσ and average MSEs of ˆσ over 10000 replications. From Table 1, we observe that as sample size increases for almost all the meth-ods, the bias and the MSE of the estimates reduce appreciably. We note that most of the estimates usually overestimate σ for n = 20, 40, and 60. For n = 80, 100 (large sample), the MLEs, MMEs and LMEs underestimate σ. Comparing the biases, it is observed that the MLEs and MMEs perform well for all sample sizes considered. We also observe the PCEs have the maximum biases. With respect to the MSEs, the MLEs, AMLEs and MMEs work better than the other estimates. The WLSEs work better than the LSEs in all cases considered. For a clear observation, the average biases and average MSEs of the different estimates of the scale parameter σ when μ is known are presented in Figures 1 and 2.
8.2. Estimation of μ and σ when both are unknown
In this subsection, we consider the estimation of μ and σ when both of them are unknown. The ˆμM LEcan be obtained from (2.3) and the ˆσM LEcan be obtained
from the fixed point solution of (2.5). The ˆσAM LEcan be obtained directly from
(3.1). The ˆμM M E and ˆσM M E can be obtained directly from (4.3) and (4.4).
The ˆσLM E can be obtained from (7.7) and then the ˆμLM E can be obtained
from (7.8). The LSEs and WLSEs can be obtained by minimizing (6.2) and (6.4), respectively, with respect to μ and σ. The PCEs can be obtained directly from (5.3) and (5.4). We take n = 20, 40, 60, 80, 100 and then generate random
samples of size n from the standard half logistic HL(μ = 0, σ = 1) distribution. We report the average biases of bμ and bσ and also the corresponding average MSEs in Table 2.
From Table 2, the average biases and average MSEs decrease as sample size increases. Therefore all estimates are asymptotically unbiased and consistent. We observe that the MLEs, MMEs, and AMLEs of μ overestimate μ while the other estimates underestimate μ. We also observe that the MLEs, MMEs, LMEs and AMLEs of σ underestimate σ while the other estimates overestimate σ.
Figure 1. Average biases of the different estimators of σ for different sample size n when μ is known
Figure 2. Average MSEs of the different estimators of σ for different sample size n when μ is known
Figure 3.Average biases of the different estimators of μ for different sample size n
Figure 4. Average biases of the differente stimators of σ for different sample size n
Figure 5. Average MSEs of the different estimators of μ for different sample size n
Figure 6. Average MSEs of the different estimators of σ for different sample size n
Comparing the performance of all the estimates, it is clear that as far as biases are concerned, LSEs work the best for all the sample sizes considered for
esti-mating μ while LME’s are the best for estiesti-mating σ . Considering the MSEs, the WLSEs provide the best results for estimating μ when n ≤ 40 while the MLEs and AMLEs work well otherwise. For estimating σ, the MLEs and AMLEs provide the best results for all the sample sizes considered. For better under-standing, the average biases of the different estimates of μ and σ are presented in Figures 3 and 4 for different sample sizes. Figures 5 and 6 show that the average MSEs of the different estimates of μ and σ for different sample sizes. 9. Acknowledgements
The authors wish to thank the editor and the referee for their helpful comments which led to improvement of this paper. This work is partially supported by the "Research Center of Algebratic Hyperstructures and Fuzzy Mathematics of the University of Mazandaran".
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