On Inference about Tilt Parameter in MarshallOlkin Family of Distributions
 DOI
 10.2991/jsta.2018.17.2.6How to use a DOI?
 Keywords
 Tilt parameter; MarshallOlkin distribution; Maximum likelihood estimation; Maximum spacing estimation; Leastsquares estimation; coverage probability; score test; Insulating fluid data
 Abstract
Marshall and Olkin [
] introduced a method for constructing a new distribution by adding a new parameter, called tilt parameter, to a parent distribution. It is observed that adding this parameter leads to a more flexible model than the parent model. In this paper, different estimators for tilt parameter as a major parameter are presented. Their performances are compared using Monte Carlo simulations. Hypothesis testing and interval estimation of tilt parameter using Rao score test is discussed.  Copyright
 Copyright © 2018, the Authors. Published by Atlantis Press.
 Open Access
 This is an open access article under the CC BYNC license (http://creativecommons.org/licences/bync/4.0/).
1. Introduction
Let X be a random variable with cumulative distribution function G(x ) and probability density function g(x ). Ref. 11 proposed a method for adding a new parameter to a distribution family. If Ḡ(x ) denote the survival function of X then survival function of MarshallOlkin family of distributions defined by:
If X is a random variable with survival function (1.1) we write X~MO(α). In literature, α is called tilt parameter.G(x) may be have some parameters. The probability density function and the cumulative distribution function is related by (1.1) are given by:
Several new distributions have been introduced from this method. Adding a new parameter leads to a more flexible model than baseline model. A generalized version of a distribution often has nice structural properties in application. For example the exponential distribution has a fixed failure rate function and so this distribution doesn’t have a good fitting to the data in many reliability applications. But a generalized exponential model, such as MarshallOlkin exponential, has a failure rate with different shapes for different values of parameters. In MarshallOlkin distribution family, tilt parameter makes this nice property. This is the motivation for considering statistical inferences of tilt parameter in this paper.
Maximum likelihood and moment estimation of tilt parameter for a specific parent distribution have been studied by several authors. For further discussions see Ref. 11, Ref. 5, Ref. 6 and Ref. 15. Ref. 8 presented MLE and Bayesian estimation of tilt parameter in a general class of MarshallOlkin distribution. Also they obtained some estimators for reliability of a system by this distribution. Ref. 2 considered different estimators of parameters of MarshallOlkin exponential distribution. In addition Ref. 7 found the estimation of reliability from MarshallOlkin extended Lomax distribution.
In this paper we will discuss several methods for estimating tilt parameter in MarshallOlkin distribution that will be denoted by MO(α). Also we will discuss hypothesis testing to tilt parameter. The rest of paper is organized as follow: In Section 2, the maximum likelihood estimation is investigated. In Section 3, the estimation of tilt parameter is discussed by using maximum spacing method. Least square and weighted least square estimators are discussed in Section 4. Hypothesis testing based on score test statistic and confidence interval for tilt parameter are proposed in Section 5. In Section 6, simulation results and comparison of estimators are provided. Also the coverage probabilities of confidence intervals and Rao Score test statistic are obtained. In a real dataset the statistical inferences about a particular distribution in MarshallOlkin family of distributions, are discussed in section 7.
2. Maximum Likelihood Estimation
Let X _{1},…X _{n} be a random sample of size n from MO(α). The likelihood function L(α) can be written as
And loglikelihood function is given by
So
The fisher information of α is given by
Using change of variables:
For example, let the parent distribution be exponential with survival function
Substituting (2.4) in (1.1), we have MarshallOlkin extended exponential that is noted by Ref. 11. The probability density function of this distribution is given by
As customary, a random variable X with the density function (2.5) will be denoted by MOEE (α,λ). In this paper we focused on inference about tilt parameter, but since it is not reasonable and practical to consider one parameter for the new model and considering all other parameters to be known involved in the model, we suppose α and λ in MOEE (α,λ) are unknown. Thus by calculating loglikelihood function of (2.5) in a random samples, we have
These equations should be solved simultaneously to obtain maximum likelihood estimators. Statistical software can be used to solve them numerically using iterative methods.
3. Maximum Spacing Estimation
Maximum spacing (MSP) method is introduced by Ref. 3 as an alternative to maximum likelihood method. Ref. 13 derived MSP method from an approximation of the KullbackLeibler divergence (KLD). Again let x_{1},…x_{n} be a random sample from a distribution function F(x,θ). Suppose f (x,θ) is the probability density function. KullbackLeibler divergence between F(x,θ) and F(x,θ_{0}) is given by
The KLD is 0 if and only if F(x,θ) = F(x,θ_{0}) for all x. For estimating θ_{0} a perfect method should make the divergence between the model and the true distribution as small as possible. In applications, this can be checked by estimating H(F_{θ},F_{θ0}) by
So by minimizing (3.1) with respect to θ , the estimator of θ_{0} can be found, that is the wellknown MLE. But in some continuous distribution, logf (x _{i}), i = 1,…, n , is not bounded above. Ref. 13 suggested another approximation of the KLD, namely
The estimator that obtained by minimizing (3.2) is called MSP estimator of θ_{0}. In regular problems, minimizing (3.2) is approximately equivalent to maximizing the loglikelihood function. It is clear that minimizing (3.2) is equivalent to maximizing:
When the likelihood function of θ is unbounded or in distributions with a parameterdependent lower bound such as threeparameter lognormal, weibull and gamma, the MSP estimator (MSPE) has been shown to have better performance than the maximum likelihood estimator (MLE). For more details, see Ref. 13 and Ref. 1. Ref. 4 showed that in small samples, MSPE is more efficient than the MLE. Based on Ref. 4, using (3.3) instead of a maximizing loglikelihood, three different problems can be solved as the same time. (i) We can test a proposed model is correct or not. (ii) An estimation of an unknown parameter can be obtained and (iii) By using approximation theory we can obtain a confidence region for unknown parameter. In section 6, we obtained MSPEs, when X has a MarshallOlkin exponential distribution.
4. Least Squares and Weighted Least Squares Estimation
The least squares and weighted least squares estimators were originally introduced by Ref. 16 to estimate the parameters of Beta distributions. It is intuitively obvious and has long been known that:
Thus the least squares estimator (LSE) of an unknown parameter can be obtained by minimizing
With respect to unknown parameter.
Similarly the weighted least squares estimator (WLSE) can be obtained by minimizing
5. Hypothesis Testing and Confidence Intervals
For completeness purposes, in this section, we briefly discuss hypothesis testing for null hypothesis H_{0} : α = 1 against H_{1} : α ≠ 1, in a MarshallOlkin family of distribution when the parent distribution doesn’t have any unknown parameter. There are different method for this purpose based on likelihood function, such as likelihood ratio test, score test and Wald test. Because of some advantage, we use score test for testing H_{0}. In addition we propose two approximate confidence intervals for tilt parameter.
5.1. Score Test for α = 1
Suppose ℓ(α) is loglikelihood function and
Where I(α) is the fisher information of tilt parameter.
U(α) and I(α) is presented in section two. Under null hypothesis, S has asymptotically chisquare distribution with 1 degree of freedom, so the null hypothesis is rejected when
And under null hypothesis
The generalization version of (5.1) is
Under null hypothesis, test statistic in (5.4) has asymptotically chisquare distribution with k degree of freedom when k is the number of components of θ. The score test statistic is useful because it is simple to compute and depends only on estimates of parameters under null hypothesis. Also the score test has the same local efficiency as the Likelihood Ratio test. Furthermore the distribution of score test statistic is not affected by parameters being on the boundary of the parameter space under null hypothesis. For further discussion about score tests see Ref. 4.
Again, let X~MOEE(α, λ). It is interested to test H_{0} : α = 1 against H_{1} : α ≠ 1 with score test method. At the first suppose λ is known, so from (5.3), the null hypothesis is rejected when
But when λ be unknown parameter, using (2.6) and (2.7) we have
So score test statistic for H_{0} : α = 1 in presence λ as a nuisance parameter is given by (5.4) where
5.2. Confidence Interval for α
In this section we assume all parameters expect than tilt parameter in MarshallOlkin extended distribution be known. The normal approximation of the MLE of α can be used for constructing approximate confidence intervals. Under conditions that are fulfilled for the parameters in the interior of the parameter space, we have
Where
On the other hand for obtaining confidence interval for α, it is interested to use score test statistic that is discussed in previous subsection. According to (5.1) the approximate confidence interval for tilt parameter, when there is no any other unknown parameter in the model is obtained from:
Equation (5.8) can be used for obtaining confidence interval for tilt parameter in MarshallOlkin extended exponential distribution.
6. Simulations
It is presented different estimators of tilt parameter that discussed in previous sections. In this section we compare the performance of these estimators by using Monte Carlo simulations. The biases and root mean square errors (RMSEs) of different estimators of α and λ in a MarshallOlkin extended exponential distribution are presented in Table 1. These criteria were computed by simulating samples of size n = 10 and 30, each sample replicated 5000 times. The values of the α are 0.25; 1 and 2.5. In all cases we take λ = 1. The different shapes of density of MOEE(α, λ) are shown in Fig 1. We can observe that when λ is fixed, the skewness of density gets to small value with increasing α.
α  Estimators  bias  RMSE  






0.25  MLE  0.2165  0.4239  0.1898  0.5616 
MSPE  0.0056  −0.2104  0.1086  0.6506  
LSE  0.7524  −0.0020  0.8521  0.2820  
WLSE  0.0243  −0.1797  0.1292  0.7192  
1  MLE  0.3232  0.1930  0.5220  0.2652 
MSPE  −0.1402  −0.2099  0.5051  0.3254  
LSE  0.0025  −0.0012  0.2889  0.2826  
WLSE  −0.0658  −0.1782  0.5572  0.3560  
2.5  MLE  −0.7504  −0.0914  0.7827  0.0838 
MSPE  −1.1159  −0.2977  1.7021  0.1795  
LSE  −1.4984  0.0024  2.5261  0.2882  
WLSE  −1.0617  −0.3055  1.5987  0.1867 
biases and RMSEs for different estimators of parameters of MarshallOlkin Exponential when λ=1 and n=10.
From Table 1 and 2,it is observed that the MSP performs the best among all methods to estimates a for small values of a since a is a shape parameter and based on figure 1 we can see that in these cases the density is skewed. As noted before MSP method have good performance when the distribution is skewed or heavytailed. For estimating λ, the LS method is the best for small values of α. But when α = 2.5 the ML method is the best among all methods. In addition with increasing sample size, the performance of the MSPEs gets to the MLEs.
α  Estimators  bias  RMSE  






0.25  MLE  0.1068  0.2480  0.0581  0.3455 
MSPE  −0.0169  −0.1553  0.0386  0.3530  
LSE  0.7535  −0.0059  0.8497  0.2780  
WLSE  0.0315  −0.0249  0.0582  0.4571 
1  MLE  0.2059  0.0920  0.3395  0.1210 
MSPE  −0.0893  −0.1233  0.2875  0.1309  
LSE  0.0001  0.0024  0.2878  0.2879  
WLSE  0.0577  −0.0290  0.3496  0.1478  
2.5  MLE  −0.6633  −0.1020  0.5447  0.0346 
MSPE  −0.8444  −0.1892  0.9223  0.0656  
LSE  −1.5011  0.0061  2.5312  0.2869  
WLSE  −0.7684  −0.1797  0.7737  0.0596 
biases and RMSEs for different estimators of parameters of MarshallOlkin Exponential when λ=1 and n=30.
In table 3, For different values of sample size and α, we determined the coverage probabilities of the 90%, 95% and 99% confidence intervals for α by two methods: Confidence interval based on an asymptotic normal pivotal quantity that is obtained from (5.7) and confidence interval based on score test method that is denoted in (5.8). In all cases we assume X~MOEE(α,λ).
Sample size  α  90% CI  95% CI  99% CI  

ML  SC  ML  SC  ML  SC  
n=10  0.25  88.39  90.71  90.94  95.35  94.42  99.09 
1  87.71  90.22  90.79  94.96  94.38  99.08  
2.5  88.06  90.58  90.32  95.10  94.72  99.15  
n=30  0.25  89.80  90.39  93.43  94.89  96.89  99.20 
1  89.22  89.91  93.84  95.50  96.99  99.05  
2.5  89.45  90.11  93.33  94.77  97.08  99.07  
n=50  0.25  90.20  90.38  93.94  94.97  97.95  99.12 
1  90.22  89.80  94.15  95.16  97.99  99.18  
2.5  89.94  89.83  93.88  94.88  97.72  98.95 
Coverage probabilities (in %) of confidence interval based on maximum likelihood (ML) and score test (SC) methods for α when λ = 1.
From Table 3, it is clear that the SC confidence interval (based on score test statistic) seems to have considerably higher coverage probabilities compared to the ML confidence interval that is based on asymptotic distribution of MLE.
7. Real Data
For further discussions, we analyze times to breakdown (in minutes) of an insulating fluid subjected to high voltage stress, which was reported by Ref. 12 (p. 462). We use group 3 of data for our goal.
Plots of the estimated density functions of MarshallOlkin extended exponential and exponential models based on MLEs are given in Fig 2. It is evident that the MOEE model provides a better fit than the old model.
In Table 4, the MLEs of the model parameters and some statistics, such as negative loglikelihood and Akaike Information Criterion (AIC), are listed. From this table, the MOEE distribution has lower 2LogLik and AIC values than Exponential, and so it could be chosen as the better model. In addition the Score statistic for testing the hypothesis H_{0} : α = 1 against H_{1} : α ≠ 1 or equally H_{0} : Exp(λ) against H_{1} : MOEE(α,λ) is
Estimates  Statistics  

distribution  α  λ  −2LogLik  AIC 
MOEE  4.7352  1.1510  29.1428  33.14 
Exponential  1  0.5720  31.1694  33.16 
MLEs and the measures 2LogLik and AIC
8. Conclusions
Ref. 11 proposed a simple generalization of a baseline distribution function by adding a tilt parameter α> 0 in order to obtain a larger class of distribution functions, which contains the parent distribution when α = 1. In this paper we investigated statistical inference about tilt parameter. We calculated different estimator for tilt parameter and studied their performances. Also we discussed about hypothesis testing and interval estimation of tilt parameter based on score test statistic. Finally in a real dataset, we fitted a MarshallOlkin extended exponential and obtain MLEs of its parameters.
References
Cite this article
TY  JOUR AU  Mostafa Tamandi PY  2018 DA  2018/06/30 TI  On Inference about Tilt Parameter in MarshallOlkin Family of Distributions JO  Journal of Statistical Theory and Applications SP  261 EP  270 VL  17 IS  2 SN  22141766 UR  https://doi.org/10.2991/jsta.2018.17.2.6 DO  10.2991/jsta.2018.17.2.6 ID  Tamandi2018 ER 