Maximum $\log_q$ Likelihood Estimation for Parameters of Weibull Distribution and Properties: Monte Carlo Simulation

The maximum ${\log}_q$ likelihood estimation method is a generalization of the known maximum $\log$ likelihood method to overcome the problem for modeling non-identical observations (inliers and outliers). The parameter $q$ is a tuning constant to manage the modeling capability. Weibull is a flexible and popular distribution for problems in engineering. In this study, this method is used to estimate the parameters of Weibull distribution when non-identical observations exist. Since the main idea is based on modeling capability of objective function $\rho(x;\boldsymbol{\theta})=\log_q\big[f(x;\boldsymbol{\theta})\big]$, we observe that the finiteness of score functions cannot play a role in the robust estimation for inliers. The properties of Weibull distribution are examined. In the numerical experiment, the parameters of Weibull distribution are estimated by $\log_q$ and its special form, $\log$, likelihood methods if the different designs of contamination into underlying Weibull distribution are applied. The optimization is performed via genetic algorithm. The modeling competence of $\rho(x;\boldsymbol{\theta})$ and insensitiveness to non-identical observations are observed by Monte Carlo simulation. The value of $q$ can be chosen by use of the mean squared error in simulation and the $p$-value of Kolmogorov-Smirnov test statistic used for evaluation of fitting competence. Thus, we can overcome the problem about determining of the value of $q$ for real data sets.


I. INTRODUCTION
After the study of a real-world phenomenon or the realization of an experiment, it may be desirable to model the experimental data by means of a proposed parametric model f (x; θ).In other words, the experimental data set is assumed to be a member of a parametric model.However, it cannot be a realistic assumption for the real world which will be modeled only with certain values of parameters in a model.The observations can be mixed with a different parameter of the same distribution or a different distribution.That is, a contamination exists into majority of the arXiv:2012.08294v1[stat.ME] 15 Dec 2020 distribution, which leads to have non-identical observations.The type of contamination is defined as deviant observations.In other words, one or more observations are made to behave differently from what it is present when creating a deviant observation.Deviant observations can be divided into inlier and outlier deviations.The data set may have both inward/inlier and outward/outlier deviations at the same time.Inward deviations are generated by short-tailed distributions and outward deviations by thick-tailed distributions.For inward deviations, it can also be realized by generating random numbers from uniform distribution in the closed range [a, b].In fact, deviant observations in the data set indicate that an assumption trusting on identically distributed random variables will not be realistic to model a phenomenia ( [1,2]).If the assumption showing that the data set includes identically distributed random observations is violated, robust estimation method for the parameters of model f (x; θ) have been applied by use of the different objective functions.
Robust methods trust the used objective function ( [3]).Deformed algebras such as Tsallis and Kaniakadis statistics are important to derive a neighborhood of a parametric model in order to overcome the problem which will occur when the assumption for ideniticality is violated ( [4][5][6]).
The origin of robust estimation method was started by biologist at 18 th century ( [3]).The main working principle is based on the estimating equations (EEs) ( [7,8]).EEs can be derived by use of maximum composite likelihood estimation method.Tsallis q-entropy is creator of deformed logarithm (log q ).In the direction of estimation method, log q from Tsallis q-entropy has been studied recently.For this aim, maximum log q likelihood estimation (MLqE) method are studied for outward observations.MLqE which is a generalization of the maximum likelihood (MLE) method [2] is used to obtain robust and also efficient estimators ( [9,10]).Different deformed logarithms can be obtained from entropy functions ( [4]).However, the deformed or the generalized logarithms should map the probability density function f (x i ; θ) as one-to-one and overlay ( [11,12]).
Kaniakadis' deformed logarithm (log κ ) can have same property with log q due to fact that (α, β)difference operator in fractional calculus (FC) is used to generate entropies.The other genaralized entropies can have same role in the estimation procedure ( [4,13]).Note that the performance of efficiency can be managed by using the generalized entropies and generalized logarithms from FC.
MLqE is simple and its computational implementation is not heavy.No extra condition is requried to apply MLqE for estimation when it is compared with divergences ( [14]).The parameters of Gamma distribution are estimated by MLqE method ( [15]).As an another approach for the robust estimation, divergences have been used.Minimization of entropies and divergences are equivalent to maximization of the generalized maximum likelihood estimation method ( [12,16]).
FC has been started to play an important role in the estimations of parameters of a probability density (p.d.) function f (x; θ).Tsallis distributions (q-distributions) have encountered a large success because of their remarkable agreement with experimental data.The parameter q behaves as a microscope and generates neigborhoods of p.d. function f (x; θ).Thus, the different behaviour of p.d. function f (x; θ) can be explored by use of log q (see [5,6] and references therein).After we use log q from Tsallis q-entropy to estimate robustly the parameters θ of underlying distribution f (x; θ), that is, we will have estimators which are not affected by inliers and outliers in a data set, the optimization of n i=1 log q f (x i ; θ) according to the parameters θ is an another challeging problem for the estimation process if we have a nonlinear function.The optimization is performed by means of genetic algorithm (GA) which is not attached to local points.Thus, we will have estimators.
GA mimicing evolutionary biology as a stochasticity is a derivative-independent method.We use hybrid method in 'ga' module in MATLAB2013a to decrease the computational error as well.
The aim is to estimate robustly shape and scale parameters of Weibull distribution when the different types of contamination into artifical data set are added.The properties of Weibull distribution are examined extensively.The roles of Λ from the deformed algebras and DPD, etc. as an objective function ρ, are observed easily by using the illustrative representations (see Fig. 1), which is a key to analyze their roles on p.d. function f (x; θ).Outliers and inliers are added into the artificial data sets at the simulation and the different types of contaminations to real data sets are applied to observe the robustness and the modeling capability of ρ(x; θ) = log q f (x; θ) .The value of parameter q is determined according to the p-value of KS test statistic of estimates of parameters for Weibull distribution.Fisher information based on log q is used to evaluate the variance-covariance matrix of estimators derived by MLqE.
The organization of study is as follows.Section II includes Weibull distribution and its properties which indicate that Weibull can be used for the modeling fruitfully.We provide the essential tools to pass the modeling sketch of estimation procedure such as convexity, (concavity), entropies, etc [37] in the M-estimation ( [18]).We also provide main tools to get the elements of Fisher information matrix.Section III introduces estimation methods [19] and estimating equations ( [7,8]).Section IV introduces the tools and robustness to outliers and also we propose a tool based on score function for robustness to inliers.Section V provides the tool used for optimization and numerical experiments.The last section VI is given for conclusions.

II. WEIBULL DISTRIBUTION
Weibull distribution is chosen because it has many applications in the field of the applied science.Further, it has many features such as existence of cumulative distribution (c.d.) function, moments and entropies, etc ( [20,21]).We say that a non-negative random variable X has a Weibull distribution with vector parameter θ = (α, β), denoted by X ∼ Weibull(θ), if its probability density function is given by where α is the shape parameter and β is the scale parameter ( [22]).

A. Properties
If X ∼ Weibull(θ) then the following properties are satisfied (see [23] for review of modified Weibull distributions).
1) Asymptotic behavior of f (x; θ).The behavior of f (x; θ) with x → 0 or x → ∞ is as follows: 2) Monotonicity, unimodality, concavity and convexity of f (x; θ).The point x is a mode of the Weibull density, if and only if it is the solution of the following equation Solving this equation, we get the following critical point A simple calculation shows that Note that and Therefore, the following properties follow immediately: For α > 1, • f (x; θ) increases as x → x 0 and decreases thereafter.
• The point x 0 is the unique mode for the Weibull density.
• The mode is non-existent.
3) Reliability.If F (t; θ) denotes the cumulative distribution function of X, then the reliability function is written as 4) Hazard rate.The hazard rate is given by The function H(t; θ) is increasing when α > 1, decreasing when α < 1 and constant when α = 1.
6) Moment of the residual life.For each t 0, we have The proof of this identity is immediate since where R(t; θ) and E 1 {X t} X n are given in Items 3) and 5), respectively.

7)
First main tool.By using the formula of Item (6) in [20], we have • Real moments.By taking r = 1 in the first main tool, we reach where Γ(z) is the complete gamma function.Taking t → 0 in the moment of the residual life φ n (t; θ) we see that the entire moments are justified by the above identity.
• Tsallis entropy.As an immediate application of the first main tool, we have that if q(α − 1) > −1 then the Tsallis entropy [24] is given by then the quadratic entropy can be written as . * Shannon entropy.By combining the Tsallis entropy with the well known relation H 1 (X) = lim q→1 S q (X), we have that the Shannon entropy can be written as where γ = − dΓ(x) dx x=1 ≈ 0.57721 is the Euler-Mascheroni constant.
8) Moment generating function.By applying Fubini's Theorem we have that the moment generating function, M X (t) = E[exp(tX)], can be expressed as follows

9)
Light-tailed distribution.From Item 8) it follows that, if α 1 then there exists t 0 > 0 such that P(X > x) exp(−t 0 x) for x large enough.
10) Second main tool.A simple change of variable shows that is the polygamma function of order m.
Indeed, by taking the change of variable w = r x β α we have, for each By combining the following known formulas with the identity (2), the formula for the expectation E X s log(X)f r−1 (X; θ) follows.
• In the particular case when r = 1 we have 11) Third main tool.Analogously to the proof of Item 10), a simple change of variable shows that, for each s > −α, The proof of this formula follows by taking the change of variable w = r x β α and by combining the formulas (3) and ( 4) with the following formula, for each • In the particular case when r = 1 we have, for s > −α,

III. INFERENCE: ESTIMATION METHODS AND FISHER INFORMATION A. Maximum likelihood estimation method
Maximum likelihood is the standard approach in parametric estimation, mainly due to the desirable asymptotic properties of consistency, efficiency and asymptotic normality under some regularity conditions ( [2]).
• The log-likelihood function for θ is given by A standard calculation shows that the first-order partial derivatives of l(θ; x) are x α i log(x i ), ( 5) The second-order partial derivatives of l(θ; x) can be written as x α i log 2 (x i ), x α i log(x i ).
• The log q -likelihood function for θ is given by Note that the first-order partial derivatives of l q (θ; x) are

B. Fisher information
The Fisher information matrix is defined by where n is sample size.E is integral for partial derivatives of log(L) according to parameters and it is taken by probability density function f (x; θ).The subscript in E represents second-order partial derivatives of log(L) according to parameters α and β.In other words, if X ∼ Weibull(θ), by using Items 7), 10) and 11) of Subsection II A, we have dx x=1 ≈ 0.57721 is the Euler-Mascheroni constant.Then the matrix ( 7) is given by α is known and β > 1} be the parameter space.Then, with probability approaching 1, as n → ∞, the likelihood equation d l(θ;x) dβ = 0 has a consistent solution, denoted by β.
Proof.If X ∼ Weibull(θ), a simple calculation shows that 3. There exits a function H(x) such that for all θ ∈ Θ, because β > 1, and Hence, by [25] the proof follows.
C. Estimating equations derived by objective functions in the composite likelihood The maximum composite likelihood estimation (MCLE) is a generalization of maximum likelihood estimation (MLE), MCLE is given by w i ∈ R is a weight function, x = (x 1 , x 2 , . . ., x n ) and f (x; θ) is a p.d. function.MCLE can cover density power divergence and its generalized forms ( [14,19]).The M-estimators are obtained by optimizing the objective function: When w i = 1 and log is replaced by log q , we have MLqE.log , lim κ→0 log κ (f ) = log(f ) and log q and log κ are deformed logarithm of log.q ∈ R\{1}, κ ∈ R and γ ≥ 0 are tuning constants used to adjust robustness and also efficiency ([4, 10, 26]).The concavity property of Λ is examined by [11,27] and references therein to use Λ for the estimation process accurately.
The density power divergence (DPD) as an objective function between f (x; θ) and g(x) which is free from parameter was proposed and the reorganized form of DPD is given by [28]: Let us try to get log q from DPD after algebraic rearrangement and γ = 1 − q > 0. Since q < 1, DPD puts a restriction for the values of tuning constant when we compare with log q .If equation ( 10) is rewritten for γ = 1 − q, then we have the following expression: where σ and µ represent scale and location for n i=1 log q f (x i ; θ) .When we consider to apply the optimization for DPD and maximum log q -likelihood, σ and µ will change where the optimized region is.In other words, there is an equivalence between DPD and log q if they are optimized according to parameters θ.Further, the integral value of I = f (x; θ)dx in DPD depends on Gamma function if f is Weibull (see [20]).The arguman of Gamma function, i.e.Γ, r > 0, has to be a positive and requires that the values of parameters remain within certain values, which is disadvantegous for a case in estimation.In addition, the computation time [30] for numerical integration can be high according to the used p.d. function which is not tractable to calculate and get an expression for the result of integral.MCLE method is useful to get robust estimators for parameters.As a generalized form of MCLE, we can use MLqE which is for robust estimation, because MLqE is a simple method and does not have extra conditions.If I does not exist, it is mandatory to show that I is finite for the values of parameters in a p.d. function by use of tools in ( [21]).Otherwise, f in I does not have a finite value, which shows that we will not have θ.

DPD for estimation of parameters of Weibull distribution does not work properly, as introduced by ([30]).
Let us derive the estimating equations (EEs) to examine the role of objective functions.EEs are obtained after taking derivatives of objective functions, ρ = Λ(f ), according to parameters θ.
For all of parameters, we have a system of EEs.The rearrangement forms of EEs are as follows: where I = Z(x i ; θ)f (x i ; θ) 1+γ dx.Eqs. ( 12)-( 15) are the weighted score function with w q , w κ , w ∝ and w γ .Z( which is disadvantegous because of the poorness in modeling capability ( [28]).Let us rewrite the system of estimating equations (EE) given by the following form: Note that w q is bigger than 1 if q > 1, which shows an advantage for us when we compare with Z as an extended form of Z is produced, which shows us that the information gained from the joint role of w and Z at same time is managed by not only Z from f /f but also the different values of w from f 1−q .For this reason, log q (f ) as an objective function is used to perform an efficient fitting on a data set.Thus, it is seen that w can manage to produce the efficient estimators θ from equation (16).Note that the chosen w affects the estimations of parameters which are shape and scale.After solving the systems of the estimating equations according to parameters, the estimators θ are also obtained instead of optimizing the objective function ( [29,31]).

D. Investigation for behaviour of objective functions in the composite likelihood
Estimation is performed when we assume that the empirical data sets are a member of an objective function ρ.For this reason, we can remove integral or summation from relative entropy or divergence to have function part of analytical expression.Since f is on the closed inverval [0, 1], we have advantage to display the behaviour of objective functions ρ derived by Λ(f ) and the values of tuning parameters are chosen as the interval [0, 1].Λ can be chosen as log q , log κ , log(∝ +f ), etc ( [12]).Let us display their behaviours via When we look at plots in Fig. 1, we can observe that log q is better than other functions from Λ ( [26,32]).
Thus, we test the role of Λ on p.d. function f .E. q-Fisher information Let us remind the definition of Fisher information based on log q ([11]).
Let us rewrite the Equation ( 17) for the parameters α and β as follows The elements of Fisher Information (FI) matrix based on log q ( q F ) can be written as the following form.
• Letting n = 1 in (6), we get . By using Item 7) of Subsection II A, we reach • Multiplying ( 5) and ( 6) with n = 1, we obtain . By using Items 7) and 10) of Subsection II A, we have Note that arguments in log and Γ functions should be positive.

IV. ROBUSTNESS
Influence function is a measure which is used to evaluate the robustness of M-estimators ( [3]).
Robustness trusts on the finiteness of score functions from ∂ρ(x;θ) ∂θ when x goes to infinity, i.e. lim x→∞ ∂ρ(x;θ) ∂θ .By using same way from robustness to outliers, the finiteness of score functions should be tested for the case in which we have lim x→0 ∂ρ(x;θ) ∂θ .Thus, we will imply the robustness to inlier observations in a data set.
A. Examination of score functions of parameters in objective function log q (f ) In order to get MLqE of parameters α and β, log q is applied to f (x; θ).Thus, we have objective function ρ(x; θ) = log q f (x; θ) .The score functions derived by ρ(x; θ) for the corresponding parameters are given in the following order: A simple calculation shows that +∞ for α ≥ 1 and q > 1, 0 for α ≥ 1 and 0 < q < 1, 0 for 0 < α < 1 and q > 1, +∞ for 0 < α < 1 and 0 < q < 1, −∞ for α ≥ 1 and q > 1, 0 for α ≥ 1 and 0 < q < 1, −∞ for 0 < α < 1 and q > 1, 0 for 0 < α < 1 and 0 < q < 1, +∞ for α ≥ 1 and q > 1, 0 for α ≥ 1 and 0 < q < 1, +∞ for 0 < α < 1 and q > 1, 0 for 0 < α < 1 and 0 < q < 1, Briefly, the limits above can be written using the following tables: the influence function of the estimators α and β is finite if q ∈ (0, 1) (see Table II).However, one can show that there are cases for log(f ) in which the score functions of parameters of f are finite or infinite according to property of f ( [33]).In addition, for an arbitrary f , we cannot get FI and the definition of influence function also includes the inverse of Fisher.There can be cases in which an element of FI matrix is not defined for some values of parameters such as Γ(r), r > 0, a/b, b = 0, etc and the inverse of FI matrix cannot exist.However, we can get estimates of parameters for these cases in which FI and its inverse does not exist.Instead of calling robustness and trusting on the tools in robustness (M.Thompson, e-mail communication, March 2, 2016), the main approach should be based on the modelling competence of an arbitrary function.Further, Table I includes the infinity cases.The modeling is carried out for the finite sample size.The robustness is for the case in which we take limit at the values which are zero and infinity.However, it is expected that the robustness should be supported by simulation.There is an open question: Even if the score functions are infinite for limit values at zero and infinity, can we perform a modeling capability on a finite sample size?Yes and we can find a counter example against the robustness theory from simulation results (see Case 4 in Table V).

V. OPTIMIZATION AND NUMERICAL EXPERIMENTS
A. Optimization via genetic algorithm for Λ(f ) The genetic algorithm (GA) can be applied to solve a variety of optimization problems that are unsuitable for standard optimization algorithms which include problems where the objective function is undifferentiated in Radon Nikodym derivative, highly nonlinear, discontinuous, (absolutely) continuous, non-smooth, even stochastic and random subsets from the real line.GA method is preferred to ensure that such objective functions converge to a global point.When log q is used as the objective function, GA method has been used by ([11]).The codes used to get estimates of MLqE are given by Appendix VI.

Monte Carlo simulation
Contamination makes a disorder in the identically distributed random variables represented by X 1 , X 2 , . . ., X n .If these random variables are disrtibuted non-identically, then we express the structure of non-identicality from mixing of two p.d. functions f 0 and f 1 , as given by following form: is the contaminated distribution.The constant ε is the contamination rate.f 0 is the underlying and f 1 is contamination.f 1 can be same distribution with f 0 , but the parameter values of f 1 are different from f 0 , i.e. f 0 = Weibull(α 0 ,β 0 ) and f 1 = Weibull (α 1 ,β 1 ).We can select f 1 distribution with the given values of parameters.For example, f 1 = BurrIII(α 1 ,β 1 ).In the real world, after assuming that a data set is a member of f 0 distribution, a contamination to f 0 by means of f 1 can occur in an empirical distribution.We do not know how much rate ε of contamination into the data As it is proven theoretically by II A, Weibull distribution has one unimodal for α > 1.Note that the mode of Weibull does not exist and makes asymptotic to y-axis for α ≤ 1.The uniform and BurrIII [33] distributions have one mode as well.
For the estimation process, the modality of p.d. function f (x; θ) and the concavity of Λ are taken in account to get Λ(f ) and apply for the estimation.Thus, the cooperation between f (x; θ) and Λ for conducting an accurate modeling on a data set can be performed successfully.For example, the smoothness property of objective function ρ(x; θ) = Λ f (x; θ) is important to apply for estimation if a data set does not have several jumpings on an interval on the real line, i.e. the smoothness of data set should be provided.The structure of inliers in Monte Carlo simulation will not be strictly existing.In other words, the frequencies of artificial data sets for a narrow interval on the real line are not extremely high degree.One can observe the schema of underlying and contamination distributions in Figures 2-4.
• As it is logical to expect, M SE( θ) of MLE cannot get the smaller values when the sample size n is increased for some designs of contamination; because n = n 0 + n 1 increases sample size n 1 of a contamination, which makes more contaminated data set when it is compared with small sample sizes, such as n = 50, 100, etc.In addition, log(f ) cannot model well.However, for some designs of contamination, M SE( θ) of log(f ) can be smaller than that of log q (f ), because the used objective function is an important indicator for modeling capability.For It is difficult to know the nature of reality and also knowing modality and bimodality in an empirical distribution or a real data set.We have to assume a parametric model and estimate the parameters of underlying distribution as much as we can do.We use two real data sets which are modeled by objective functions log q f (x; α, β) and log f (x; α, β) .Contaminations are applied into real data sets.Thus, we will test the performance of MLE and MLqE when the different types of contamination exist in real data sets.Three types of contamination to the real data set were performed.These are given by the following items: This section consists of the numerical example for the application of real data set.R Version 4.0.2 and some packages such as source("http://bioconductor.org/biocLite.R") bio-cLite("GEOquery") and require(GEOquery) are used to reach the real data set.We use the real data set from "test$myMean".The sample size n for this data set is 6136.The value of tuning constant is chosen until the highest p-value of KS test statistc is obtained when q is 0.85, which means that the best values for estimates of parameters can obtained.statistic is chosen to be 10 −5 , then Weibull with MLqE(θ) provides sufficient evidence not to reject the null hypothesis H 0 .Let us focus on the p-values instead of considering whether or not the data set does really come from Weibull distribution with parameters α and β which were estimated by MLqE and MLE methods.The p-values of KS test statistic obtained from two cases which are outliers and both contaminations are very small when they are compared with that of MLqE, which shows that adding outliers to real data set makes more far from Weibull distribution with the estimates obtained from MLE.However, the p-values of KS test statistic with estimates of MLqE(θ) did not differ much for three cases which are with inliers, outliers and both contaminations.Let us focus on the comparison of MLE for without contamination and with inliers cases, p-values from 4.3225e − 07 to 2.3326e − 07 tend to be near to zero due to the fact that adding inliers affects the estimates of MLE.In MLqE, such tendency being zero is not observed because of robustness property of MLqE.

Real data application: Example 2
The data are the strengths of 1.5 cm glass fibres measured by the National Physical Laboratory, England.The sample size is n = 63 ( [35]).The distributions and references therein [36] are used to model the data set.The estimates from Weibull with log q (f ) as an objective function give the p-value which is bigger than that of distributions in ( [36]).Note that the parametric model and objective functions should be tried for the case in which we can improve the modeling competence.The insensitivities for α and β in Tables VII-IX to contaminations show not only the robustness of MLqE but also we can conclude that the modeling competence of log q (f ) is better than log(f ).  of KS test statistic should be used to determine the value of constant q for the real data set.Thus, the evaluation of fitting performance of Weibull with the estimated parameters can be tested easily.
The score functions derived by log(f ) and log q (f ) are infinite and finite for 0 < q < 1 and α ≥ 1, respectively.The results in real data show that if 0 < q < 1, then we have estimates which can be insensitive to the added inliers and outliers.When we consider on the results of simulation, we can have for inlier case in which q > 1 which shows that the score functions of parameters are infinite for q > 1 and α ≥ 1.Consequently, the numerical experiments show that robustness is not enough to imply that the best modeling is accomplished.By using the MLqE, the values of the parameters representing the majority of the distribution were estimated with small M SE( θ) for different scenarios of contaminations.Many results from simulation and the application of real data sets show that MLqE is capable to model efficiently and gives an advantage to obtain the estimates for parameters of underlying distribution.Information geometry will be used ( [29,31]); and the adopted goodness of fit test will be proposed for further advance to determine the value of q by means of tools in statistics.In our future works, we will try to find counter examples from deformation family and its generalization as theoretical results and numerical experiments for different types of contamination will be used to test their modeling competence even if their score function is infinite.

Fig. 1 :
Figs. 1(a) and 1(b) have big range for the values of functions when compared with the Figs.1(c) and 1(d).As it seen from plots in Fig. 1, the concavity property of Figs.1(a) and 1(c) can be better than Figs.1(b) and 1(d).

FIG. 1 :
FIG. 1: f ∈ [0, 1] and q, κ, ∝, γ ∈ [0, 1] abbverivated as PDF show the underlying f 0 and the contamination f 1 into underlying distribution respectively.Simulation in section V C includes the robust estimations of parameters of blue lines.

example, ( 1
− ε)Weibull(α 0 = 1,β 0 = 5) + ε Weibull(α 1 = 2,β 1 = 8).•M SE( θ) were used to compare the performance of MLqE and MLE.Depending on the structure of contamination, the unbiasedness of estimators θ obtained by MLqE were examined by simulation.However, the contamination structure, i.e. the selected f 1 distribution and its parameter values and contamination rate ε can lead to observe the biasedness in Tables test statistic from c.d. function of Weibull is reached.This approach is also supported by Figures5-9which depict the fitting of c.d. and p.d. functions of Weibull distribution with α and β ([34]).

FIG. 5 :FIG. 6 :
FIG.5: CDF, PDF and histogram when real data set does not have inliers and outliers for bioconductor test data(color online)

FIG. 9 :FIG. 10 :FIG. 11 :
FIG.9: CDF, PDF and histogram when real data set does not have inliers and outliers for 1.5 cm glass fibres(color online)

FIG. 12 :
FIG. 12: CDF, PDF and histogram when real data set has inliers and outliers for 1.5 cm glass fibres(color online)

TABLE II :
Limit values of a vector Ψ SE and V ar obtained through simulation are sampling forms of theoretical MSE and Var, respectively.If the estimators θ obtained by MLqE are unbiased, then MSE( θ)=Var( θ).For comparison between MLqE and MLE methods, Tables III-VI give the estimates of parameters α 0 and β 0 , V ar( θ) and M SE( θ).Note that other objective functions from log κ and DPD did not give 2 Bias 2 and E show the bias and the expected values obtained from the simulation, respectively.M

TABLE III :
The underlying and contamination distributions are Weibull

TABLE IV :
The underlying and contamination distributions are Weibull (continuation of the TableIII)

TABLE V :
The underlying is Weibull and contamination is Uniform distributions

TABLE VII :
The estimates of parameters α and β via MLE and MLqE methods

Table
VII shows that MLqE can be robust to inliers and ouliers at each case.The estimates of scale parameter β can be similar to each other at each case for MLE and MLqE methods.MLE and MLqE with inliers cannot be more different than that of without the contamination case.However, when MLE and MLqE methods are compared for the case of outliers, it is seen that the shape parameter α obtained by MLE is very sensitive outlier.For the real data set, the results show that MLqE is robust to outliers, because the score functions derived from log q for parameters α and β are finite.For this real data set, the sensivity of estimates of β from MLE cannot be more, however when the estimates of β is compared with that of MLqE, MLqE is insensitive to contamination in all of cases.The estimates of α from MLqE can be insensitive to contamination in all of cases.Especially, the estimates of β from MLqE for both contamination are insensitivite when compared with that of MLE.

TABLE VIII :
The p-value of KS test statistics computed by the estimates of parameters α and β via MLE and MLqE

Table
VIII shows the p-value of KS test statistics.According to p-values of KS test statistic, there does not exist enough evidence to accept the null hypothesis H 0 which shows that the real data set is a member of Weibull distribution.However, note that if the significance level of test CDF, PDF and histogram when real data set has inliers and outliers for bioconductor test data(color online)

TABLE IX :
The estimates of parameters α and β via MLE and MLqE methods TableIXshows that MLqE can be robust to inliers and ouliers at each case.For all of scenarios of contaminations which are inliers, outliers and both of them, the estimates α and β from MLE are sensitive to contamination.However, such sensivity has not been observed at MLqE.When outliers and both contaminations are examined, it is observed that the estimates from MLE are very sensitive to contamination schemas.However, the estimates from MLqE can have similar values at which there are not contaminations into data set, which shows that MLqE are robust.

TABLE X :
The p-value of KS test statistics computed by the estimates of parameters α and β via MLE and MLqE Table X shows the p-value of KS test statistics.According to p-values of Kolmogorov-Smirnov (KS) as a goodness of fit test, there exists enough evidence to accept the null hypothesis H 0 which shows that the real data set is a member of Weibull distribution.MLqE and MLE depend on log q and log functions respectively.So the importance of the used objective function has been observed when we make a comparsion between the p-values which are 0.0936 and 0.7283 of MLE and MLqE respectively.The value of tuning constant is chosen until the highest p-value of KS test statistcs is obtained when q is 0.8.Even though there is no strict changing of the values of estimates from MLqE, the p-values of KS test statistics from MLqE go to lower values.This is due to the definition of KS test statistic which uses values of x (see codes for the computation of p-value in Appendix VI).Figures 9-12 illustrate that there exist a good fitting by MLqE, i.e.CDF MLqE and PDF MLqE , as supported by p-values of KS test statistics in Table X.