View of Univariate Discrete Nadarajah and Haghighi Distribution: Properties and Different Methods of Estimation

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UNIVARIATE DISCRETE NADARAJAH AND HAGHIGHI

DISTRIBUTION: PROPERTIES AND DIFFERENT METHODS

OF ESTIMATION

Muhammad Shafqat

Department of Statistics, Quaid-i-Azam University, Islamabad 45320, Pakistan Sajid Ali

Department of Statistics, Quaid-i-Azam University, Islamabad 45320, Pakistan Ismail Shah1

Department of Statistics, Quaid-i-Azam University, Islamabad 45320, Pakistan Sanku Dey

Department of Statistics, St. Anthony’s College, Shillong-793001, Meghalaya, India

1. INTRODUCTION

Researchers in many fields regularly encounter variables that are discrete in nature. Thus, continuous models may not be appropriate or suitable when data are discrete. For ex-ample, the number of rounds fired from a weapon till the first failure, the number of deaths at a given place over a given time period, the number of cycles prior to the first failure when devices work in cycles, the marks obtained by students in an examination, the number of forest fires during a given period of time, the number of breakdowns of a computer, the length of stay (usually measured as number of days) in an observation ward of leukemia patients, etc. In all these cases, the lifetimes are not measured on con-tinuous scale but are simply counted and hence are discrete random variables. During the last few decades, several continuous lifetime distributions have been proposed, and some of them have been extensively studied and modified. However, studies on discrete distributions are comparatively less than continuous distributions and therefore more studies are required in this area. The discretization of a continuous lifetime distribution retains the same functional form of the survival function, therefore, many reliability characteristics and properties shall remain unchanged. Thus, discretization of a contin-uous lifetime model is an interesting and simple approach to derive a discrete lifetime model corresponding to the continuous one.

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During the last two decades, many continuous distributions have been discretized. For example,Nakagawa and Osaki(1975) proposed the discrete Weibull distribution,

Nekhoukhouet al.(2013) proposed a new discrete version of the generalized exponen-tial distribution (Gupta and Kundu,1999) and compared it toNekoukhouet al.(2011),

Bakouchet al.(2014) andNekoukhouet al.(2015) suggested a discrete version of the beta-exponential distribution (Nadarajah and Kotz,2006),Para and Jan(2017) proposed discrete analogue of generalized Weibull distribution. Alamatsazet al.(2016) proposed discrete generalized Rayleigh distribution and the references cited therein.

This paper introduces a new discrete probability model from the extended exponen-tial distribution introduced byNadarajah and Haghighi(2011) and provides a detailed study of its properties. The survival function of the Nadarajah and Haghighi (NH) dis-tribution is:

S(t) = exp(1 − λ(1 + t)α_), ₍₁₎

whereα,λ > 0 are the shape and the scale parameters of the NH distribution. The result-ing cumulative distribution function (CDF), probability distribution function (PDF) and hazard rate function (hrf) are given as follows:

F(t) = 1 − exp(1 − (1 + λt)α), (2)

f(t) = αλ(1 + λt)α−1_exp_{(1 − (1 + λt)}α_), ₍₃₎

h(t) = αλ(1 + λt)α−1. (4)

The hrf of NH distribution has a closed form (4) as in the case of Weibull distribution and the generalized exponential distribution. Eq. (3) reduces to the exponential distri-bution forα = 1, and has an interesting property of having zero mode and allowing increasing, decreasing and constant hrfs.

Discretization of continuous distribution can be done using different methodolo-gies. For example,Nekhoukhouet al.(2013) used a technique to convert a continuous distribution into discrete analogue. To this end, for any continuous distribution on ℜ+= [0,+∞) with f(t), one can construct a discrete counterpart supported on the set of integersN₀= 0,1,2,··· whose probability mass function (pmf) is of the form

P_t = P(T = t) = s(t) − s(t + 1), t= 0,1,2,3,··· , (5) wheres(t) is the survival function of f (t). Substituting (1) in (5), the pmf of the resulting discrete NH distribution is given by

P(T = t) = exp(1)(θ(1λ+t)α_{− θ}(1λ+t+1)α), _t= 0,1,2,3,··· , ₍₆₎

whereθ = exp(−λ)αand 0< θ < 1. The new distribution is named as the discrete NH (DNH) distribution denoted by DNH(α,λ,θ) with parameters α,λ > 0 and 0 < θ < 1.

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The objective of this article is twofold: First, we introduce a new three-parameter discrete distribution, with the aim that the new distribution will produce a better fit compared to the Poisson, negative binomial, and zero inflated Poisson distributions in certain practical situations. Additionally, we will provide a comprehensive details of the mathematical properties of the proposed new distribution. Second, we will estimate the parameters of the model using eight different methods of estimation and to provide a guideline for choosing the best estimation method for the DNH distribution, which we think would be of deep interest to applied statisticians. Also, to the best of our knowledge, no attempt have been made to compare all these estimators for the DNH distribution along with mathematical and statistical properties. Additionally, all results are new, unless otherwise indicated. Other attractive features of the DNH distribution include heavy tail and closed forms for its cumulative distribution function (CDF) and hazard rate function.

The rest of the paper is organized as follows: In Section 2, we investigate some of its mathematical properties such as the quantile function, hazard rate behavior, Lorenz and Bonferroni curves, order statistics, stochastic ordering, mean, and variance. Estimation of the parameters of the DNH distribution is discussed in Section 3 using eight different estimation methods. A simulation study is carried out to assess the performance of dif-ferent estimation methods in Section 4. Two applications of the DNH distribution and comparison with the Poisson, negative binomial and zero inflated Poisson distributions are presented in Section 5. Finally, some concluding remarks are given in Section 6.

2. PROPERTIES OF THEDNHDISTRIBUTION

This section discusses some of the mathematical quantities of the DNH distribution, in-cluding the shape of the distribution, mean and variance, quantile function, Bonferroni and Lorenz curves, order statistics and stochastic ordering.

2.1. Shape of the DNH distribution

In Figure 1, PMF (6) of theDN H(α,λ,θ) for different values of α,λ and θ has been depicted.

In Figure1(a) and Figure1(b), the shape parametersα are assumed fixed and θ = 0.25 and 0.5 are used respectively. It is observed that the frequencies remain same regardless of the choice ofθ, but, when we increase the value of α and decrease the values of λ and θ, the frequencies also increased. Contrary to this, the frequencies decreases by decreasing the value ofα and increasing the values of λ and θ as shown in Figure1(c) and Figure1(d) .

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(a)α = 0.5,θ = 0.25,λ = 1 (b)α = 0.5,θ = 0.5,λ = 1

(c)α = 5,θ = 0.01,λ = 0.05 (d)α = 1,θ = 0.5,λ = 0.5

Figure 1 – Comparison of the shape of DNH distribution for different values ofα,θ and λ.

2.2. The survival and hazard rate functions

The survival and hazard rate functions of the DNH(α,λ,θ) distribution are given as s(t) = exp(1)(θ(1 λ+t)α), _t> 0, ₍₇₎ and h(t) =θ( 1 λ+t)α− θ(1λ+t+1)α θ(1 λ+t)α , t> 0, (8)

respectively. Figure2depicts the hazard rate function of the DNH(α,λ,θ) distribution for different values ofα,λ and θ.

For different choices of the parameters of the DNH(α,λ,θ) distribution, the hazard function has different frequencies. From Figure2(a) and Figure2(b), clearly, one can see that for different values of the parameterθ, frequencies remain the same. However, Figure2(c) and Figure2(d) show that the hazard function changes with the values ofα andλ, i.e., the hazard rate decreases by increasing the values of α and λ.

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(a)α = 0.5,θ = 0.01,λ = 1 (b)α = 0.5,θ = 0.9,λ = 1

(c)α = 0.5,θ = 0.01,λ = 0.5 (d)α = 5,θ = 0.9,λ = 0.5 Figure 2 – Comparison of the hazard function of DNH distribution assuming different values of

α,λ and θ.

2.3. Mean and variance of the distribution

The mean and the variance ofDN H(α,λ,θ) distribution are given by E(T ) =X∞ t=0 t exp(1)(θ(1 λ+t)α_{− θ}(1λ+t+1)α), ₍₉₎ and Var(T ) =X∞ t=0 t2exp(1)(θ(1λ+t)α_{− θ}(1λ+t+1)α) − ( ∞ X t=0 t exp(1)(θ(1 λ+t)α_{− θ}(λ1+t+1)α))2_{, (10)}

respectively. Ifα > 0 is an integer value, ∞ in sum expressions can be replaced by α, i.e., in (9), (10) the notationP_∞

t=0 should be replaced byPα−1t=0 (Nekhoukhouet al.,

2013). Since Eqs. (9) and (10) cannot be further simplified, the mean and the variance of DN H(α,λ,θ) distribution are calculated numerically in Table1.

It is observed that the mean and variance decrease by increasing the value of param-eter α, whereas increase for values of λ. Further, the mean and variance increase by increasing the value ofθ, for instance when λ = 1, α = 0.5 and θ = 0.1 the mean and variance are 0.2274 and 0.0517, respectively. However, ifα = 0.5 and θ = 0.5, the mean and variance are 8.6068 and 74.07, respectively.

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TABLE 1

Mean and variance of the DNH distribution for different values ofα, λ and θ.

α/θ 0.1 0.2 0.5 0.7 λ = 0.5 0.5 0.123 (0.542) 0.579 (4.031) 7.598 (131.760) 25.413 (415.461) 1 0.003(0.004) 0.027(0.040) 0.679 (1.577) 3.108 (7.952) 1.5 1.0× 10−4_(1.3_{× 10}−4₎ _{0.001 (0.001)} _{0.869 (0.105)} _{0.652 (0.869)} 2 2.7× 10−9_(2.7_{× 10}−9₎ _{1.3× 10}−6_{(1.3× 10}−6₎ _{0.005 (0.005)} _{0.119 (0.124)} λ = 1 0.5 0.227 (0.052) 0.859 (0.737) 8.607 (74.070) 26.851 (720.900) 1 0.030 (0.001) 0.136 (0.018) 1.359 (1.847) 4.439 (19.712) 1.5 0.004 (1.0× 10−5₎ _{0.029 (0.001)} _{0.469 (0.219)} _{1.643 (2.701)} 2 3.0× 10−4_{(7.4 × 10}−8₎ _{0.004 (2.0× 10}−5₎ _{0.175 (0.031)} _{0.772 (0.596)} λ = 1.5 0.5 0.288 (1.005) 0.996 (5.444) 9.007 (129.318) 27.380 (364.448) 1 0.065 (0.075) 0.232 (0.295) 1.712 (2.205) 5.000 (3.332) 1.5 0.019 (0.019) 0.088 (0.085) 0.768 (0.545) 2.164 (0.227) 2 0.005 (0.004) 0.031 (0.030) 0.416 (0.283) 1.248 (0.218) 2.4. Quantile function

The quantile function plays an important role in probability distribution theory and it is also known as the percent-point function or the inverse cumulative distribution function. It is used to generate random number from a given distribution. The quantile function of the DNH distribution is given by

Q(p) = F−1_{(p) = −}1 λ+ log(1 − p) logθ − 1 logθ 1 α , (11) where 0< p < 1.

2.5. Bonferroni and Lorenz curves

The Bonferroni curve (BC) and Lorenz curve (LC) are also known as the income in-equalities and are commonly used in economics to describe the wealth of a nation. For a random variableT , with PMF and CDF p(t) and F (t) respectively and quantile func-tionF−1_{(p), the BC and the LC are given as}

B_F(p) = 1 pµ p X 0 F−1(t), (12) and L_F(p) = 1 µ p X 0 F−1(t), (13)

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respectively, whereµ is the mean of DNH. The BC and LC of DNH distribution are given below: B_F(p) = Pp t=0(−1λ+ (log(1−p)logθ − 1 logθ) 1 α) pP∞ t=0t exp(1)(θ( 1 λ+t)α− θ(1λ+t+1)α) , (14) and L_F(p) = Pp t=0(−1λ+ (log(1−p)logθ − 1 logθ) 1 α) P∞ t=0t exp(1)(θ( 1 λ+t)α− θ(λ1+t+1)α) , (15)

respectively. Eqs. (14) and (15) can be solved numerically.

2.6. Order statistics

Order statistics are required in many fields, such as climatology, engineering, industry, etc. (Arnoldet al.,1992). Joined with rank statistics, order statistics are among the most important tools in nonparametric statistics which is used to develop robust estimators. In statistics, the jth ordered value is equal to the jth order statistic of the sample. The largest order statistic is the maximum for a sample of sizen, the n− t h order statistic is, T_(n) = max(T₁,T₂,· · · , Tn). Moreover, the sample range, Ran g e(T1,T2,· · · , Tn) = T_(n)− T(1)is the difference between the maximum and the minimum values of a given data set which is also a function of order statistics. IfT₍₁₎,T₍₂₎,· · · , T_(n)denote the order statistics of a random sampleT₁,T₂,· · · , Tnfrom a population with cdfF(t) and the pmf

p(t), then the pmf of T_(j)is given by F_T

(j)(t) =

n!

(j − 1)!(n − j)!p(t)[F (t)]j−1[1 − F (t)]n− j, (16) for j= 1,2,··· , n. The pmf of the j-th order statistic of the DNH distribution is given by F_T (j)(t) = n! (j − 1)!(n − j)![exp(1)(θ( 1 λ+t)α_{− θ}(1λ+t+1)α)][(1 − exp(1)θ(1λ+t)α)]j−1 × [exp(1)θ(1λ+t)α]n− j_. ₍₁₇₎

Thus, the PMF of the first order statistic is obtained by j= 1 in (17). The resulting cdf is given as: F_T (1)(t) = n! (n − 1)![exp(1)(θ( 1 λ+t)α_{− θ}(1λ+t+1)α)][exp(1)θ(1λ+t)α]n−1_. ₍₁₈₎

Similarly, the PMF of then− t h order statistic is obtained by j = n, which is given as: F_T (n)(t) = n! (n − 1)![exp(1)(θ( 1 λ+t)α_{− θ}(1λ+t+1)α)][(1 − exp(1)θ(1λ+t)α)]n−1_. ₍₁₉₎

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2.7. Stochastic ordering

Stochastic ordering is an interesting and important area of statistics and it is basically the concept concerning of one random variable being bigger than another random vari-able. Stochastic ordering, especially hazard rate ordering, mean residual life ordering and likelihood ratio ordering have the following relationship:

1. Hazard rate order(T₁≤h rT2) if hT1(t) ≤ hT2(x) for all t.

2. Mean residual life order(T₁≤ml rT2) if mT₁≤ mT₂for all t. 3. Likelihood ratio order(T₁≤l r T2) if

f_T1(t)

f_T2(t)is a decreasing functiont .

THEOREM1. Let T₁∼ DN H (α₁,λ,θ) and T₂∼ DN H (α₂,λ,θ) be two independent

random variables. Ifα₁≤ α2then(T1≤l rT2) for all t.

PROOF. To prove the theorem, it is enough to show that fT1(t)

f_T2(t)is decreasing int . f_T 1(t) f_T 2(t) =exp(1)θ( 1 λ+t)α1− θ(1λ+t+1)α1 exp(1)θ(1 λ+t)α2− θ(1λ+t+1)α2 (20) Now, d d tlog f_T 1(t) f_T 2(t) = d d tlog θ(1 λ+t)α1− θ(1λ+t+1)α1 θ(1 λ+t)α2_{− θ}(1λ+t+1)α2 < 0 (21) ifα₁< α₂,∀ t > 0. Hence, for α1< α2,T1≤l rT2,∀ t. 2 3. PARAMETER ESTIMATION

This section describes eight methods of parameter estimation for the DNH distribution. To this end, assume that t= (t₁,· · · , tn) is a random sample of size n from the DNH distribution and we are interested to estimate theα, λ and θ using the information of the observed sample.

3.1. Maximum likelihood estimation

Method of the maximum likelihood estimation is widely used estimation method in statistics (Casella and Berger,2002). This method of estimation has many attractive properties like consistency, asymptotic efficiency, etc. The likelihood function of the probability mass function (6) is given by

L(t;α,θ,λ) =Yn i=1

p(ti) = exp(1)(θ(

1

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logL(t;α,θ,λ) =Xn i=0 log exp(1)(θ(1 λ+ti)α− θ( 1 λ+ti+1)α). ₍₂₃₎

Taking partial derivatives of logL(t;α,θ,λ) with respect to α,λ and θ, and equating the resulting equations to zero, one can obtain the normal equations to find the ML estimators. ∂ log L(t;α,θ) ∂ α = 0, n X i=1 log(θ(1+t)α

logθ log(1 + t)(1 + t)α− θ(2+t)αlogθ log(2 + t)(2 + t)α= 0, (24) and ∂ log L(t;α,θ) ∂ θ = 0, n X i=1 log(θ(−1+(1+t)α₎ (1 + t)α_{− θ}(−1+(2+t)α₎ (2 + t)α_{= 0.} ₍₂₅₎ Ifλ 6= 1 then ∂ log L(t;α,λθ) ∂ λ = 0, n X i=1 −αθ(λ1+ti)α_logθ(1 λ+ ti)α−1+ αθ(1+ 1 λ+ti)α_logθ(1 +1 λ+ ti)α−1 θ(1 λ+ti)α− θ(1+1λ+ti)α = 0. (26)

The solution of above normal Eqs. (24), (25) and (26) cannot be obtained in closed form, hence require an iterative method like Newton Raphson to compute MLEs of (α,λ,θ). 3.2. Method of least squares

To estimate parameters by minimizing the sum of square of residuals, a standard ap-proach is the least squares estimators (Swainet al.,1988). For the estimation of param-etersα,λ and θ of DNH distribution, the least squares estimators can be obtained by minimizing: n X i=1 F(t_(i)) − i n+ 1 2 , (27)

with respect to unknown parametersα,λ and θ. In other words, ˆα_LSE, ˆλ_LSE and ˆθ_LSE are obtained by minimizing

n X i=1 1− exp(1)(θ(1λ+ti)α) − i n+ 1 2 (28)

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with respect toα,λ and θ.

Taking partial derivative with respect toα,λ and θ, the following are obtained n X i=1 −2 exp(1)θ(λ1+ti)α(1 − i n+ 1− exp(1)θ (1 λ+ti)α)log[θ]log[1 λ+ ti]( 1 λ+ ti), (29) n X i=1 2 exp(1)αθ(1 λ+ti)α(1 − i n+1− exp(1)θ( 1 λ+ti)α)log[θ](1 λ+ ti)α−1 λ2 , (30) n X i=1 −2 exp(1)αθ−1+(λ1+ti)α(1 − i n+ 1− exp(1)θ (1 λ+ti)α)(1 λ+ ti)α, (31) respectively. The weighted least squares estimators of the parameters of DNH distribu-tion are obtained by minimizing

n X i=1 w_i F(t_(i)) − i n+ 1 2 (32)

with respect to parameters, wherew_i=_V_(T1

i) =

(n+1)2_(n+2)

i(n−i+1) are the weights. Thus, ˆαW LSE, ˆ

λW LSEand ˆθW LSEcan be obtained by minimizing n X i=1 (n + 1)2_{(n + 2)} n− i + 1 1− exp(1)(θ(1λ+ti)α) − i n+ 1 2 (33)

with respect toα,λ and θ.

3.3. Method of percentile estimation

The method of percentile estimation was originally introduced by Kao(1959) and it has been used for the parameter estimation of several distributions. In this study, we use this method to estimate the parameters of the DNH distribution. Since F(t) = 1− exp(1)θ(1λ+t)α andQ(p) = −1 λ+ log(1−p) logθ −log1θ 1_α

, letT_ibe the i-th order statistic, i.e.,T₍₁₎< T₍₂₎< ··· < T_(n). Ifp_idenote an estimate ofF(t_(i)), then the estimate of α,λ andθ can be obtained by minimizing

n X i=1 t_(i)−−1 λ+ ( log(1 − p_i) − 1 logθ ) 1 α 2 (34)

with respect toα,λ and θ. The estimators of α,λ and θ are obtained by solving the following non-linear equations

n X

i=1

2 log[−1+log(1− pi)

logθ ](−1+log(1− plogθ i))

1 α(1 λ− (−1+log(1− plogθ i)) 1 α + t_i) α2 = 0, (35)

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n X i=1 −2[1_λ− (−1+log[1− pi] log[θ] ) 1 α + t_i] λ2 = 0, (36) n X i=1 2(−1 + log(1 − p_i))(−1+log(1− pi) logθ )−1+ 1 α(1 λ− (−1+log(1− plogθ i))) 1 α+ti αθ(logθ)2 = 0. (37)

The resulting estimators are denoted as the percentile estimators. Many estimators of p_i are proposed in the literature (Mannet al.,1974). In this work we considerp_i=_n+1i . 3.4. Method of maximum product spacings

An alternative to MLE,Cheng and Amin (1983) suggested the method of maximum product of spacing (MPS) for parameter estimation. Independently, it was introduced byRanneby(1984) as an approximation for the Kullback-Leibler information measure. For a random sample with uniform spacings from the DNH distribution are given by:

D_i= F (ti|α, λ, θ) − F (ti−1|α, λ, θ), (38) wherei= 1,2,··· , n.Cheng and Amin(1983) suggestedPn+1

i=1Di(α,λ,θ) = 1. The esti-mators under the MPS are ˆα_{M P S}, ˆλ_{M P S}and ˆθ_{M P S}obtained by maximizing the following geometric mean of the spacings with respect to parameters,

G(α,λ,θ) =Yn+1 i=1 D_i(α,λ,θ) 1 n+1 , (39)

or alternatively, by maximizing the function

H(α,λ,θ) = 1 n+ 1 n+1 X i=1 logD_i(α,λ,θ), (40) ˆ

αM P S, ˆλM P Sand ˆθM P Scan be obtained by solving the non-linear equations

∂ ∂ αH(α,λ,θ) = 1 n+ 1 n+1 X i=1 1 D_i(α,λ,θ)[δ1(ti|α, λ, θ)] = 0, (41) ∂ ∂ λH(α,λ,θ) = 1 n+ 1 n+1 X i=1 1 D_i(α,λ,θ)[δ3(ti|α, λ, θ)] = 0, (42) ∂ ∂ θH(α,λ,θ) = 1 n+ 1 n+1 X i=1 1 D_i(α,λ,θ)[δ2(ti|α, λ, θ)] = 0, (43)

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where δ1(ti|α, λ, θ) =θ( 1 λ+ti−1)αlog[θ]log[1 λ+ ti−1]( 1 λ+ ti−1)α+ θ( 1 λ+ti)α_× log[θ]log[1 λ+ ti]( 1 λ+ ti)α, (44) δ2(ti|α, λ, θ) = − 1 λ2[αθ (1 λ+ti−1)αlog[θ](1 λ+ ti−1)α−1+ αθ( 1 λ+ti)α_× log[θ](1 λ+ ti)α−1], (45) δ3(ti|α, λ, θ) = θ−1+( 1 λ+ti)α(1 λ+ ti−1)α+ θ−1+( 1 λ+ti)α(1 λ+ ti)α, (46)

Cheng and Amin(1983) proved that maximizing the product of spacings is as efficient and consistent as the MLE estimators under different conditions.

3.5. Cramèr-von-Mises method of estimation

This is another parameter estimation method.Macdonald(1971) proved that the Cramèr-von-Mises has a smaller bias as compare to other minimum distance type estimators. Assuming ˆα_{C M E}, ˆλ_{C M E} and ˆθ_{C M E}, the Cramèr-von-Mises estimators are obtained by minimizing C(α,λ,θ) = 1 12n+ n X i=1 [F (t_i) −2i− 1 2n ] 2_. ₍₄₇₎

To obtain the estimators of DNH distribution, the following non-linear equations need to be solved numerically. n X i=1 [F (ti) − 2_i− 1 2n ]δ1(ti|α, λ, θ) = 0, (48) n X i=1 [F (ti) − 2_i− 1 2n ]δ2(ti|α, λ, θ) = 0, (49) n X i=1 [F (ti) − 2_i− 1 2n ]δ3(ti|α, λ, θ) = 0, (50)

whereδ₁(.|α,λ,θ), δ₂(.|α,λ,θ) and δ₃(.|α,λ,θ) are given by Eqs. (44), (45) and (46), respectively.

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3.6. Method of Anderson-Darling and right-tail Anderson-Darling

The Anderson-Darling (AD) test was proposed byAnderson and Darling(1952) as an alternative to other statistical tests for detecting sample distribution departure from the normality. A nice feature of the AD test is that it converge towards the asymptote very quickly (Anderson and Darling,1954;Pettitt,1976). Let ˆα_{AD E}, ˆλ_{AD E} and ˆθ_{AD E} are the AD estimators obtained by minimizing, with respect toα,λ and θ, as given below:

A(α,θ) = −n − 1 n

n X

i=1

(2i− 1)[log F (ti:n|α, θ) + log ¯F(tn+1−i:n|α, θ)]. (51) These estimators can also be obtained by solving the non-linear equations:

The estimates of the Right-tail Anderson-Darling ˆα_{RT AD E}, ˆλ_{RT AD E} and ˆθ_{RT AD E} of the parametersα,λ and θ respectively, are obtained by minimizing

R(α,θ) =n 2 − 2 n X i=1 F(t_i:n|α, λ, θ) − 1 n n X i=1

(2i − 1)log ¯F(t_n+1−i:n|α, λ, θ). (55) Solving the following non-linear equations:

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4. MONTECARLO SIMULATION STUDY

In this section, a Monte Carlo simulation study is conducted to compare the perfor-mance of the different methods of estimation. To be more precise, the perforperfor-mance of different estimators is evaluated on the basis of bias, root mean squared error, the aver-age absolute difference between the theoretical and empirical estimate of the distribution functions, and the maximum absolute difference between the theoretical and the empir-ical distribution functions. Methods are compared for sample sizesn = 10,20,50,75, and 100. To this end, first we generate five thousand independent samples of size n from the DNH distribution with different parameter configurations. In particular, we considerα = 1,5; λ = 1,1.5, and θ = 0.1,0.3,0.7. It is observed that 5,000 repetitions are sufficiently large enough to have stable results. Next, the parameters are estimated using the method of maximum likelihood. Then for all other methods, we have used the maximum likelihood estimates as the initial values. In addition, the same randomly generated samples are used to compute and compare the other estimation methods. The results of the simulation studies are tabulated in Tables2to6.

To generate the pseudo-random numbers from the DNH distribution, the following quantile function is used:

Q(p) = −1 λ+ _log(1 − p) logθ − 1 logθ _α1 ,

wherep∼ UNIF(0, 1). The formulae to calculate the bias, root mean-squared error, and the average absolute difference between the theoretical and the empirical estimate of the distribution functions, and the maximum absolute difference between the theoretical and empirical distribution functions are given as follows:

Bias(ˆα) = 1 R R X i=1 (ˆα_i− α), Bias(ˆλ) = 1 R R X i=1 (ˆλ_i− λ), (59) Bias( ˆθ) = 1 R R X i=1 ( ˆθ_i− θ), RMSE(ˆα) = v u u t1 R R X i=1 (ˆα − α)2_, ₍₆₀₎ RMSE(ˆλ) = v u u t1 R R X i=1 (ˆλ− λ)2_, ₍₆₁₎ RMSE( ˆθ) = v u u t 1 R R X i=1 ( ˆθ − θ)2_, ₍₆₂₎ D_{ab s}(ˆα) = 1 (nR) R X i=1 n X j=1 |F (ti j|α, λ, θ) − F (ti j|ˆα, ˆλ, ˆθ)|, (63)

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D_max(ˆα) = 1 R R X i=1 max j |F (ti j|α, λ, θ) − F (ti j|ˆα, ˆλ, ˆθ)|. (64) Using the above measures, we conducted a simulation study and the results are tabulated in Table2to Table9. Partial sum of the ranks shown in row with labelP Rank s. A su-perscript shows the rank of every estimations among the all considered eight estimation methods. For instance, Table2presents the bias of MLE, Bias(ˆα) as 6.3128_{for sample} sizen = 10. This shows that Bias(ˆα) calculated using MLE method is 6.312 and it is ranked 8t h among all consider estimators which are used in this study. The following conclusion are drawn from Table2to Table9.

1. All the discussed methods of estimation have consistency property, i.e., the bias and RMSE decrease with the sample size n.

2. It is observed that for a givenθ, Bias(ˆα) generally increases as the value of α increases. This pattern is observed among all eight estimation methods.

3. In case of RMSE, all estimators produce minimal RMSE for parameter ˆα as compared to ˆθ and ˆλ.

4. Comparing the D_{ab s} and D_max for eight considered estimation methods, D_{ab s} is smaller as compared toD_max. Further, it gets smaller as sample sizen becomes large.

5. As the performance of different methods of estimation is concerned, it is observed that the MPS method performs better on the basis of least biases and RMSE as compared to other considered methods. Cramèr-von-Mises (CVM) is the next best method of estimation, followed by percentile estimators (PCE). The ML estima-tion method ranked 8th while weighted least squared (WLS) estimaestima-tion ranked 5th, right-tail Anderson-Darling ranked 6th among the eight methods of estima-tion.

6. For all sample sizes andα, MPS performed uniformly the best while percentile and weighted least square methods also performed better forα ≤ 1. However, it is also observed that their performance is degraded for large value ofα, e.g., α = 5. 7. All assumed estimation methods perform better whenλ 6= 1 as compared to λ = 1. 8. In most methods, parameters converge quickly whenλ 6= 1 as compared to λ = 1.

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TABLE 2

Simulation results forα = 0.5, θ = 0.5 and λ = 1.

n Est. MLE LSE WLS PCE MPS CVM AD RAD

10 Bias(ˆα) 6.3128 _-0.0595 _-0.0474 _-0.0313 _-0.5007 _-0.0051 _0.2996 _-0.0111 RMSE(ˆα) 6.3128 _0.1353 _0.1332 _0.1855 _0.5006 _0.1397 _1.3827 _0.1261 Bias( ˆθ) 0.4978 _-0.0566 _-0.0505 _-0.0364 _-0.2637 _-0.0283 _0.0241 _-0.0272 RMSE( ˆθ) 0.4978 _0.0804 _0.0722 _0.1445 _0.2637 _0.0671 _0.1536 _0.0733 Dabs 0.9998 _0.1104 _0.1033 _0.1435 _0.2937 _0.1032 _0.1796 _0.1011 Dmax 0.9998 _0.1664 _0.1543 _0.2535 _0.5557 _0.1511 _0.2916 _0.1542 PRanks 488 ₂₆4 ₁₉3 ₂₇5 ₄₁7 ₁₂2 ₃₂6 ₁₁1 20 Bias(ˆα) -0.1867 _-0.0566 _-0.0413 _-0.0464 _0.2448 _-0.0292 _0.0475 _-0.0261 RMSE(ˆα) 0.6918 _0.0964 _0.0882 _0.1465 _0.2486 _0.0893 _0.6517 _0.0831 Bias( ˆθ) 531.9058 _-0.0536 _-0.0444 _-0.0505 _-0.4977 _-0.0383 _-0.0201 _-0.0352 RMSE( ˆθ) 934.6278 _0.0644 _0.0521 _0.1336 _0.4977 _0.0532 _0.0965 _0.0573 Dabs 0.9998 _0.0844 _0.0751 _0.1226 _0.5457 _0.0803 _0.1035 _0.0782 Dmax 0.9998 _0.1334 _0.1161 _0.2256 _0.9987 _0.1213 _0.1675 _0.1212 PRanks 478 ₂₈4.5 ₁₂2 ₃₂6 ₄₂7 ₁₆3 ₂₈4.5 ₁₁1 50 Bias(ˆα) -0.3138 _-0.0546 _-0.0341 _-0.0505 _0.2687 _-0.0444 _-0.0424 _-0.0352 RMSE(ˆα) 0.5938 _0.0714 _0.0571 _0.1156 _0.2697 _0.0653 _0.0855 _0.0582 Bias( ˆθ) 508.6848 _-0.0505 _-0.0371 _-0.0556 _-0.5007 _-0.0443 _-0.0454 _-0.0392 RMSE( ˆθ) 655.3658 _0.0554 _0.0391 _0.1166 _0.5007 _0.0503 _0.0555 _0.0482 Dabs 0.9998 _0.0644 _0.0541 _0.0996 _0.5447 _0.0623 _0.0655 _0.0592 Dmax 0.9998 _0.1104 _0.0881 _0.1886 _0.9987 _0.1033 _0.1105 _0.0982 PRanks 488 ₂₇4.5 ₆1 ₃₅6 ₄₂7 ₁₉3 ₂₇4.5 ₁₂2 75 Bias(ˆα) -0.3628 _-0.0526 _-0.0291 _-0.0475 _0.2757 _-0.0453 _-0.0464 _-0.0362 RMSE(ˆα 0.5918 _0.0655 _0.0481 _0.1046 _0.2767 _0.0603 _0.0614 _0.0522 Bias( ˆθ) 726.6568 _-0.0495 _-0.0341 _-0.0546 _-0.5007 _-0.0453 _-0.0484 _-0.0402 RMSE( ˆθ) 1146.1578 _0.0525 _0.0361 _0.1086 _0.5007 _0.0493 _0.0514 _0.0462 Dabs 0.9998 _0.0594 _0.0491 _0.0916 _0.5457 _0.0583 _0.0594 _0.0552 Dmax 0.9998 _0.1065 _0.0801 _0.1736 _0.9997 _0.1003 _0.1054 _0.0942 PRanks 488 ₃₀5 ₆1 ₃₅6 ₄₂7 ₁₈3 ₂₅4 ₁₂2 100 Bias(ˆα) -0.4448 _-0.0526 _-0.0271 _-0.0463 _0.2787 _-0.0474 _-0.0485 _-0.0372 RMSE(ˆα) 0.5798 _0.0625 _0.0421 _0.0966 _0.2797 _0.0584 _0.0573 _0.0492 Bias( ˆθ) 2379.1268 _-0.0495 _-0.0321 _-0.0536 _-0.5007 _-0.0463 _-0.0484 _-0.0402 RMSE( ˆθ) 3125.4648 _0.0515 _0.0341 _0.1036 _0.5007 _0.0483 _0.0504 _0.0442 Dabs 0.9998 _0.0564 _0.0451 _0.0866 _0.5457 _0.0553 _0.0575 _0.0522 Dmax 0.9998 _0.1034 _0.0761 _0.1646 _0.9997 _0.0993 _0.1045 _0.0912 PRanks 488 ₂₉5 ₆1 ₃₃6 ₄₂7 ₂₀3 ₂₆4 ₁₂2

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TABLE 3

Simulation results forα = 1, θ = 0.7 and λ = 1.

10 Bias(ˆα) -1.1227 _-0.1965 _-0.1724 _-0.5666 _-0.0491 _-0.1013 _1.8388 _-0.0792 RMSE(ˆα) 1.2577 _0.2944 _0.3005 _0.5686 _0.0581 _0.2693 _3.8048 _0.2452 Bias( ˆθ) 619.5398 _-0.1184 _-0.1103 _-0.4036 _-0.6977 _-0.0802 _0.1295 _-0.0701 RMSE( ˆθ) 777.3028 _0.1454 _0.1433 _0.4106 _0.6977 _0.1192 _0.1915 _0.1101 Dabs 0.9998 _0.1464 _0.1423 _0.3305 _0.5997 _0.1382 _0.4726 _0.1351 Dmax 0.9998 _0.2634 _0.2613 _0.6555 _0.9987 _0.2352 _0.7316 _0.2201 PRanks 468 ₂₅4 ₂₁3 ₃₄6 ₃₀5 ₁₄2 ₃₈7 ₈1 20 Bias(ˆα) 5.6208 _-0.2004 _-0.2005 _-0.5336 _-0.0491 _-0.1543 _1.0027 _-0.1072 RMSE(ˆα) 5.6328 _0.2454 _0.2585 _0.5366 _0.0571 _0.2173 _2.3967 _0.1832 Bias( ˆθ) 3.9008 _-0.1164 _-0.1185 _-0.4536 _-0.6997 _-0.0972 _0.1093 _-0.0781 RMSE( ˆθ) 113.9988 _0.1293 _0.1354 _0.4616 _0.6997 _0.1132 _0.1735 _0.0971 Dabs 0.9998 _0.1264 _0.1263 _0.4116 _0.6017 _0.1212 _0.3675 _0.1171 Dmax 0.9998 _0.2583 _0.2704 _0.8526 _0.9997 _0.2342 _0.5675 _0.2091 PRanks 488 ₂₂3 ₂₆4 ₃₆7 ₃₀5 ₁₄2 ₃₂6 ₈1 50 Bias(ˆα) 7.1038 _-0.2004 _-0.2415 _-0.4896 _-0.0441 _-0.1823 _0.5567 _-0.1222 RMSE(ˆα) 7.1378 _0.2174 _0.2525 _0.4906 _0.0531 _0.2013 _1.2547 _0.1512 Bias( ˆθ) 7.9528 _-0.1134 _-0.1345 _-0.5206 _-0.7007 _-0.1063 _0.0801 _-0.0822 RMSE( ˆθ) 144.6768 _0.1183 _0.1384 _0.5216 _0.7007 _0.1112 _0.1485 _0.0891 Dabs 0.9998 _0.1113 _0.1194 _0.5136 _0.6007 _0.1092 _0.2665 _0.1051 Dmax 0.9998 _0.2553 _0.2934 _0.9966 _0.9987 _0.2432 _0.4235 _0.2071 PRanks 488 ₂₁3 ₂₇4 ₃₆7 ₃₀5.5 ₁₅2 ₃₀5.5 ₉1 75 Bias(ˆα) 7.0368 _-0.1994 _-0.2505 _-0.6317 _-0.0431 _-0.1873 _0.5116 _-0.1252 RMSE(ˆα) 7.0788 _0.2104 _0.2555 _0.6316 _0.0551 _0.1993 _1.1817 _0.1442 Bias( ˆθ) 5.4208 _-0.1134 _-0.1385 _-0.2176 _-0.7007 _-0.1083 _0.0711 _-0.0832 RMSE( ˆθ) 87.8658 _0.1163 _0.1404 _0.2176 _0.7007 _0.1112 _0.1425 _0.0881 Dabs 0.9998 _0.1093 _0.1194 _0.2145 _0.6017 _0.1082 _0.2486 _0.1041 Dmax 0.9998 _0.2553 _0.3014 _0.4196 _0.9987 _0.2472 _0.3945 _0.2081 PRanks 488 ₂₁3 ₂₇4 ₃₆7 ₃₀5.5 ₁₅2 ₃₀5.5 ₉1 100 Bias(ˆα) 7.6378 _-0.1994 _-0.2545 _-0.6317 _-0.0381 _-0.1903 _0.4506 _-0.1262 RMSE(ˆα) 7.6758 _0.2074 _0.2575 _0.6316 _0.0541 _0.1993 _1.1017 _0.1412 Bias( ˆθ) 6.5818 _-0.1124 _-0.1405 _-0.2176 _-0.7007 _-0.1093 _0.0591 _-0.0832 RMSE( ˆθ) 114.7188 _0.1153 _0.1415 _0.2176 _0.7007 _0.1112 _0.1324 _0.0871 Dabs 0.9998 _0.1073 _0.1194 _0.2145 _0.6017 _0.1062 _0.2246 _0.1031 Dmax 0.9988 _0.2553 _0.3044 _0.4206 _0.9957 _0.2482 _0.3535 _0.2081 PRanks 488 ₂₁3 ₂₈4 ₃₆7 ₃₀6 ₁₅2 ₂₉5 ₉1

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TABLE 4

10 Bias(ˆα) 0.3571 _{- 4.490}5 _-4.4022 _-4.6586 _-5.0007 _-4.4043 _4.773e+518 _-4.4104 RMSE(ˆα) 2.1741 _4.4905 _4.4022 _4.6636 _5.0007 _4.4043 _3.116e+528 _4.4104 Bias( ˆθ) -1215.2737 _-0.0341 _-0.0432 _-0.0595 _0.1206 _-0.0454 _-5.023e+518 _-0.0453 RMSE( ˆθ) 6284.9507 _0.0411 _0.0472 _0.0605 _0.1206 _0.0534 _3.267e+528 _0.0533 Dabs 0.1856 _0.0211 _0.0282 _0.0415 _0.9877 _0.0294 _0.9998 _0.0293 Dmax 0.4856 _0.0971 _0.1172 _0.1515 _0.9987 _0.1274 _0.9998 _0.1273 PRanks 285 ₁₄2 ₁₂1 ₃₂6 ₄₀7 ₂₂4 ₄₈8 ₂₀3 20 Bias(ˆα) 4.0311 _-4.4905 _-4.3802 _-4.5566 _-5.0007 _-4.3984 _3.370e+508 _-4.3823 RMSE(ˆα) 5.2497 _4.4904 _4.3801 _4.5585 _5.0006 _4.3983 _9.111e+518 _4.3822 Bias( ˆθ) -109.8957 _-0.0432 _-0.0575 _-0.0716 _0.0201 _-0.0504 _-3.451e+508 _-0.0503 RMSE( ˆθ) 2594.3347 _0.0471 _0.0585 _0.0716 _0.0201 _0.0534 _9.296e+518 _0.0533 Dabs 0.1566 _0.0281 _0.0394 _0.0515 _.7 _0.0323 _0.9998 _0.0322 Dmax 0.4746 _0.1181 _0.1554 _0.1925 _0.9987 _0.1353 _0.9998 _0.1352 PRanks 347 ₁₅1.5 ₂₁3.5 ₃₃6 ₂₉5 ₂₁3.5 ₄₈8 ₁₅1.5 50 Bias(ˆα) 4.5436 _-4.4905 _-4.3882 _-4.4694 _-5.0008 _-4.3923 _-4.8577 _-4.3821 RMSE(ˆα) 7.9868 _4.4905 _4.3882 _4.4704 _5.0007 _4.3923 _4.8576 _4.3821 Bias( ˆθ) 0.1738 _-0.0472 _-0.0676 _0.0797 _0.0201 _-0.0504 _-0.0545 _-0.0503 RMSE( ˆθ) 0.1848 _0.0482 _0.0676 _0.0807 _0.0201 _0.0514 _0.0565 _0.0513 Dabs 0.1407 _0.0331 _0.0495 _0.0586 _0.9998 _0.0353 _0.0394 _0.0352 Dmax 0.4717 _0.1281 _0.1835 _0.2166 _0.9998 _0.1353 _0.1474 _0.1352 PRanks 448 ₁₆2 ₂₆4 ₃₄7 ₃₃6 ₂₀3 ₃₁5 ₁₂1 75 Bias(ˆα) 4.7456 _-4.4905 _-4.3953 _-4.4434 _-5.0008 _-4.3832 _-4.8537 _-4.3801 RMSE(ˆα) 8.4248 _4.4905 _4.3953 _4.4444 _5.0007 _4.3832 _4.8536 _4.3801 Bias( ˆθ) 0.1728 _-0.0482 _-0.0706 _-0.0827 _0.0201 _-0.0503 _-0.0565 _-0.0504 RMSE( ˆθ) 0.1808 _0.0492 _0.0716 _0.0827 _0.0201 _0.0513 _0.0585 _0.0514 Dabs 0.1357 _0.0341 _0.0525 _0.0606 _0.9998 _0.0362 _0.0414 _0.0363 Dmax 0.4687 _0.1311 _0.1925 _0.2226 _0.9998 _0.1362 _0.1534 _0.1363 PRanks 448 ₁₆2.5 ₂₈4 ₃₄7 ₃₃6 ₁₄1 ₃₁5 ₁₆2.5 100 Bias(ˆα) 4.7456 _-4.4905 _-4.3953 _-4.4434 _-5.0008 _-4.3832 _-4.8537 _-4.3801 RMSE(ˆα) 8.4248 _4.4905 _4.3953 _4.4444 _5.0007 _4.3832 _4.8536 _4.3801 Bias( ˆθ) 0.1728 _-0.0482 _-0.0706 _-0.0827 _0.0201 _-0.0503 _-0.0565 _-0.0504 RMSE( ˆθ) 0.1808 _0.0492 _0.0716 _0.0827 _0.0201 _0.0513 _0.0585 _0.0514 Dabs 0.1357 _0.0341 _0.0525 _0.0606 _0.9998 _0.0362 _0.0414 _0.0363 Dmax 0.4687 _0.1311 _0.1925 _0.2226 _0.9998 _0.1362 _0.1534 _0.1363 PRanks 448 ₁₆2.5 ₂₈4 ₃₄7 ₃₃6 ₁₄1 ₃₁5 ₁₆2.5

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TABLE 5

n Est. MLE LSE WLS PCE MPS CVM AD RTAD

10 Bias(ˆα) 1.7166 _-0.0101 _0.6083 _5259.7478 _-4.9997 _0.6235 _0.4302 _0.6194 RMSE(ˆα) 2.7336 _0.0101 _0.6133 _15005.5018 _4.9997 _0.6264 _0.5232 _0.6505 Bias( ˆθ) 0.5148 _-0.1472 _-0.1666 _-0.2057 _0.0411 _-0.1503 _-0.1525 _-0.1504 RMSE( ˆθ) 0.5298 _0.1482 _0.1686 _0.2107 _0.0421 _0.1523 _0.1575 _0.1524 Dabs 0.9998 _0.3191 _0.3595 _0.4447 _0.4246 _0.3253 _0.3252 _0.3254 Dmax 0.9977 _0.4001 _0.4515 _0.5586 _0.9998 _0.4082 _0.4144 _0.4083 PRanks 437.5 ₈1 ₂₈5 ₄₃7.5 ₃₀6 ₂₀2.5 ₂₀2.5 ₂₄4 20 Bias(ˆα) 1.3066 _-0.0101 _0.5754 _407.4388 _-5.0007 _0.5703 _0.4692 _0.7825 RMSE(ˆα) 1.6836 _0.0101 _0.5894 _2900.7988 _5.0007 _0.5753 _0.5612 _0.8295 Bias( ˆθ) 0.5168 _-0.1482 _-0.1806 _-0.2287 _0.0411 _-0.1504 _-0.1493 _-0.1505 RMSE( ˆθ) 0.5238 _0.1492 _0.1816 _0.2287 _0.0411 _0.1513 _0.1515 _0.1514 Dabs 0.9998 _0.3252 _0.3955 _0.5037 _0.4216 _0.3293 _0.3241 _0.3294 Dmax 0.9987 _0.4031 _0.4895 _0.6206 _0.9998 _0.4083 _0.4042 _0.4084 PRanks 437.5 ₉1 ₃₀5.5 ₄₃7.5 ₃₀5.5 ₁₉3 ₁₅2 ₂₇4 50 Bias(ˆα) 1.1296 _-0.0101 _0.4502 _8.1358 _-5.0007 _0.6004 _0.5663 _0.7325 RMSE(ˆα) 1.2565 _0.0101 _0.4922 _779.6428 _5.0007 _0.6053 _1.9396 _0.7894 Bias( ˆθ) 0.5158 _-0.1493 _-0.1976 _-0.2367 _0.0411 _-0.1505 _-0.1472 _-0.1504 RMSE( ˆθ) 0.5188 _0.1503 _0.1976 _0.2367 _0.0411 _0.1505 _0.1482 _0.1504 Dabs 0.9998 _0.3302 _0.4356 _0.5237 _0.4225 _0.3314 _0.3231 _0.3313 Dmax 0.9997 _0.4062 _0.5355 _0.6426 _0.9998 _0.4084 _0.3991 _0.4083 PRanks 427 ₁₂1 ₂₇5 ₄₃8 ₂₉6 ₂₅4 ₁₅2 ₂₃3 75 Bias(ˆα) 1.1316 _-0.0101 _0.6353 _-2.9087 _-5.0008 _0.6102 _0.6824 _0.7645 RMSE(ˆα) 1.2495 _0.0101 _0.7113 _2.9496 _5.0008 _0.6142 _3.3767 _0.8344 Bias( ˆθ) 0.5158 _-0.1503 _-0.2036 _-0.2397 _0.0411 _-0.1505 _-0.1462 _-0.1504 RMSE( ˆθ) 0.5178 _0.1503 _0.2036 _0.2397 _0.0411 _0.1505 _0.1472 _0.1504 Dabs 0.9998 _0.3312 _0.4496 _0.5287 _0.4225 _0.3324 _0.3231 _0.3323 Dmax 0.9977 _0.4072 _0.5525 _0.6496 _0.9998 _0.4084 _0.3981 _0.4083 PRanks 428 ₁₂1 ₂₉5 ₄₀7 ₃₁6 ₂₂3 ₁₇2 ₂₃4 100 Bias(ˆα) 1.1756 _-0.0101 _0.6924 _-2.9047 _-5.0008 _0.5982 _1.1275 _0.6443 RMSE(ˆα) 1.2995 _0.0101 _0.7444 _2.9316 _5.0007 _0.6022 _13.0488 _0.6863 Bias( ˆθ) 0.5168 _-0.1503 _-0.2076 _-0.2407 _0.0411 _-0.1505 _-0.1452 _-0.1504 RMSE( ˆθ) 0.5178 _0.1503 _0.2076 _0.2407 _0.0411 _0.1505 _0.1472 _0.1504 Dabs 0.9998 _0.3312 _0.4586 _0.5317 _0.4215 _0.3324 _0.3241 _0.3323 Dmax 0.9997 _0.4072 _0.5635 _0.6526 _0.9998 _0.4084 _0.3981 _0.4083 PRanks 428 ₁₂1 ₃₁6 ₄₀7 ₃₀5 ₂₂4 ₁₉2 ₂₀3

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TABLE 6

10 Bias(ˆα) 6.1847 _0.1001 _1.1173 _116.5728 _0.9232 _1.1184 _1.1395 _1.2206 RMSE(ˆα) 6.1877 _0.1001 _1.1194 _116.7338 _0.9722 _1.1183 _1.1695 _1.2226 Bias( ˆθ) 0.4512 _-0.5165 _-0.5166 _-0.6028 _-0.5073 _-0.5167 _-0.4041 _-0.5164 RMSE( ˆθ) 10.9238 _0.5163 _0.5164 _0.6027 _0.5266 _0.5165 _0.4041 _0.5162 20 Bias(ˆα) 6.3847 _0.1001 _1.1173 _194.5678 _0.9152 _1.1174 _1.2145 _1.2166 RMSE(ˆα) 6.3967 _0.1001 _1.1173 _194.2868 _0.9742 _1.1184 _1.8206 _1.2255 Bias( ˆθ) 1.0478 _-0.5164 _-0.5163 _-0.5957 _-0.5716 _-0.5165 _-0.4041 _-0.5161 RMSE( ˆθ) 23.7308 _0.5164 _0.5163 _0.5957 _0.5896 _0.5165 _0.4051 _0.5161 50 Bias(ˆα) 6.5327 _0.1001 _1.1164 _203.2608 _0.8772 _1.1143 _1.6426 _1.2085 RMSE(ˆα) 6.5657 _0.1001 _1.1174 _204.8568 _0.9552 _1.1173 _5.6356 _1.2315 Bias( ˆθ) 5.9368 _-0.5165 _-0.5163 _-0.4902 _-0.6417 _-0.5166 _-0.4041 _-0.5164 RMSE( ˆθ) 106.5688 _0.5165 _0.5163 _0.4902 _0.6527 _0.5166 _0.4041 _0.5164 75 Bias(ˆα) 7.2827 _0.1001 _1.1154 _267.4168 _0.8442 _1.1143 _2.0546 _1.2065 RMSE(ˆα) 7.3256 _0.1001 _1.1164 _268.0298 _0.9292 _1.1163 _8.0177 _1.2335 Bias( ˆθ) 3.0248 _-0.5165 _-0.5163 _-0.4252 _-0.6617 _-0.5166 _-0.4031 _-0.5164 RMSE( ˆθ) 46.7668 _0.5165 _0.5163 _0.4252 _0.6697 _0.5166 _0.4031 _0.5164 100 Bias(ˆα) 8.3487 _0.1001 _1.1154 _267.1708 _0.7992 _1.1143 _3.6536 _1.2085 RMSE(ˆα) 8.3856 _0.1001 _1.1163 _268.4118 _0.8912 _1.1164 _15.4187 _1.2315 Bias( ˆθ) 57.7898 _-0.5165 _-0.5163 _-0.4252 _-0.6757 _-0.5166 _-0.4001 _-0.5164 RMSE( ˆθ) 104.6328 _0.5165 _0.5163 _0.4252 _0.6807 _0.5166 _0.4011 _0.5164

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TABLE 7

Simulation results forα = 0.5, θ = 0.5 and λ = 0.5.

10 Bias(ˆα) -0.4617 _0.1293 _0.1554 _0.1242 _-0.0911 _0.2405 _0.5508 _0.2496 RMSE(ˆα) 0.6537 _0.3552 _0.4314 _0.4033 _0.1501 _0.4345 _1.7108 _0.5156 Bias( ˆθ) 987.1768 _0.0774 _0.0733 _0.0671 _-0.0712 _0.1347 _0.0965 _0.1166 RMSE( ˆθ) 1723.5888 _0.2423 _0.2524 _0.2756 _0.1031 _0.2685 _0.2332 _0.2767 Bias( ˆλ) 91.8515 _0.9762 _93000.0467 _10.9364 _6.6713 _0.6271 _20499.0526 _169552.3218 RMSE( ˆλ) 122.7544 _2.7522 _470922.7967 _25.0663 _464.0565 _2.3771 _266742.6896 _655496.4908 Dabs 0.9987 _0.1125 _0.1064 _0.9998 _0.0921 _0.1053 _0.2806 _0.1052 Dmax 0.9987 _0.1755 _0.1654 _0.9998 _0.1531 _0.1642 _0.4626 _0.1653 PRanks 538 ₂₆2 ₃₇5 ₃₅4 ₁₅1 ₂₉3 ₄₇7 ₄₆6 20 Bias(ˆα) -0.5238 _0.0912 _0.0923 _0.0541 _-0.0994 _0.1516 _0.2047 _0.1275 RMSE(ˆα) 0.6017 _0.2762 _0.3124 _0.3043 _0.1071 _0.3185 _1.0458 _0.3376 Bias( ˆθ) 1010.8258 _0.0604 _0.0483 _0.0252 _-0.0766 _0.0947 _0.0241 _0.0695 RMSE( ˆθ) 1963.2728 _0.2023 _0.2044 _0.2417 _0.0791 _0.2206 _0.1692 _0.2155 Bias( ˆλ) 91.2796 _0.4023 _3652.3757 _13.4705 _0.1301 _0.2502 _1.5514 _7398.2338 RMSE( ˆλ) 126.1946 _1.8423 _23610.3947 _31.0004 _0.1441 _1.6952 _50.8145 _3876.9278 Dabs 0.9987 _0.0865 _0.0792 _0.9998 _0.0711 _0.0824 _0.1786 _0.0823 Dmax 0.9987 _0.1435 _0.1281 _0.9998 _0.1292 _0.1333 _0.2936 _0.1354 PRanks 578 ₂₇2 ₃₁3 ₃₈5 ₁₇1 ₃₅4 ₃₉6 ₄₄7 50 Bias(ˆα) -0.5858 _0.0515 _0.0203 _0.0011 _-0.0947 _0.0756 _-0.0062 _0.0474 RMSE(ˆα) 0.5858 _0.1834 _0.1642 _0.2006 _0.1001 _0.2005 _0.2737 _0.1743 Bias( ˆθ) 2342.4438 _0.0395 _0.0071 _-0.0082 _-0.0667 _0.0566 _-0.0323 _0.0344 RMSE( ˆθ) 3848.0288 _0.1495 _0.1323 _0.1897 _0.0681 _0.1586 _0.1072 _0.1424 Bias( ˆλ) 167.9936 _0.2143 _8097.5537 _19.3315 _0.1272 _0.1041 _0.2254 _9906.8498 RMSE( ˆλ) 207.3566 _1.9744 _90739.3617 _40.6695 _0.1421 _1.7443 _1.2592 _112496.2808 Dabs 0.9997 _0.0655 _0.0571 _0.9998 _0.0582 _0.0644 _0.0936 _0.0633 Dmax 0.9997 _0.1215 _0.0961 _0.9998 _0.1092 _0.1154 _0.1576 _0.1153 PRanks 588 ₃₆5 ₂₅2 ₄₂7 ₂₃1 ₃₅4 ₃₂3 ₃₇6 75 Bias(ˆα) -0.5858 _0.0365 _-0.0011 _-0.0092 _-0.0907 _0.0526 _-0.0314 _0.0303 RMSE(ˆα) 0.5868 _0.1454 _0.1162 _0.1686 _0.0961 _0.1555 _0.2247 _0.1293 Bias( ˆθ) 1718.6128 _0.0294 _-0.0091 _-0.0172 _-0.0627 _0.0405 _-0.0466 _0.0243 RMSE( ˆθ) 3159.9318 _0.1245 _0.1043 _0.1717 _0.0651 _0.1296 _0.0892 _0.1154 Bias( ˆλ) 132.4176 _-0.0051 _1710.1647 _19.2995 _0.1273 _-0.0572 _0.2074 _3176.8078 RMSE( ˆλ) 173.8356 _1.3924 _25716.2687 _44.0445 _0.1421 _1.3133 _0.9952 _52977.8008 Dabs 0.9997 _0.0605 _0.0511 _0.9998 _0.0542 _0.0592 _0.0736 _0.0583 Dmax 0.9997 _0.1165 _0.0871 _0.9998 _0.1022 _0.1122 _0.1276 _0.1113 PRanks 588 ₃₃3 ₂₃1 ₄₃7 ₂₄2 ₃₅4.5 ₃₇6 ₃₅4.5 100 Bias(ˆα) -0.5858 _0.0284 _-0.0111 _-0.0152 _-0.0877 _0.0405 _-0.0456 _0.0233 RMSE(ˆα) 0.5898 _0.1205 _0.0921 _0.1517 _0.0932 _0.1276 _0.1184 _0.1033 Bias( ˆθ) 2693.6038 _0.0244 _-0.0171 _-0.0213 _-0.0597 _0.0335 _-0.0516 _0.0192 RMSE( ˆθ) 5072.6598 _0.1065 _0.0883 _0.1597 _0.0611 _0.1116 _0.0802 _0.0984 Bias( ˆλ) 168.6726 _-0.1372 _971.2498 _23.1585 _0.1261 _-0.1733 _0.1734 _438.5407 RMSE( ˆλ) 232.4316 _0.9644 _17087.0048 _49.2975 _0.1421 _0.8753 _0.4322 _13122.3987 Dabs 0.9987 _0.0565 _0.0471 _0.9998 _0.0522 _0.0554 _0.0656 _0.0553 Dmax 0.9987 _0.1145 _0.0811 _0.9998 _0.0962 _0.1104 _0.1156 _0.1103 PRanks 588 ₃₄4 ₂₄2 ₄₅7 ₂₃1 ₃₆5.5 ₃₆5.5 ₃₂3

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TABLE 8

Simulation results forα = 0.5, θ = 0.7 and λ = 0.7.

10 Bias(ˆα) 0.7718 _0.2813 _0.3655 _0.1641 _-0.3494 _0.446 _0.1752 _0.477

RMSE(ˆα) 1.0467 _0.5162 _0.6635 _0.5543 _0.3551 _0.6264 _1.6198 _0.786

Bias( ˆθ) 0.3418 _0.1733 _0.1814 _0.0692 _-0.1966 _0.2377 _-0.0391 _0.195

RMSE( ˆθ) 0.3528 _0.2963 _0.3185 _0.3044 _0.2012 _0.3266 _0.1571 _0.337

Bias( ˆλ) 8.903e078 _-0.2091 _16887.025 _10.6684 _0.2142 _-0.3513 _5.599e077 _48334.626

RMSE( ˆλ) 2.889e097 _1.3433 _22504.095 _23.8984 _0.2311 _1.1952 _2.978e098 _48296.096

Dabs 0.9998 _0.1644 _0.1593 _0.1665 _0.1956 _0.1562 _0.9987 _0.1551 Dmax 0.9998 _0.3665 _0.3634 _0.3423 _0.4296 _0.3362 _0.9987 _0.3261 PRanks 628 ₂₄1 ₃₆5 ₂₆2 ₂₈3 ₃₂4 ₄₁7 ₃₉6 20 Bias(ˆα) 0.5638 _0.3634 _0.4987 _0.0852 _-0.3383 _0.4606 _-0.0311 _0.3695 RMSE(ˆα) 0.6317 _0.5143 _0.7048 _0.4412 _0.3431 _0.5875 _0.5794 _0.6146 Bias( ˆθ) 0.3278 _0.2325 _0.2616 _0.0321 _-0.1964 _0.2707 _-0.0732 _0.1873 RMSE( ˆθ) 0.3327 _0.3005 _0.3388 _0.2693 _0.1992 _0.3246 _0.1231 _0.2894 Bias( ˆλ) 467478.5678 _-0.5782 _498.8916 _13.7454 _0.2101 _-0.6173 _227.9455 _5173.6327 RMSE( ˆλ) 684114.3568 _0.8653 _18904.1956 _31.2434 _0.2241 _0.8452 _2806.7885 _11367.5907 Dabs 0.9547 _0.1425 _0.1371 _0.9998 _0.1876 _0.1383 _0.1485 _0.1372 Dmax 0.9997 _0.3863 _0.3874 _0.9998 _0.4576 _0.3652 _0.3885 _0.3501 PRanks 608 ₂₉3 ₄₆7 ₃₂4 ₂₄1 ₃₄5 ₂₈2 ₃₅6 50 Bias(ˆα) 0.4797 _0.4205 _0.5568 _0.0151 _-0.3134 _0.4656 _-0.0822 _0.2783 RMSE(ˆα) 0.5026 _0.4985 _0.6788 _0.3092 _0.3183 _0.5357 _0.2261 _0.4324 Bias( ˆθ) 0.3198 _0.2805 _0.3157 _-0.0031 _-0.1904 _0.2986 _-0.0892 _0.1813 RMSE( ˆθ) 0.3216 _0.3095 _0.3518 _0.2173 _0.1912 _0.3237 _0.1301 _0.2454 Bias( ˆλ) 1398.8288 _-0.7244 _-0.5982 _15.9906 _0.2011 _-0.7345 _19.7187 _-0.6033 RMSE( ˆλ) 8789.7648 _0.7623 _10.5085 _40.0366 _0.2171 _0.7502 _53.6537 _0.9104 Dabs 0.9457 _0.1264 _0.1201 _0.9998 _0.1786 _0.1243 _0.1395 _0.1242 Dmax 0.9987 _0.3964 _0.3953 _0.9998 _0.4475 _0.3872 _0.4566 _0.3701 PRanks 578 ₃₅4.5 ₄₂7 ₃₅4.5 ₂₆2 ₃₈6 ₃₁3 ₂₄1 75 Bias(ˆα) 0.4647 _0.4315 _0.5428 _0.0011 _-0.3004 _0.4626_- _0.0952 _0.2543 RMSE(ˆα) 0.4795 _0.4886 _0.6398 _0.2632 _0.3053 _0.5157 _0.1571 _0.3744 Bias( ˆθ) 0.3177 _0.2915 _0.3198 _-0.0101 _-0.1854 _0.3036 _-0.0962 _0.1803 RMSE( ˆθ) 0.3186 _0.3115 _0.3468 _0.1953 _0.1862 _0.3217 _0.1391 _0.2294 Bias( ˆλ) 1306.1678 _-0.7393 _-0.7585 _16.3256 _0.1941 _-0.7444 _31.0217 _-0.6252 RMSE( ˆλ) 3659.3668 _0.7433 _0.7655 _44.4576 _0.2111 _0.7484 _72.9547 _0.6432 Dabs 0.9438 _0.1224 _0.1161 _0.1305 _0.1747 _0.1213 _0.1466 _0.1212 Dmax 0.9998 _0.3965 _0.3934 _0.3902 _0.4356 _0.3903 _0.5297 _0.3741 PRanks 578 ₃₆5 ₄₇7 ₂₆2 ₂₈3 ₄₀6 ₃₃4 ₂₁1 100 Bias(ˆα) 0.4556 _0.4355 _0.5298 _-0.0071 _-0.2874 _0.4607 _-0.1032 _0.2403 RMSE(ˆα) 0.4675 _0.4826 _0.6148 _0.2352 _0.2933 _0.5047 _0.1621 _0.3374 Bias( ˆθ) 0.3167 _0.2965 _0.3208 _-0.0141 _-0.1814 _0.3066 _-0.1042 _0.1783 RMSE( ˆθ) 0.3176 _0.3125 _0.3438 _0.1823 _0.1822 _0.3217 _0.1471 _0.2204 Bias( ˆλ) 9870.3298 _-0.7443 _-0.7585 _14.9926 _0.1871 _-0.7484 _39.4157 _-0.6312 RMSE( ˆλ) 1817.7638 _0.7473 _0.7645 _42.8736 _0.2071 _0.7514 _84.6837 _0.6442 Dabs 0.9428 _0.1204 _0.1131 _0.1265 _0.1727 _0.1203 _0.1526 _0.1192 Dmax 0.9998 _0.3965 _0.3903 _0.3802 _0.4256 _0.3924 _0.5797 _0.3751 PRanks 568 ₃₆5 ₄₆7 ₂₆2 ₂₈3 ₄₂6 ₃₃4 ₂₁1