Research article

The usage of logistic regression and artificial neural networks for evaluation and predicting property-liability insurers' solvency in Egypt

  • Unlike prior solvency prediction studies conducted in Egypt, this study aims to set up a real picture of companies' financial performance in the Egyptian insurance market. Therefore, 11 financial ratios commonly used by NAIC, AM BEST Company, and S & P Global Ratings were calculated for all property-liability insurance companies in Egypt from 2010 to 2020. They have been used to measure those companies' financial performance efficiency levels by comparing these ratios with the international standard limits. The financial analysis results for those companies revealed that property-liability insurers in Egypt do not have the same level of financial performance efficiency where those companies are classified into three groups: excellent, good, and poor. Furthermore, this paper investigates using the stepwise logistic regression model to determine the most factors among these selected financial ratios that influence those companies' financial performance. The results suggest that only three ratios were statistically significant predictors: "Risk retention rate", "Insurance account receivable to total assets", and "Net profit after tax to total assets". Finally, this paper presents the multi-layers artificial neural network with a backpropagation algorithm as a new solvency prediction model with perfect classifying accuracy of 100%. The trained ANN could predict the next fiscal year with a prediction accuracy of 91.67%, and this percent is a good and favorable result comparing to other solvency prediction models used in Egypt.

    Citation: Eid Elghaly Hassan, Diping Zhang. The usage of logistic regression and artificial neural networks for evaluation and predicting property-liability insurers' solvency in Egypt[J]. Data Science in Finance and Economics, 2021, 1(3): 215-234. doi: 10.3934/DSFE.2021012

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  • Unlike prior solvency prediction studies conducted in Egypt, this study aims to set up a real picture of companies' financial performance in the Egyptian insurance market. Therefore, 11 financial ratios commonly used by NAIC, AM BEST Company, and S & P Global Ratings were calculated for all property-liability insurance companies in Egypt from 2010 to 2020. They have been used to measure those companies' financial performance efficiency levels by comparing these ratios with the international standard limits. The financial analysis results for those companies revealed that property-liability insurers in Egypt do not have the same level of financial performance efficiency where those companies are classified into three groups: excellent, good, and poor. Furthermore, this paper investigates using the stepwise logistic regression model to determine the most factors among these selected financial ratios that influence those companies' financial performance. The results suggest that only three ratios were statistically significant predictors: "Risk retention rate", "Insurance account receivable to total assets", and "Net profit after tax to total assets". Finally, this paper presents the multi-layers artificial neural network with a backpropagation algorithm as a new solvency prediction model with perfect classifying accuracy of 100%. The trained ANN could predict the next fiscal year with a prediction accuracy of 91.67%, and this percent is a good and favorable result comparing to other solvency prediction models used in Egypt.



    Asymmetric datasets modeling utilizing generalized continuous distributions is quite an interesting area of research in recent times, and quite a number of generalized continuous distributions have been proposed, established and utilized in different fields of study. Besides with the new progressions in probability distribution theory, more vigorous forms of continuous distributions have been proposed utilizing the exponentiated method, transformer-transformer method, beta-generated method and gamma-generated method, among others. Shockingly, these recent distributions have been valuable in resolving issues in different fields of study other than in GARCH-type volatility models. Along these lines, it is important to develop new conditional innovation distributions capable of handling both significant skewness and excess kurtosis present in financial asset returns. Jones (2001) proposed a simplified extension of the symmetric Student's-t distribution known as the skew-t distribution to handle real-life asymmetric datasets. Jones and Faddy (2003) described the distribution as a simplified skew-t distribution and provided some important structural properties. Some other forms of the skew-t distribution that have been developed can be found in the literature (Johnson et al., 1995; Sahu et al., 2003; Azzalini and Capitanio, 2003). More so, the statistical literature consists several extended forms of the skew-t distribution. For example, the generalized hyperbolic skew-t distribution (Aas and Haff, 2006), beta skew-t distribution (Shittu et al., 2014), Balakrishnan skew-t distribution (Shafiei and Doostparast, 2014), Kumaraswamy skew-t distribution (Khamis et al., 2017), exponentiated skew-t distribution (Dikko and Agboola, 2017), generalized alpha skew-t distribution (Altun et al., 2018), odd exponentiated skew-t distribution (Adubisi et al., 2021a), and type I half-logistic skew-t distribution (Adubisi et al., 2021b).

    For any baseline distribution with parameter vector ζ, Cordeiro et al. (2014) proposed the exponentiated half-logistic-G (EHL-G) family based on the T-X method (Alzaatreh et al., 2013). Given that k(t) and K(t) are the probability density function (pdf) and cumulative distribution function (cdf) of a random variable (r.v) T[c,d] for [c,d](,+), respectively. Let R[H(y;ζ)] be a function of the cdf of another r. v Y satisfying the following conditions: [i] R[H(y;ζ)][c,d]; [ii] R[H(y;ζ)]c as y and R[H(y;ζ)]d as y; and [iii] R[H(y;ζ)] is differentiable and monotonically non-decreasing. The cdf of the T-X method is given by

    F(y)=R[H(y;ζ)]ck(t)dt=K(R[H(y;ζ)]), (1)

    Setting k(t)=2αφeαt[1eαt]φ1(1+eαt)φ+1,t>0, where α,φ>0 are the shape parameters, and

    21R[H(y;ζ)]= log [1H(y;ζ)], the cdf of the EHL-G family is given by

    F(y;α,φ,ζ)={1[1H(y;ζ)]α1+[1H(y;ζ)]α}φ, (2)

    and the corresponding pdf to Equation 2 is

    f(y;α,φ,ζ)=2αφh(y;ζ)[1H(y;ζ)]α1{1[1H(y;ζ)]α}φ1{1+[1H(y;ζ)]α}φ+1, (3)

    where α,φ>0 are two additional shape parameters. H(y;ζ) and h(y;ζ) are the baseline cdf and pdf.

    In this research work, a hybrid form of the skew-t distribution developed by Jones and Faddy (2003) using the exponentiated half logistic-G family of distributions is proposed. The new three-parameter model called the exponentiated half logistic skew t (EHLST) distribution appropriate for modeling right or left skewed and heavy-tailed datasets is developed. By inserting the skew-t cdf H(y;κ)=12(1+yκ+y2) into Equation 2 leads to

    F(y)={1[1(12(1+yκ+y2))]α1+[1(12(1+yκ+y2))]α}φ, (4)

    The corresponding pdf to Equation 4 is

    f(y)=2αφκ[1(12(1+yκ+y2))]α1{1[1(12(1+yκ+y2))]α}φ12(κ+y2)32{1+[1(12(1+yκ+y2))]α}φ+1, (5)

    where α>0 and φ>0 are two shape parameters, and κ denote the skew parameter. From now onward, YEHLST(α,φ,κ) denotes a random variable with pdf Equation 5. The hazard rate function (hrf) of Y is

    η(y)=αφ(κ(κ+y2)32)[1(12(1+yκ+y2))]α1{1[1(12(1+yκ+y2))]α}φ1({1+[1(12(1+yκ+y2))]α}φ{1[1(12(1+yκ+y2))]α}φ{1+[1(12(1+yκ+y2))]α}). (6)

    Detailed properties of the EHLST model can be found in Adubisi et al. (2021c). Lately, there has been a key interest in the comparison of different estimation procedures for the parameter estimation of several distributions. For example, the extended exponential geometric (Louzada et al., 2016), Binomial exponential 2 (Bakouch et al., 2017), Poisson exponential (Rodrigues et al., 2018), type-I half-logistic Top-Leone (ZeinEldin et al., 2019), polynomial exponential (Chesneau et al., 2020), Fréchet (Ramos et al., 2020), odd exponential half-logistic exponential (Aldahlan and Afify, 2020), among others. The motivation in developing the new model is to create a more flexible heavy-tailed distribution with right-skewed, left-skewed, symmetric and unimodal features capable of handling the stylized features in financial datasets. The aim of this work is to estimate the parameters of the EHLST distribution using different frequentist estimation criterions such as the maximum likelihood, Anderson-Darling, maximum product of spacing, ordinary least squares, Cramer-von mises and weighted least squares. These estimation criterions using extensive Monte Carlo simulation are compared to discourse their performance. Also, Altun (2019) noted that the assumption of the conditional innovation density of the GARCH volatility model directly impacts on the accuracy of volatility predictions. Hence, the standardized form of the EHLST model is also derived to serve as an alternative conditional innovation density in modeling and forecasting the volatility of the inflation rates in sub-Sahara Africa, specifically Nigeria inflation rate using the GARCH-type volatility models. The rest of this research work is designed as follows. The quantile function of the EHLST is derived in Section 2. Six estimation criterions for estimating the parameters of the EHLST model are presented in Section 3. Simulation study to compare these criterions and real-life application are presented in Section 4. The application of the EHLST density in volatility modeling is illustrated via GARCH-type models in Section 5. The empirical results of the GARCH process are presented in Section 6. Conclusion of the work in Section 7.

    Figures 1 and 2 provides plots of the pdf and hazard rate function of the EHLST distribution for selected values of the parameters. The plot in figure 2, shows that the hazard rate function can be J-shape, increasing and decreasing.

    Figure 1.  The EHLST density (pdf) plots for selected parameter values.
    Figure 2.  The EHLST hazard rate function (hrf) plots for selected parameter values.

    The quantile function, Q(u)=F1(u) of Y can be derived by inverting Equation 4. The quantile function of the EHLST distribution takes the form:

    Q(u)=κ12[12(1u1φ1+u1φ)1α][1(12(1u1φ1+u1φ)1α)2]12,0u1. (7)

    The quantile function in Equation 7 can define various quantile measures such as Bowley's skewness and Moor's kurtosis of the EHLST distribution. Figure 3 depicts the three-dimensional plots of these skewness and kurtosis measures. It is clear that the skewness was increasing in φ and decreasing in α while the kurtosis is increasing in α and decreasing in φ, and were independent of κ.

    Figure 3.  Bowley's skewness and Moor's kurtosis plots of the EHLST distribution.

    This section discusses six estimation criterions, these are the maximum likelihood (ML), maximum product of spacing (MPS), Anderson-Darling (ANDA), ordinary least squares (OLS), weighted least squares (WLS) and Cramer-von Mises (CVM) to estimate the new model parameters.

    The ML estimation is considered the best for estimating distribution parameters given its excellent characteristics. Let y1,,yT be the observed values of sizes (T) from the EHLST distribution. The log-likelihood function derived from Equation 6 is given by

    l(α,φ,κ)=T ln α+T ln φ+T ln κ32Tk=1 ln (κ+y2k)+(α1)Tk=1 ln (zk)+(φ1)Tk=1 ln (1(zk)α)(φ+1)Tk=1 ln (1+(zk)α) (8)
    where zk=(112(1+ykκ+y2k)) (9)

    Let ˆαMLE,ˆφMLE and ˆκMLE be the ML estimates of α,φ and κ. These can be obtained numerically by maximizing l(α,φ,κ) or solving the following differential equations:

    Uα=Tα+Tk=1 ln (zk)(φ1)Tk=1(zk)α ln (zk){1(zk)α}(φ+1)Tk=1(zk)α ln (zk){1+(zk)α}=0, (10)
    Uφ=Tφ+Tk=1 ln (1(zk)α)Tk=1 ln {1+(zk)α}=0, (11)

    and

    Uη=Tκ32Tk=11(κ+y2k)+(α1)Tk=1yk4(κ+y2k)32(zk)α(φ1)Tk=1yk(zk)α4(κ+y2k)32(zk){1+(zk)α}α(φ+1)Tk=1yk(zk)α4(κ+y2k)32(zk){1+(zk)α}=0 (12)

    Let y(1:T),y(2:T),,y(T:T) be the ordered sample of size (T) from the EHLST distribution with cdf in Equation 5. The OLS estimates ˆαOLS,ˆφOLS and ˆκOLS of α,φ and κ can be obtained numerically by minimizing

    OL(α,φ,κ)=Tk=1[{1[1(12(1+y(k:T)κ+y2(k:T)))]α1+[1(12(1+y(k:T)κ+y2(k:T)))]α}φ(k,T)]2, (13)

    where (k,T)=(T+1k)(T+1), with respect to α,φ and κ.Also, the model estimates can be obtained by solving the following differential equations:

    OL(α,φ,κ)α=Tk=1[{1[z(k:T)]α1+[z(k:T)]α}φ(k,T)]Φ1(y(k:T)|α,φ,κ)=0, (14)
    OL(α,φ,κ)φ=Tk=1[{1[z(k:T)]α1+[z(k:T)]α}φ(k,T)]Φ2(y(k:T)|α,φ,κ)=0, (15)

    and

    OL(α,φ,κ)κ=Tk=1[{1[z(k:T)]α1+[z(k:T)]α}φ(k,T)]Φ3(y(k:T)|α,φ,κ)=0, (16)

    where z(k:T) is the order transformed value of zk given by Equation 9, and

    Φ1(y(k:T  )  |α,φ,κ)={(1(z(k:T )   )α1+(z(k:T )   )α)φφ((z(k:T )   )α log (z(k:T )   )1+(z(k:T )   )α[1(z(k:T )   )α  ](z(k:T )   )α log (z(k:T )   )[1+(z(k:T )   )α  ]  2)[1+(z(k:T )   )α  ]}1(z(k:T )   )α (17)
    Φ2(y(k:T  )  |α,φ,κ)=(1(z(k:T )   )α1+(z(k:T )   )α)φ log (1(z(k:T )   )α1+(z(k:T )   )α) (18)

    and

    Φ3(y(k:T  )  |α,φ,κ)={(1(z(k:T )   )α1+(z(k:T )   )α)φφ(αy(z(k:T )   )α4(κ+y2(k:T )   )32(z(k:T )   )[1+(z(k:T )   )α  ]αy[1(z(k:T )   )α  ](z(k:T )   )α4[1+(z(k:T )   )α  ]2(κ+y2(k:T )   )32(z(k:T )   ))[1+(z(k:T )   )α  ]}1(z(k:T )   )α (19)

    Applying the same symbolization as in the preceding subsection, the WLS estimates ˆαWLS,ˆφWLS and ˆκWLS of α,φ and κ can be obtained numerically by minimizing

    WL(α,φ,κ)=Tk=1Ψ(k,T)[{1[1(12(1+y(k:T  )  κ+y2(k:T  )  ))  ]α1+[1(12(1+y(k:T  )  κ+y2(k:T  )  ))  ]α}φ(k,T)  ]2, (20)

    where Ψ(k,T)=(T+1)2(T+2)k(Tk+1), relative to α,φ and κ. Also, the model estimates can be obtained by solving the following differential equations:

    WL(α,φ,κ)α=Tk=1Ψ(k,T)[{1[z(k:T)]α1+[z(k:T)]α}φ(k,T)]Φ1(y(k:T)|α,φ,κ)=0, (21)
    WL(α,φ,κ)φ=Tk=1Ψ(k,T)[{1[z(k:T)]α1+[z(k:T)]α}φ(k,T)]Φ2(y(k:T)|α,φ,κ)=0, (22)

    and

    OL(α,φ,κ)κ=Tk=1Ψ(k,T)[{1[z(k:T)]α1+[z(k:T)]α}φ(k,T)]Φ3(y(k:T)|α,φ,κ)=0, (23)

    where z(k:T) and Φi(y(k:T)|α,φ,κ), i=1,2,3 are given by Eqs 9, 17, 18 and 19, respectively.

    Cheng and Amin(1979, 1983) proposed the MPS estimator for the estimation of unknown parameters with ordered sample y(1:T),y(2:T),,y(T:T) from the EHLST distribution and uniform spacing for this random sample is given by:

    Dk(α,φ,κ)=F(y(k:T)|α,φ,κ)F(y(k1:T)|α,φ,κ), for k=1,2,,T, (24)

    where F(y(0:T)|α,φ,κ)=0 and F(y(T+1:T)|α,φ,κ)=1. The MPS estimates ˆαMPS, ˆφMPS and ˆκMPS of α,φ, and κ can be obtained by maximizing

    MP(α,φ,κ)=1T+1T+1k=1 log Dk(α,φ,κ) (25)

    relative to α,φ, and κ. Also, the estimates can be obtained by solving the following differential equations:

    MP(α,φ,κ)α=T+1k=11Dk(α,φ,κ)Φ1(y(k:T)|α,φ,κ)Φ1(y(k1:T)|α,φ,κ)=0, (26)
    MP(α,φ,κ)φ=T+1k=11Dk(α,φ,κ)Φ2(y(k:T)|α,φ,κ)Φ2(y(k1:T)|α,φ,κ)=0, (27)

    and

    MP(α,φ,κ)κ=T+1k=11Dk(α,φ,κ)Φ3(y(k:T)|α,φ,κ)Φ3(y(k1:T)|α,φ,κ)=0 (28)

    where Φi(.|α,φ,κ), i=1,2,3 are specified in Equation s 17, 18 and 19, respectively.

    The ANDA estimates ˆαANDA, ˆφANDA and ˆκANDA of the EHLST parameters can be obtained by minimizing relative to α,φ, and κ, the function

    AD(α,φ,κ)=T1TTk=1(2k1) log [F(y(k:T)|α,φ,κ)]+ log [ˉF(y(T+1k:n)|α,φ,κ)], (29)

    Also, estimates obtained by solving the following nonlinear equations:

    AD(α,φ,κ)α=Tk=1(2k1)[Φ1(y(k:T  )    |α,φ,κ)F(y(k:T  )    |α,φ,κ)Φ1(y(T+1k:T  )    |α,φ,κ)ˉF(y(T+1k:T  )    |α,φ,κ)]=0, (30)
    AD(α,φ,κ)φ=Tk=1(2k1)[Φ2(y(k:T  )    |α,φ,κ)F(y(k:T  )    |α,φ,κ)Φ2(y(T+1k:T  )    |α,φ,κ)ˉF(y(T+1k:T  )    |α,φ,κ)]=0, (31)

    and

    AD(α,φ,κ)κ=Tk=1(2k1)[Φ3(y(k:T  )    |α,φ,κ)F(y(k:T  )    |α,φ,κ)Φ3(y(T+1k:T  )    |α,φ,κ)ˉF(y(T+1k:T  )    |α,φ,κ)]=0, (32)

    where Φi(.|α,φ,κ), i=1,2,3 are specified in Eqs 17, 18 and 19, respectively.

    The CVM estimates ˆαCVM, ˆφCVM and ˆκCVM of the EHLST parameters can be obtained by minimizing relative to α,φ, and κ, the function

    CV(α,φ,κ)=112T+Tk=1[{1[1(12(1+y(k:T  )    κ+y2(k:T  )    ))]α1+[1(12(1+y(k:T  )    κ+y2(k:T  )    ))]α}φ{2(k1)+12T}]2, (33)

    Solving the following nonlinear equations, the CVM estimates can also be obtained as follows:

    OL(α,φ,κ)α=Tk=1[{1[z(k:T)]α1+[z(k:T)]α}φ{2(k1)+12T}]Φ1(y(k:T)|α,φ,κ)=0, (34)
    OL(α,φ,κ)φ=Tk=1[{1[z(k:T)]α1+[z(k:T)]α}φ{2(k1)+12T}]Φ2(y(k:T)|α,φ,κ)=0, (35)

    and

    OL(α,φ,κ)κ=Tk=1[{1[z(k:T)]α1+[z(k:T)]α}φ{2(k1)+12T}]Φ3(y(k:T)|α,φ,κ)=0, (36)

    where z(k:T) and Φi(y(k:T)|α,φ,κ), i=1,2,3 are given by Eqs 9, 17, 18 and 19, respectively.

    The performance of the six estimators defined in Section 5 are compared using Monte Carlo simulations. The parameter combinations (Comb), i.e., Comb 1 (α=1.0,φ=1.3,κ=0.5), Comb 2 (α=1.3,φ=1.7,κ=1.0), Comb 3 (α=2.0,φ=2.5,κ=1.5), and Comb 4 (α=2.5,φ=3.0,κ=2.0). The datasets are produced from the EHLST distribution under these combinations by selecting n = 10, 50,100 and 200. For each process, 1000 random samples from the EHLST distribution are produced using the quantile function in Eq 6. The parameter estimates performance are evaluated through the average values (AVEs), average absolute biases (AVABs), mean square errors (MSEs) and root mean square errors (RMSEs) for the different sample sizes (n). For the estimates, the AVAB, MSE and RMSE are computed using

    AVAB(ˆϑi)=110001000j=1|ˆϑi,jϑi|,MSE(ˆϑi)=110001000j=1(ˆϑi,jϑi)2,
    RMSE(ˆϑi)=110001000j=1(ˆϑi,jϑi)2

    where, ˆϑ=(ˆα,ˆφ,ˆκ) and ϑ=(α,φ,κ). The partial and overall ranks of the six estimators for the combinations to determine the best method for estimating the EHLST parameters are provided in Table 1. Based on the results, it is realistic to use the ML and MPS methods for estimating the unknown parameters of the EHLST distribution.

    Table 1.  Partial and overall ranks of the six estimators.
    Parameter (pa) n ML MPS ANDA CVM OLS WLS
    α=1.0,φ=1.3,κ=0.5 10 2 5 6 4 1 3
    50 2 1 6 5 4 3
    100 2 1 4 6 5 3
    200 3 1 2 6 5 4
    α=1.3,φ=1.7,κ=1.0 10 2 5 6 4 3 1
    50 2 1 4.5 6 4.5 3
    100 3 1 2 6 5 4
    200 3 1 2 6 5 4
    α=2.0,φ=2.5,κ=1.5 10 1 5 6 4 2 3
    50 2 1 3 6 5 4
    100 2 1 3 6 4 5
    200 2 1 3 6 5 4
    α=2.5,φ=3.0,κ=2.0 10 1 5 6 4 2 3
    50 1.5 1.5 6 6 3 4
    100 1.5 1.5 5 6 3 4
    200 2 1 3 6 6 4
    ranks 32 33 67.5 87 62.5 56
    Overall rank 1 2 5 6 4 3

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    Tables 25 provides the AVEs, AVABs, MSEs and RMSEs of the six estimators. More so, these tables show the rank of each estimator among all the estimators in each row, the superscripts are the rank indicators and ranks is the partial sum of the ranks for each column and sample size (n).

    Table 2.  Simulation results of the estimators for α=1.0,φ=1.3,κ=0.5.
    n Measures Pa. ML MPS ANDA CVM OLS WLS
    10 AVE α 1.4661(5) 1.4066(4) 1.6544(6) 1.3840(3) 1.1710(2) 1.1590(1)
    φ 2.4233(3) 4.8935(5) 8.1802(6) 2.6132(4) 1.9540(1) 2.0060(2)
    κ 1.1683(1) 2.3008(5) 2.4332(6) 1.3781(4) 1.2180(2) 1.2700(3)
    AVAB α 0.4661(5) 0.4066(4) 0.6544(6) 0.3840(3) 0.1713(2) 0.1594(1)
    φ 1.1233(3) 3.5935(5) 6.8802(6) 1.3132(4) 0.6536(1) 0.7063(2)
    κ 0.6683(1) 1.8008(5) 1.9332(6) 0.8781(4) 0.7178(2) 0.7701(3)
    MSE α 0.8648(3) 2.3453(5) 3.3241(6) 1.0758(4) 0.6714(1) 0.7021(2)
    φ 4.1873(1) 172.4338(5) 472.9212(6) 19.7630(4) 15.0648(2) 15.7997(3)
    κ 2.4706(1) 35.4217(5) 36.2999(6) 6.9764(3) 5.3805(2) 7.9437(4)
    RMSE α 0.9300(3) 1.5314(5) 1.8232(6) 1.0372(4) 0.8194(1) 0.8379(2)
    φ 2.0463(1) 13.1314(5) 21.7468(6) 4.4456(4) 3.8813(2) 3.9749(3)
    κ 1.5718(1) 5.9516(6) 6.0249(7) 2.6413(3) 2.3196(2) 2.8185(4)
    ranks 28(2) 49(5) 73(6) 44(4) 20(1) 29(3)
    50 AVE α 1.1687(5) 0.9853(1) 1.1595(3) 1.2278(6) 1.1672(4) 1.1501(2)
    φ 1.5906(2) 1.3920(1) 2.1469(6) 1.8565(5) 1.7203(4) 1.6610(3)
    κ 0.7185(2) 0.6255(1) 0.8535(5) 0.8632(6) 0.8289(4) 0.7911(3)
    AVAB α 0.1687(5) 0.0147(1) 0.1595(3) 0.2278(6) 0.1672(4) 0.1501(2)
    φ 0.2906(2) 0.0920(1) 0.8469(6) 0.5565(5) 0.4203(4) 0.3610(3)
    κ 0.2185(2) 0.1255(1) 0.3535(4) 0.3632(5) 1.3812(6) 0.2911(3)
    MSE α 0.2171(1) 0.2513(2) 0.5655(6) 0.4314(5) 0.3657(4) 0.3091(3)
    φ 0.5725(1) 2.5172(2) 38.3736(6) 3.8641(3) 3.9756(4) 4.4224(5)
    κ 0.3991(1) 0.9812(2) 3.3622(6) 1.5146(4) 1.3812(3) 1.8538(5)
    RMSE α 0.4659(1) 0.5013(2) 0.7520(6) 0.6568(5) 0.6047(4) 0.5559(3)
    φ 0.7567(1) 1.5866(2) 6.1946(6) 1.9657(3) 1.9939(4) 2.1029(5)
    κ 0.6317(1) 0.9905(2) 1.8336(6) 1.2307(4) 1.1752(3) 1.3616(5)
    ranks 24(2) 18(1) 63(6) 57(5) 48(4) 42(3)
    100 AVE α 1.0768(4) 0.9461(1) 1.0561(2) 1.1320(6) 1.0928(5) 1.0652(3)
    φ 1.4224(2) 1.2463(1) 1.5157(4) 1.6517(6) 1.5443(5) 1.4523(3)
    κ 0.6022(2) 0.4996(1) 0.6226(4) 0.7279(6) 0.6878(5) 0.6203(3)
    AVAB α 0.0768(4) 0.0539(1) 0.0561(2) 0.1320(6) 0.0928(5) 0.0652(3)
    φ 0.1224(2) 0.0537(1) 0.2157(4) 0.3517(6) 0.2443(5) 0.1523(3)
    κ 0.1022(2) 0.0004(1) 0.1226(5) 0.2279(6) 0.1878(3) 0.1203(4)
    MSE α 0.0831(2) 0.0682(1) 0.1692(4) 0.2889(6) 0.2350(5) 0.1393(3)
    φ 0.1652(2) 0.1829(3) 6.2211(6) 3.6319(5) 2.1739(4) 1.4473(1)
    κ 0.1402(1) 0.1508(2) 1.0192(5) 1.1513(6) 0.8198(4) 0.3606(3)
    RMSE α 0.2882(2) 0.2612(1) 0.4113(4) 0.5375(6) 0.4848(5) 0.3732(3)
    φ 0.4065(1) 0.4277(2) 2.4942(6) 1.9058(5) 1.4744(4) 1.2030(3)
    κ 0.3744(1) 0.3883(2) 1.0096(5) 1.0730(6) 0.9054(4) 0.6005(3)
    ranks 25(2) 17(1) 51(4) 70(6) 54(5) 35(3)
    200 AVE α 1.0360(4) 0.9541(1) 1.0240(2) 1.0588(6) 1.0415(5) 1.0285(3)
    φ 1.3576(3) 1.2478(1) 1.3482(2) 1.4147(6) 1.3846(5) 1.3593(4)
    κ 0.5436(3) 0.4775(1) 0.5384(2) 0.5821(6) 0.5722(5) 0.5493(4)
    AVAB α 0.0360(3) 0.0459(5) 0.0240(1) 0.0588(6) 0.0415(4) 0.0285(2)
    φ 0.0576(3) 0.0522(2) 0.0482(1) 0.1147(6) 0.0846(5) 0.0593(4)
    κ 0.0436(3) 0.0225(1) 0.0384(2) 0.0821(6) 0.0722(5) 0.0493(4)
    MSE α 0.0329(2) 0.0270(1) 0.0441(3) 0.0917(6) 0.0824(5) 0.0567(4)
    φ 0.0624(2) 0.0455(1) 0.0849(3) 0.2907(6) 0.2382(5) 0.1839(4)
    κ 0.0417(2) 0.0291(1) 0.0552(3) 0.1570(6) 0.1399(5) 0.0947(4)
    RMSE α 0.1813(2) 0.1645(1) 0.2101(3) 0.3028(6) 0.2871(5) 0.2380(4)
    φ 0.2498(2) 0.2133(1) 0.2914(3) 0.5392(6) 0.4881(5) 0.4289(4)
    κ 0.2041(2) 0.1706(1) 0.2350(3) 0.3963(6) 0.3741(5) 0.3077(4)
    ranks 31(3) 17(1) 28(2) 82(6) 59(5) 45(4)

     | Show Table
    DownLoad: CSV
    Table 3.  Simulation results of the estimators for α=0.8,φ=1.5,κ=1.0.
    n Measures Pa. ML MPS ANDA CVM OLS WLS
    10 AVE α 1.1024(3) 1.1175(5) 1.4136(6) 1.0918(4) 0.9769(2) 0.9432(1)
    φ 2.7852(4) 5.2337(5) 9.8213(6) 2.7427(3) 2.0152(2) 1.8648(1)
    κ 1.9428(1) 4.5719(5) 6.0877(6) 2.7247(3) 2.9305(4) 2.4634(2)
    AVAB α 0.3024(4) 0.3175(5) 0.6136(6) 0.2918(3) 0.1769(2) 0.1432(1)
    φ 1.2852(4) 3.7337(5) 8.3213(6) 1.2427(3) 0.5152(2) 0.3648(1)
    κ 0.9428(1) 3.5719(5) 5.0877(6) 1.7247(3) 1.9305(4) 1.4634(2)
    MSE α 0.5045(3) 1.4348(5) 2.5202(6) 0.5695(4) 0.4175(2) 0.3182(1)
    φ 7.3621(3) 151.9961(5) 634.1503(6) 16.9030(4) 6.2433(2) 3.1990(1)
    κ 7.4544(1) 153.2414(5) 298.9136(6) 33.2122(3) 40.8208(4) 13.8077(2)
    RMSE α 0.7103(3) 1.1978(5) 1.5875(6) 0.7546(4) 0.6461(2) 0.5641(1)
    φ 2.7133(3) 12.3287(5) 25.1823(6) 4.1113(4) 2.4987(2) 1.7886(1)
    κ 2.7303(1) 12.3791(5) 17.2891(6) 5.7630(3) 6.3891(4) 3.7159(2)
    ranks 31(2) 60(5) 72(6) 41(4) 32(3) 16(1)
    50 AVE α 0.8833(4) 0.7896(1) 0.8815(3) 0.9182(6) 0.8904(5) 0.8811(2)
    φ 1.6996(3) 1.4800(1) 1.8893(6) 1.7904(5) 1.7388(4) 1.6990(2)
    κ 1.2704(2) 1.1696(1) 1.4091(3) 1.4440(5) 1.4889(6) 1.4167(4)
    AVAB α 0.0833(4) 0.0104(1) 0.0815(3) 0.1182(6) 0.0904(5) 0.0811(2)
    φ 0.1996(3) 0.0200(1) 0.3893(6) 0.2904(5) 0.2388(4) 0.1990(2)
    κ 0.2704(2) 0.1696(1) 0.4091(3) 0.4440(5) 0.4889(6) 0.4167(4)
    MSE α 0.0794(2) 0.0681(1) 0.1771(6) 0.1309(4) 0.1464(5) 0.1199(3)
    φ 0.3262(2) 0.3085(1) 20.5402(6) 0.7886(3) 1.7903(5) 1.0740(4)
    κ 0.9907(1) 1.1406(2) 7.5480(6) 1.9706(3) 4.2142(5) 3.7125(4)
    RMSE α 0.2819(2) 0.2610(1) 0.4209(6) 0.3618(4) 0.3826(5) 0.3462(3)
    φ 0.5711(2) 0.5554(1) 4.5321(6) 0.8880(3) 1.3380(5) 1.0363(4)
    κ 0.9907(1) 1.0680(2) 2.7474(6) 1.4038(3) 2.0528(5) 1.9268(4)
    ranks 28(2) 14(1) 60(4.5) 62(6) 60(4.5) 38(3)
    100 AVE α 0.8324(5) 0.7737(1) 0.8214(2) 0.8476(6) 0.8292(4) 0.8239(3)
    φ 1.5765(5) 1.4469(1) 1.5593(3) 1.6181(6) 1.5676(4) 1.5566(2)
    κ 1.1040(3) 1.0233(1) 1.0901(2) 1.1638(6) 1.1483(5) 1.1139(4)
    AVAB α 0.0324(5) 0.0263(3) 0.0214(1) 0.0476(6) 0.0292(4) 0.0239(2)
    φ 0.0765(5) 0.0531(1) 0.0593(3) 0.1181(6) 0.0676(4) 0.0566(2)
    κ 0.1040(3) 0.0233(1) 0.0901(2) 0.1638(6) 0.1483(5) 0.1139(4)
    MSE α 0.0258(2) 0.0222(1) 0.0298(3) 0.0507(6) 0.0445(5) 0.0323(4)
    φ 0.0948(2) 0.0745(1) 0.1090(3) 0.2038(6) 0.1641(5) 0.1151(4)
    κ 0.2745(2) 0.0233(1) 0.2965(3) 0.5554(6) 0.4886(5) 0.3328(4)
    RMSE α 0.1607(2) 0.1491(1) 0.1727(3) 0.2251(6) 0.2109(5) 0.1797(4)
    φ 0.3079(2) 0.2729(1) 0.3301(3) 0.4514(6) 0.4051(5) 0.3392(4)
    κ 0.5240(2) 0.4679(1) 0.5445(3) 0.7453(6) 0.6990(5) 0.5769(4)
    ranks 38(3) 14(1) 31(2) 72(6) 56(5) 41(4)
    200 AVE α 0.8155(5) 0.7798(1) 0.8117(3) 0.8206(6) 0.8122(4) 0.8102(2)
    φ 1.5382(5) 1.4621(1) 1.5325(3) 1.5549(6) 1.5328(4) 1.5277(2)
    κ 1.0486(3) 0.9959(1) 1.0469(2) 1.0692(6) 1.0643(5) 1.0491(4)
    AVAB α 0.0155(4) 0.0202(5) 0.0117(2) 0.0206(6) 0.0122(3) 0.0102(1)
    φ 0.0382(5) 0.0359(4) 0.0325(2) 0.0549(6) 0.0328(3) 0.0277(1)
    κ 0.0486(3) 0.0041(1) 0.0469(2) 0.0692(6) 0.0643(5) 0.0491(4)
    MSE α 0.0110(2) 0.0109(1) 0.0141(3) 0.0204(6) 0.0193(5) 0.0144(4)
    φ 0.0380(2) 0.0359(1) 0.0460(3) 0.0648(6) 0.0597(5) 0.0466(4)
    κ 0.1041(2) 0.0949(1) 0.1258(3) 0.1858(6) 0.1794(5) 0.1280(4)
    RMSE α 0.1049(1.5) 0.1049(1.5) 0.1186(3) 0.1427(6) 0.1390(5) 0.1198(4)
    φ 0.1950(2) 0.1894(1) 0.2144(3) 0.2546(6) 0.2443(5) 0.2158(4)
    κ 0.3227(2) 0.3080(1) 0.3547(3) 0.4310(6) 0.4236(5) 0.3578(4)
    ranks 36.5(3) 19.5(1) 32(2) 72(6) 64(5) 38(4)

     | Show Table
    DownLoad: CSV
    Table 4.  Simulation results of the estimators for α=2.0,φ=1.0,κ=0.7.
    n Measures Pa. ML MPS ANDA CVM OLS WLS
    10 AVE α 2.8336(3) 2.5944(1) 3.1558(5) 3.3131(6) 2.8120(2) 2.9230(4)
    φ 2.0036(1) 8.0269(5) 19.3064(6) 5.4068(4) 4.1680(2) 4.4550(3)
    κ 1.4579(1) 3.2311(5) 3.6263(6) 3.0043(4) 2.8950(2) 2.9920(3)
    AVAB α 0.8336(3) 0.5944(1) 1.1558(5) 1.3131(6) 0.8124(2) 0.9231(4)
    φ 1.0036(1) 7.0269(5) 18.3064(6) 4.4068(4) 3.1675(2) 3.4554(3)
    κ 0.7579(1) 2.5311(5) 2.9263(6) 2.3043(4) 2.1952(2) 2.2918(3)
    MSE α 2.1274(1) 6.7714(5) 9.2849(6) 5.0302(4) 3.5630(2) 3.9500(3)
    φ 3.3320(1) 648.1073(5) 3383.9749(6) 204.2319(4) 177.4580(3) 152.0100(2)
    κ 2.8070(1) 49.8495(5) 60.4505(6) 28.6379(3) 27.8840(2) 33.0500(4)
    RMSE α 1.4586(1) 2.6022(5) 3.0471(6) 2.2428(4) 1.8870(2) 1.9870(3)
    φ 1.8254(1) 25.4580(5) 58.1719(6) 14.2910(4) 13.3210(3) 12.3290(2)
    κ 1.6754(1) 7.0404(5) 7.7750(6) 5.3514(3) 5.2800(2) 5.7490(4)
    ranks 16(1) 52(5) 70(6) 50(4) 26(2) 38(3)
    50 AVE α 2.2547(3) 1.8578(1) 2.1777(2) 2.5788(6) 2.4490(5) 2.3520(4)
    φ 1.2645(2) 1.0723(1) 1.5965(3) 2.3717(6) 2.1870(5) 2.0330(4)
    κ 0.9587(2) 0.7912(1) 1.0157(3) 1.4816(6) 1.4490(5) 1.2750(4)
    AVAB α 0.2547(3) 0.1422(1) 0.1777(2) 0.5788(6) 0.4486(5) 0.3518(4)
    φ 0.2645(2) 0.0723(1) 0.5965(3) 1.3717(6) 1.1873(5) 1.0332(4)
    κ 0.2587(2) 0.0912(1) 0.3157(3) 0.7816(6) 0.7489(5) 0.5751(4)
    MSE α 0.4982(1) 0.6380(2) 0.8915(3) 1.6694(6) 1.5060(5) 1.1900(4)
    φ 0.4962(1) 1.4375(2) 32.6387(5) 30.5760(4) 28.1190(3) 49.6000(6)
    κ 0.5600(1) 0.7722(2) 1.7465(3) 3.9829(5) 4.1170(6) 3.2910(4)
    RMSE α 0.7058(1) 0.7988(2) 0.9442(3) 1.2921(6) 1.2270(5) 1.0910(4)
    φ 0.7044(1) 1.1990(2) 5.7130(5) 5.5296(4) 5.3030(3) 7.0430(6)
    κ 0.7483(1) 0.8788(2) 1.3215(3) 1.9957(5) 2.0290(6) 1.8140(4)
    ranks 20(2) 18(1) 28(3) 64(6) 58(5) 52(4)
    100 AVE α 2.1444(3) 1.8589(1) 2.0854(2) 2.3221(6) 2.2590(5) 2.1566(4)
    φ 1.1384(3) 0.9579(1) 1.1305(2) 1.6180(5) 1.5510(4) 1.6937(6)
    κ 0.1510(1) 0.6947(2) 0.8403(3) 1.1163(6) 1.0970(5) 0.9608(4)
    AVAB α 0.1444(3) 0.1411(2) 0.0854(1) 0.3221(6) 0.2591(5) 0.1566(4)
    φ 0.1384(3) 0.0421(1) 0.1305(2) 0.6180(5) 0.5507(4) 0.6937(6)
    κ 0.1510(3) 0.0053(1) 0.1403(2) 0.4163(6) 0.3975(5) 0.2608(4)
    MSE α 0.2375(1) 0.2656(2) 0.3276(3) 0.8324(6) 0.7795(5) 0.5226(4)
    φ 0.1731(2) 0.1389(1) 0.3441(3) 9.3730(5) 9.3187(4) 168.1577(6)
    κ 0.2335(2) 0.2183(1) 0.3689(3) 1.5424(5) 1.5302(4) 1.6901(6)
    RMSE α 0.4873(1) 0.5154(2) 0.5724(3) 0.9123(6) 0.8829(5) 0.7229(4)
    φ 0.4161(2) 0.3728(1) 0.5866(3) 3.0615(5) 3.0526(4) 12.9676(6)
    κ 0.4832(2) 0.4672(1) 0.6074(3) 1.2420(5) 1.2370(4) 1.3001(6)
    ranks 26(2) 16(1) 30(3) 66(6) 54(4) 60(5)
    200 AVE α 2.0655(4) 1.8856(1) 2.0368(2) 2.1254(6) 2.0920(5) 2.0566(3)
    φ 1.0615(3) 0.9484(1) 1.0528(2) 1.1499(6) 1.1250(5) 1.0714(4)
    κ 0.7647(3) 0.6606(1) 0.7589(2) 0.8454(6) 0.8340(5) 0.7822(4)
    AVAB α 0.0655(3) 0.1144(5) 0.0368(1) 0.1254(6) 0.0922(4) 0.0566(2)
    φ 0.0615(3) 0.0516(1) 0.0528(2) 0.1499(6) 0.1246(5) 0.0714(4)
    κ 0.0647(3) 0.0394(1) 0.0589(2) 0.1454(6) 0.1340(5) 0.0822(4)
    MSE α 0.1055(1) 0.1232(2) 0.1489(3) 0.2735(6) 0.2627(5) 0.1648(4)
    φ 0.0582(2) 0.0498(1) 0.0816(3) 0.2593(6) 0.2387(5) 0.1083(4)
    κ 0.0765(2) 0.0688(1) 0.1080(3) 0.2400(6) 0.2333(5) 0.1343(4)
    RMSE α 0.3248(1) 0.3509(2) 0.3858(3) 0.5229(6) 0.5126(5) 0.4059(4)
    φ 0.2412(2) 0.2231(1) 0.2857(3) 0.5092(6) 0.4886(5) 0.3290(4)
    κ 0.2766(2) 0.2622(1) 0.3286(3) 0.4899(6) 0.4830(5) 0.3665(4)
    ranks 25(2) 18(1) 29(3) 72(6) 59(5) 45(4)

     | Show Table
    DownLoad: CSV
    Table 5.  Simulation results of the estimators for α=1.5,φ=2.0,κ=1.0.
    n Measures Pa. ML MPS ANDA CVM OLS WLS
    10 AVE α 1.9113(4) 1.8080(3) 2.1893(6) 2.0102(5) 1.6790(1) 1.7030(2)
    φ 3.4196(3) 6.6547(5) 13.4896(6) 4.6929(4) 3.1980(1.5) 3.1980(1.5)
    κ 1.6351(1) 2.9747(5) 3.4465(6) 2.4926(4) 2.2030(3) 2.1720(2)
    AVAB α 0.4113(4) 0.3080(3) 0.6893(6) 0.5102(5) 0.1787(1) 0.2032(2)
    φ 1.4196(3) 4.6547(5) 11.4896(6) 2.6929(4) 1.1978(1) 1.1980(2)
    κ 0.6351(1) 1.9747(5) 2.4465(6) 1.4926(4) 1.2034(3) 1.1716(2)
    MSE α 0.8424(1) 2.6556(5) 4.2739(6) 1.6544(4) 1.0310(2) 1.0990(3)
    φ 6.2459(1) 242.8573(5) 1099.2808(6) 63.1670(4) 34.3340(2) 36.4950(3)
    κ 2.8238(1) 37.3047(5) 47.8131(6) 17.0243(4) 12.9340(3) 10.7970(2)
    RMSE α 0.9178(1) 1.6296(5) 2.0673(6) 1.2862(4) 1.0150(3) 1.0480(2)
    φ 2.4992(1) 15.5839(5) 33.1554(6) 7.9478(4) 5.8600(2) 6.0410(3)
    κ 1.6804(1) 6.1078(5) 6.9147(6) 4.1261(4) 3.5960(3) 3.2860(2)
    ranks 22(1) 56(5) 72(6) 50(4) 25.5(2) 26.5(3)
    50 AVE α 1.6952(2) 1.4342(1) 1.8042(5) 1.8992(6) 1.7690(3) 1.8000(4)
    φ 2.4984(2) 2.1884(1) 4.9115(6) 3.7973(4) 3.1380(3) 3.3830(5)
    κ 1.3226(2) 1.1667(1) 1.9301(6) 1.8651(5) 1.7060(3) 1.7400(4)
    AVAB α 0.1952(2) 0.0658(1) 0.3042(5) 0.3992(6) 0.2692(3) 0.3001(4)
    φ 0.4984(2) 0.1884(1) 2.9115(6) 1.7973(5) 1.1377(3) 1.3826(4)
    κ 0.3226(2) 0.1667(1) 0.9301(6) 0.8651(5) 0.7060(3) 0.7402(4)
    MSE α 0.3337(1) 0.5070(2) 1.5355(6) 1.0502(5) 0.7593(3) 0.8669(4)
    φ 1.5498(1) 4.0449(2) 155.5950(6) 60.8746(5) 17.0518(3) 29.0772(4)
    κ 0.9269(1) 1.7944(2) 11.4674(6) 6.4988(5) 4.3962(3) 5.1792(4)
    RMSE α 0.5776(1) 0.7120(2) 1.2391(6) 1.0248(5) 0.8714(3) 0.9311(4)
    φ 1.2449(1) 2.0112(2) 12.4738(6) 7.8022(5) 4.1294(4) 5.3923(3)
    κ 0.9627(1) 1.3396(2) 3.3864(6) 2.5493(5) 2.0967(3) 2.2758(4)
    ranks 18(1.5) 18(1.5) 70(6) 61(5) 37(3) 48(4)
    100 AVE α 1.6264(2) 1.3950(1) 1.6954(3) 1.7879(6) 1.7060(4) 1.7390(5)
    φ 2.3116(2) 1.9354(1) 3.4363(6) 3.2511(5) 2.8230(3) 3.0510(4)
    κ 1.2179(2) 0.9918(1) 1.5569(4) 1.6311(6) 1.5010(3) 1.5570(5)
    AVAB α 0.1264(2) 0.1050(1) 0.1954(3) 0.2879(6) 0.2062(4) 0.2393(5)
    φ 0.3116(2) 0.0646(1) 1.4363(6) 1.2511(5) 0.8234(3) 1.0510(4)
    κ 0.2179(2) 0.0082(1) 0.5569(5) 0.6311(6) 0.5009(3) 0.5567(4)
    MSE α 0.2014(1) 0.4738(2) 0.8545(6) 0.7787(5) 0.5939(3) 0.6521(4)
    φ 0.8984(1) 0.9734(2) 56.0050(6) 25.6484(5) 9.4794(3) 21.8101(4)
    κ 0.5434(1) 0.6022(2) 5.2431(6) 3.9995(5) 2.7885(3) 3.5394(4)
    RMSE α 0.4488(1) 0.4738(2) 0.9244(6) 0.8824(5) 0.7706(3) 0.8075(4)
    φ 0.9478(1) 0.9734(2) 7.4836(6) 5.0644(5) 3.0789(3) 4.6701(4)
    κ 0.7372(1) 0.7760(2) 2.2898(6) 1.9999(5) 1.6699(3) 1.8813(4)
    ranks 18(1.5) 18(1.5) 63(5) 64(6) 38(3) 51(4)
    200 AVE α 1.5676(2) 1.4003(1) 1.5936(3) 1.6640(6) 1.6350(5) 1.6210(4)
    φ 2.1555(2) 1.8840(1) 2.4471(3) 2.5978(6) 2.5470(5) 2.4900(4)
    κ 1.1155(2) 0.9290(1) 1.2360(3) 1.3342(6) 1.2990(5) 1.2790(4)
    AVAB α 0.0676(1) 0.0997(3) 0.0936(2) 0.1640(6) 0.1254(5) 0.1210(4)
    φ 0.1555(2) 0.1160(1) 0.4471(3) 0.5978(6) 0.5470(5) 0.4895(4)
    κ 0.1155(2) 0.0710(1) 0.2360(3) 0.3342(6) 0.2988(5) 0.2790(4)
    MSE α 0.0927(1) 0.0972(2) 0.3192(3) 0.4075(6) 0.3785(5) 0.3419(4)
    φ 0.3186(2) 0.2844(1) 8.2814(5) 5.1207(3) 10.0634(6) 5.8513(4)
    κ 0.2216(2) 0.1781(1) 1.5037(5) 1.4824(4) 1.5757(6) 1.4326(3)
    RMSE α 0.3045(2) 0.3117(1) 0.5650(3) 0.6384(6) 0.6152(5) 0.5847(4)
    φ 0.5645(2) 0.5333(1) 2.8777(5) 2.2629(3) 3.1723(6) 2.4190(4)
    κ 0.4708(2) 0.4221(1) 1.2263(5) 1.2175(4) 1.2553(6) 1.1969(3)
    ranks 22(2) 15(1) 43(3) 62(5) 64(6) 46(4)

     | Show Table
    DownLoad: CSV

    The results in Tables 25 show that the mean square errors (MSEs) decrease for all parameter combinations as the sample size (n) increases, that is, all the estimators are quite consistent since their RMSE tend to zero for large sample size (n). Hence, the estimates of the EHLST parameters provide credible MSEs and low AVABs for all the parameter combinations using the six estimation methods for large sample size (n). Additionally, the average absolute biases (AVABs) approach zero as the sample size (n) increase, proving that all the estimators are quite asymptotically unbiased.

    The model evaluation in relation to some existing distributions from the same family of distributions using the inflation rates (INF) monthly dataset is performed. The few existing distributions from the same family are the exponentiated half logistic Lomax (EHLL), exponentiated half logistic Fréchet (EHLF) and exponentiated half logistic log-logistic (EHLLL). The model parameters are estimated using the maximum likelihood and maximum product of spacing estimation procedures. Using the R-software package "AdequacyModel", the following performance measures: negative log-likelihood, Akaike Information Criterion (AIC), Kolmogorov-Smirnov (K-S) statistic and its p-value including the ML estimates (MLEs) and standard errors (SEs) of the parameters are provided in Table 6 while the MPS estimates (MPSEs), log-likelihood and other measures are provided in Table 7. The results in Tables 6 and 7 show that the new model is quite competitive with the existing four-parameter distributions in fitting the inflation rates dataset.

    Table 6.  MLE, Standard error (SE) and performance measures.
    Model Par MLE (SE) -LL AIC K-S p-value
    EHLST ˆα 3.2407(0.5773) 588.256 1182.513 0.070 0.263
    ˆφ 67.8208(27.2116)
    ˆκ 217.1097(81.7760)
    EHLL ˆα 4.2120(2.2915) 590.130 1188.260 0.077 0.165
    ˆβ 9.9874(1.6320)
    ˆa 4.2120(2.2951)
    ˆb 0.0186(0.0047)
    EHLF ˆα 1.2744(0.4305) 585.933 1179.867 0.069 0.267
    ˆλ 26.4325(15.3670)
    ˆa 0.7669(0.1661)
    ˆb 48.0005(23.9203)
    EHLLL ˆα 0.5109(0.4213) 583.900 1175.800 0.052 0.627
    ˆλ 0.7105(0.6145)
    ˆa 11.3008(0.9864)
    ˆb 7.5646(5.2527)

     | Show Table
    DownLoad: CSV
    Table 7.  MPSE, Standard error (SE) and performance measures.
    Model Par MPS (SE) LL AIC K-S p-value
    EHLST ˆα 3.1763(0.0286) −590.015 1186.000 0.069 0.269
    ˆφ 64.0223(1.7981)
    ˆκ 218.3550(4.2330)
    EHLL ˆα 3.3253(0.0749) −592.43 1192.900 0.089 0.072
    ˆβ 9.9379(0.1238)
    ˆa 3.3075(0.0745)
    ˆb 0.0318(0.0007)
    EHLF ˆα 1.4956(0.3689) −587.234 1182.500 0.068 0.300
    ˆλ 30.5011(13.4873)
    ˆa 0.7660(0.0970)
    ˆb 50.1275(26.6395)
    EHLLL ˆα 0.5220(0.2593) −585.169 1178.300 0.059 0.460
    ˆλ 0.8105(0.58140)
    ˆa 11.5673(0.7104)
    ˆb 6.9976 (7.3261)

     | Show Table
    DownLoad: CSV

    The standardized form of the EHLST density function is derived using the transformed random variable z=yμh2, where E(z)=0 and var(z)=1. Thus, the standardized EHLST density function with z=εh2 and zε=1h2 is given by

    f(ε;α,φ,κ,h2t)=(1{h2t}12)2αφκ[1(12(1+εth2tκ+(εth2t)2))]α1{1[1(12(1+εth2tκ+(εth2t)2))]α}φ12(κ+(εth2t)2)32{1+[1(12(1+εth2tκ+(εth2t)2))]α}φ+1 (37)

    The log-likelihood function (LL) is given by

    LL(ξ)=n log α+n log φ+n log κ32nt=1 log (κ+(εth2t)2)+(α1)nt=1 log {1(12(1+εth2tκ+(εth2t)2))}
    +(φ1)nt=1 log {1[1(12(1+εth2tκ+(εth2t)2))]ϑ} (38)
    (φ+1)nt=1 log {1+[1(12(1+εth2tκ+(εth2t)2))]ϑ}12nt=1 log (h2t)

    where ξ=(α,φ,κ,h2t) and h2t denote the GARCH-type volatility models with unknown vector parameters.

    The applicability of the standardized EHLST density function as the conditional innovation density in GARCH-type volatility models are demonstrated using the monthly all-items Nigeria inflation rates (year on change) dataset from January 2003 to August 2021. The dataset can be sourced from the Central Bank of Nigeria website at www.cbn.gov.ng.

    In financial time series, volatility modeling is a dynamic direction for both advisors and financial expert. Engle (1982) and Bollerslev (1986) introduced the autoregressive conditional heteroscedasticity (ARCH) and the generalized ARCH (GARCH) models for volatility modeling. The log-return is denoted as rt, and the GARCH (1, 1) model is expressed as follows:

    rt=μ+εt,εt=zth2t,zti.i.d.
    h2t=ω+η1ε2t1+η2h2t1, (39)

    where ω>0, η1>0, η2>0, zt is the conditional innovation distribution with E(zt)=0 and var(zt)=1. The conditional variance and mean are denoted as h2t and μ, respectively. In this research work, the GARCH model (Bollerslev, 1986) and GJRGARCH model (Glosten et al., 1993) are considered. The GARCH model is used as the benchmark in this research. The GJRGARCH (1, 1) model conditional variance functional form is expressed as:

    h2t=ω+η1ε2t1+η3It1ε2t1+η2h2t1 (40)

    where η3 is the leverage effect, and It1 is an indicator function expressed as:

    It1={1,ifεt1<0,0,ifεt10,

    The GJRGARCH model checks for leverage effects and when the parameter η3 is positive it implies that negative shocks have a larger effect on volatility than positive shocks. The common conditional innovations are provided in Appendix A.

    The accuracy of both models is appraised using the modified information criteria of Brooks and Burke (2003). The modified Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), criteria are given by

    AIC=2cT2LLTBIC=c log e(T)T2LLT (41)

    where T is the number of observations, c is number of estimated parameters, and LL denotes the log-likelihood value. The model with the least AIC and BIC values is regarded as the finest model in terms of the specified conditional innovation density.

    The predictive ability of the GARCH-type models is evaluated using the mean square error (MSE), root mean square root (RMSE), and mean absolute error (MAE). The MSE, RMSE, and MAE for the volatility forecasts are given by

    MSE=1TTt=1(ˆhtht)2,RMSE=1TTt=1(ˆhtht)2,MAE=1TTt=1|ˆhtht|

    where ht and ˆht represent the realized volatility and predicted volatility, and T is the sample size. The model with the least MSE, RMSE, and MAE values is regarded as the most suitable for predicting the volatility of the financial dataset.

    To assess the ability of the GARCH-type volatility models using the EHLST density in relation to the commonly existing densities in modeling and predicting volatility, the all-items Nigeria inflation rates is utilized. The utilized data consist of 224 inflation rates from January 2003 to August 2021. Table 8 presents the summary statistics of 223 monthly Inflation rate (INF) log-returns. Negative skewness and excess kurtosis are quite evident, leading to large Jarque-Bera (JB) test statistic value (p < 0.001) indicating that the monthly log-returns are non-normally distributed. More so, the Lagrange-multiplier (LM) test specifies the prevalence of ARCH effects in the conditional variance. The graphical plots of the log-returns, squared log-returns and absolute log-returns including their autocorrelation functions are depicted in Figure 4.

    Table 8.  Summary statistics for the inflation rate (INF) log-returns.
    INF
    Number of observations 223
    Mean 0.2121
    Median 0.4906
    Minimum −104.1454
    Maximum 53.2217
    Standard Deviation 14.3681
    Skewness −1.3377
    Kurtosis 13.1953
    Jarque-Bera 1721.3 (p < 0.001)
    ARCH (2) 15.406 (p < 0.001)

     | Show Table
    DownLoad: CSV
    Figure 4.  Inflation (INF) monthly log-returns, squared and absolute log-returns and sample autocorrelations.

    The plots in Figure 4(b, c) show evidence of volatility clustering; that is low volatility values followed by low volatility values and high volatility values followed by high volatility values. More so, the monthly log-returns show no clear evidence of serial correlation, but the squared and absolute log-returns are positively autocorrelated as depicted in Figures 4(e, f).

    The GARCH-type volatility models are estimated under the normal (NORM), Student-t (STD), generalized error (GED) and their skewed versions, generalized hyperbolic (GHYP), Johnson (SU) reparametrized (JSU), generalized hyperbolic skew-Student (GHST), Normal inverse Gaussian (NIG) and EHLST innovation densities. To acquire the parameter estimates of both models under NORM, STD, GED, SNORM, SSTD, SGED, GHYP, JSU, GHST and NIG densities, the rugarch package in R-software is used. The Optim function in R-software is utilized for the estimation of the model's parameter with the EHLST innovation density. Table 9 shows the GARCH (1, 1) and GJRGARCH (1, 1) models parameter estimates under eleven innovation densities.

    Table 9.  Parameter estimates of the models for the INF log-returns under eleven innovation densities.
    GARCH (1, 1)
    Pa. NORM STD GED SNORM SSTD SGED GHYP JSU GHST NIG EHLST
    μ 0.6403 0.5656 0.6504'***' 0.5158 0.3304 0.3277'***' 0.4667 0.3513 0.1605 0.3589 0.1824
    ω 0.0900 0.7026 1.0661 0.1468'*' 0.9007 1.4755'***' 0.8444 1.0241'.' 0.9637'.' 1.2821'.' 0.1891
    η1 0.0602'*' 0.2777'*' 0.2536'.' 0.0586'***' 0.3115'*' 0.2957'***' 0.2904'*' 0.3100'*' 0.2876'*' 0.3088'*' 0.1112
    η2 0.9292'***' 0.7213'***' 0.7454'***' 0.9289'***' 0.6875'***' 0.7033'***' 0.7086'***' 0.6889'***' 0.7114'***' 0.6902'***' 0.6069'***'
    η3 - - - - - - - - - - -
    α - - - 0.9423'***' - - 0.2500 - 8.1967'***' 0.6823'.' 1.7778'***'
    ξ - - - - 0.8956'***' - - - - - -
    ν - 4.1458'***' 0.9782'***' - 4.1743'***' 0.8941'***' - - - - -
    η - - - - - 0.9696'***' - - - - -
    β - - - - - - -0.3260 - -1.1452'' -0.1358 -
    λ - - - - - - -1.9370'*' - - - -
    υ - - - - - - - -0.1940 - - -
    ϑ - - - - - - - 1.3328'***' - - -
    φ - - - - - - - - - - 1.2070'***'
    κ - - - - - - - - - - 6.6593
    GJRGARCH (1, 1)
    Pa. NORM STD GED SNORM SSTD SGED GHYP JSU GHST NIG EHLST
    μ 0.8364'.' 0.5669'.' 0.6504'***' 0.7550'' 0.3233 0.3215'***' 0.4650 0.3452 -0.0555 0.3574 0.2001
    ω 0.0000 0.6843 1.1503 0.0000 0.8795 1.4925'***' 0.8253 1.0071 1.2293'.' 1.2783'' 0.2750
    η1 0.0864'*' 0.2833'*' 0.2414'.' 0.0804'*' 0.3254'*' 0.2893'***' 0.2978'*' 0.3194'*' 0.2800'*' 0.3105'.' 0.1692
    η2 0.9397'***' 0.7255'***' 0.7322'***' 0.9402'***' 0.6912'***' 0.7004'***' 0.7125'***' 0.6918'***' 0.7322'***' 0.6908'***' 0.6113'***'
    η3 −0.0643 −0.0195 0.0508 −0.0562 −0.0371 0.0200 −0.0232 −0.0258 −0.0292 −0.0049 −0.0004
    α - - - 0.9651'***' - - 0.2500 - 8.1943'***' 0.6824'*' 1.7571'***'
    ξ - - - - 0.8923'***' - - - - - -
    ν - 4.1432'***' 0.9821'***' - 4.1776'***' 0.8923'***' - - - - -
    η - - - - - 0.9694'***' - - - - -
    β - - - - - - −0.3324 - -1.8837'.' −0.1365 -
    λ - - - - - - −1.9378'**' - - - -
    υ - - - - - - - -0.1988 - - -
    ϑ - - - - - - - 1.3332'***' - - -
    φ - - - - - - - - - - 1.1858'***'
    κ - - - - - - - - - - 4.2176
    Significance level 0'***', 0.01'*', 0.05'*', 0.1'', 1

     | Show Table
    DownLoad: CSV

    The parameter estimates of the conditional variance for the GARCH-EHLST are statistically significant at various standard levels including the leverage effect parameter η3. Hence, the impact of the shocks is asymmetric in nature which implies that the impact of positive shocks is higher than negative shocks of the same magnitude on volatility. Tables 10 provides vital statistics for model selection, which shows that the GARCH-EHLST has the highest log-likelihood, and least AIC and BIC values relative to other models. Hence, the GARCH-EHLST model is selected as the best model for modeling the volatility of the financial data series.

    Table 10.  Comparison of the models for selection.
    Model Log-Likelihood AIC BIC
    GARCH-NORM −808.135 7.2837 7.3448
    GARCH-STD −772.170 6.9701 7.0665
    GARCH-GED −776.818 7.0118 7.0882
    GARCH-SNORM −807.730 7.2891 7.3655
    GARCH-SSTD −771.494 6.9730 7.0647
    GARCH-SGED −775.870 7.0123 7.1040
    GARCH-GHYP −772.006 6.9866 7.0936
    GARCH-JSU −772.083 6.9783 7.0700
    GARCH-GHST −778.603 7.0368 7.1285
    GARCH- NIG −773.341 6.9896 7.0813
    GARCH- EHLST −768.492 6.9551 7.0620
    GJRGARCH-NORM −807.280 7.2850 7.3614
    GJRGARCH-STD −772.159 6.9790 7.0900
    GJRGARCH-GED −776.764 7.0203 7.1120
    GJRGARCH-SNORM −807.129 7.2926 7.3843
    GJRGARCH-SSTD −771.455 6.9817 7.0923
    GJRGARCH-SGED −775.843 7.0210 7.1280
    GJRGARCH-GHYP −771.991 6.9954 7.1177
    GJRGARCH-JSU −772.066 6.9871 7.0941
    GJRGARCH-GHST −781.977 7.0760 7.1830
    GJRGARCH- NIG −773.340 6.9986 7.1055
    GJRGARCH- EHLST −768.846 6.9672 7.0895

     | Show Table
    DownLoad: CSV

    Table 11 provides the diagnostic analysis tests for the GARCH-type volatility models which indicates that the GARCH-EHLST model did well in describing the conditional variance dynamics as well as other models. Basically, the Ljung-Box statistic (p-value) indicates no sign of serial-correlation in the squared standardized residuals, and the ARCH-LM statistic indicates no extra ARCH processes in the standardized residuals which implies that the conditional variance equations are well-defined.

    Table 11.  Estimated models diagnostic tests.
    Model Ljung-Box
    Statistic
    p-value ARCH-LM Statistic p-value
    GARCH-NORM 3.331 0.9725 2.932 0.9830
    GARCH-STD 1.542 0.9988 1.715 0.9981
    GARCH-GED 1.797 0.9977 1.991 0.9964
    GARCH-SNORM 3.226 0.9756 2.848 0.9848
    GARCH-SSTD 1.735 0.9980 1.982 0.9965
    GARCH-SGED 1.891 0.9971 2.141 0.9951
    GARCH-GHYP 1.679 0.9983 1.893 0.9971
    GARCH-JSU 1.810 0.9976 2.073 0.9958
    GARCH-GHST 1.800 0.9977 2.033 0.9961
    GARCH- NIG 1.898 0.9965 2.176 0.9948
    GARCH- EHLST 1.511 0.9989 1.774 0.9978
    GJRGARCH-NORM 4.328 0.9313 3.617 0.9630
    GJRGARCH-STD 1.545 0.9988 1.714 0.9981
    GJRGARCH-GED 1.832 0.9975 2.054 0.9959
    GJRGARCH-SNORM 4.292 0.9332 3.610 0.9632
    GJRGARCH-SSTD 3.429 0.9991 1.997 0.9964
    GJRGARCH-SGED 1.903 0.9970 2.165 0.9949
    GJRGARCH-GHYP 1.684 0.9982 1.895 0.9971
    GJRGARCH-JSU 1.815 0.9976 2.075 0.9957
    GJRGARCH-GHST 1.849 0.9974 2.038 0.9960
    GJRGARCH- NIG 1.897 0.9971 2.172 0.9948
    GJRGARCH- EHLST 1.472 0.9990 1.711 0.9981

     | Show Table
    DownLoad: CSV

    The in-sample volatility predictive measures for the models are provided in Table 12, and Table 13 provides the partial and total ranking of the innovation densities based on the three predictive performance measures. The results show that the GARCH (1, 1) and GJRGARCH (1, 1) models has the least MSE, RMSE and MAE values under the EHLST innovation density. Overall, the results indicate that the GARCH-EHLST model is statistically effective in forecasting the volatility of the inflation log-returns relative to other models based on the MSE and RMSE results while the GJRGARCH-EHLST model is statistically effective based on the MAE result. Furthermore, the EHLST innovation density has the least total rank value based on the performance measures among the eleven innovation densities considered in this study.

    Table 12.  Forecasts evaluation of the estimated models.
    Model MSE RMSE MAE
    GARCH-NORM 112.720 10.617 7.011
    GARCH-STD 122.686 11.076 6.671
    GARCH-GED 122.269 11.057 6.776
    GARCH-SNORM 111.427 10.556 6.952
    GARCH-SSTD 123.930 11.132 6.650
    GARCH-SGED 123.902 11.131 6.764
    GARCH-GHYP 123.204 11.100 6.673
    GARCH-JSU 123.939 11.133 6.670
    GARCH-GHST 123.355 11.106 6.693
    GARCH- NIG 124.100 11.140 6.712
    GARCH- EHLST 109.109 10.445 5.396
    GJRGARCH-NORM 156.588 12.513 9.119
    GJRGARCH-STD 124.678 11.166 6.757
    GJRGARCH-GED 117.606 10.845 6.555
    GJRGARCH-SNORM 149.356 12.221 8.828
    GJRGARCH-SSTD 127.418 11.288 6.791
    GJRGARCH-SGED 122.197 11.054 6.686
    GJRGARCH-GHYP 125.462 11.201 6.768
    GJRGARCH-JSU 126.321 11.239 6.768
    GJRGARCH-GHST 126.353 11.241 6.911
    GJRGARCH- NIG 124.527 11.159 6.730
    GJRGARCH- EHLST 104.874 10.241 5.446

     | Show Table
    DownLoad: CSV
    Table 13.  Forecasts Evaluation–Innovation densities comparison.
    GARCH (1, 1)
    NORM STD GED SNORM SSTD SGED GHYP JSU GHST NIG EHLST
    MSE 3 5 4 2 9 8 6 10 7 11 1
    RMSE 3 5 4 2 9 8 6 10 7 11 1
    MAE 11 4 9 10 2 8 5 3 6 7 1
    TOTAL 17 14 17 14 20 24 17 23 20 29 3
    GJRGARCH (1, 1)
    NORM STD GED SNORM SSTD SGED GHYP JSU GHST NIG EHLST
    MSE 11 5 2 10 9 3 6 7 8 4 1
    RMSE 11 5 2 10 9 3 6 7 8 4 1
    MAE 11 5 2 10 8 3 6.5 6.5 9 4 1
    TOTAL 33 15 6 30 26 9 18.5 20.5 25 12 3

     | Show Table
    DownLoad: CSV

    Overall, the comparison between the volatility models in this study basically supports the use of the symmetric GARCH-EHLST model for estimating and forecasting the volatility of the inflation rates. More so, the proposed EHLST density seems generally the best innovation density for both the GARCH (1, 1) and GJRGARCH (1, 1) models.

    In this research, the estimation of the exponentiated half logistic skew-t (EHLST) distribution parameters using six classical estimation procedures, namely the maximum likelihood, maximum product of spacing, least squares, Anderson-Darling, weighted least squares and Cramer-von mises is considered. Given that the theoretical comparison between these classical estimation procedures is not quite practicable, an extensive Monte Carlo simulation study is performed to compare the estimators in terms of average absolute biases, mean square error of each estimate and average parameter estimates. The results show that the criterions performance ordering from finest to poorest, using the overall ranks is MLE, MPS, WLS, OLS, ANDA, and CVM for all the parameter combinations. The MLE outperform all the other criterions with an overall rank of 32. Consequently, the performance of the EHLST model in fitting the inflation rates, showed that the EHLST model is quite competitive against the existing four-parameter models from the same family of distributions. Furthermore, volatility modeling of the inflation log-returns using the GARCH (1, 1) and GJRGARCH (1, 1) models with EHLST innovation density relative to the normal, student-t, generalized error and their skewed versions, generalized hyperbolic, Johnson (SU) reparametrized, generalized hyperbolic skew-Student, Normal inverse Gaussian innovation densities in terms of predictive performance using three forecast performance measures was carried-out. The research findings confirm that GARCH (1, 1) and GJRGARCH (1, 1) models with EHLST innovation density is optimally the best model based on model selection criteria and forecasts performance measures among other models. Overall, the results validate the superiority of the GARCH (1, 1) and GJRGARCH (1, 1) models with EHLST innovation density in both in-and-out samples performance over other models for inflation rates volatility modeling.

    This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sector.

    All authors declare no conflicts of interest in this paper.



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