A Two-Parameter Model: Properties and Estimation under Ranked Sampling

: This study introduces a ﬂexible model with two parameters by combining the type II half-logistic-G family with the inverted Topp–Leone distribution. The proposed model is referred to as the half logistic inverted Topp–Leone (HLITL) distribution. The associated probability density function can be considered a mixture of the inverted Topp–Leone distributions. The proposed model can be deemed an acceptable model for ﬁtting the right-skewed, reversed J-shaped, and unimodal data. The statistical properties, including the moments, Bonferroni and Lorenz curves, R é nyi entropy, and quantile function, are derived. Additionally, the plots of the skewness and kurtosis measures are plotted based on the quantiles. The parameter estimators are implemented using the maximum likelihood method based on two sampling schemes: the simple random sample method and the ranked set sampling method. The proposed method is evaluated by using simulations. The results show that the maximum likelihood estimates of the parameters under ranked set sampling are more accurate than those under simple random sampling. Generally, there is good agreement between the theoretical and empirical results. Two real datasets are used to compare the HLITL model with the following models: alpha power exponential, alpha power Lindley, odd Fr é chet inverse exponential, and odd Fr é chet inverse Rayleigh models. The comparison results show that the HLITL model represents a better alternative lifetime distribution than the other competitive distributions. shown empirically that the maximum likelihood estimates taken from the observed RSS were more efﬁcient than the corresponding SRS. The merit of the HLITL distribution was evaluated experimentally using two real datasets. The results show that it performed better than the other ﬁtted distributions. Based on the obtained results, the proposed distribution can be useful for modeling data in various ﬁelds.

Significant properties of the TL distribution were discussed in [1]. The TL distribution was analyzed in many studies [2][3][4]. The inverted (or inverse) distributions are formally regarded as inverse transforms of random variables. The inverted distributions have different densities and hazard rate shapes from the associated noninverted distributions. The usefulness and applicability of inverted distributions have been used in various fields, such as survey sampling, biological sciences, life testing problems, and engineering sciences. In the literature, various inverted distributions and their applications in many areas have been studied [5][6][7][8][9][10].
In [11], the inverted Topp-Leone (ITL) was proposed using the transformation X = 1 Y − 1, where Y denotes a random variable with the CDF and PDF that are, respectively, defined as g(x; θ) = 2θx(1 + 2x) θ−1 where θ is the shape parameter and θ > 0. In [11], several properties and the estimation of the parameter of the ITL distribution were discussed based on censored samples. Recently, serious efforts have been put into developing new families of continuous probability distributions that extend the classical ones. These extended distributions have more additional parameters as well as enhanced flexibility in modeling a variety of data (see [12][13][14][15][16][17]). In [18], a highly flexible extended family, called Marshall-Olkin-G (MO-G), was proposed, and its PDF is expressed as where δ = 1 − δ and G(x) is the survival function (SF) of a baseline distribution with CDF G(x), and PDF g(x). Several new families that are related to the MO-G family have been considered in recent studies. This study considers the univariate family generated by a half-logistic random variable, as presented in [19]. The CDF of the HL-G family approaches that of the MO-G family. It is specified as follows: where λ is the shape parameter. For λ = 1 in Equation (6) and δ = 0.5 in Equation (5), the HL-G and MO-G families coincide. The CDF given by Equation (6) extends several flexible distributions as specific cases in their skewness and kurtosis, as presented in [20][21][22][23][24].
In [25], the ranked set sampling (RSS) scheme was provided as another sampling technique for simple random sampling (SRS) in situations where the measurement of the variable of interest is costly or difficult to conduct. The RSS protocol is more suitable than the SRS protocol. To obtain an RSS sample, the following steps are performed: • Draw m random samples with a size n from the desired population such that m = n; • Without taking any measurements, rank the samples in each dataset according to the criterion determined by the experimenter; • Choose a sample for true judgment by involving the smallest ordered unit in the first set and the second-smallest ordered unit in the second set. The operation is continued in this manner until the largest-ranked unit is chosen from the final set; • To collect a sample of a size of mk units, the above operations of the "cycle" may be repeated k times. For more studies about RSS, we may refer to [26][27][28][29][30].
This paper proposes a more flexible model for the ITL distribution by inducing one extra shape parameter for enhancing the goodness-of-fit to the real data. The main motivation for introducing the half-logistic inverted Topp-Leone (HLITL) is as follows: (1) obtain a more flexible PDF for right-skewed, unimodal, and reversed J-shaped data; (2) provide decreasing, increasing, and upside-down hazard rate shapes; (3) estimate the Mathematics 2021, 9, 1214 3 of 16 HLITL distribution parameters on the basis of SRS and RSS; and (4) provide significant improvement in the data modeling.
The rest of the article is organized as follows. Section 2 defines the PDF, CDF, and hazard rate function (HRF) of the HLITL distribution. Section 3 derives the structural properties of the HLITL distribution. Section 4 presents the maximum likelihood (ML) parameter estimators as well as the simulation study of SRS and RSS. Section 5 describes the real data analysis. Section 6 concludes the paper.

HLITL Model Characterizations
The ITL distribution, having the CDF and PDF defined by Equations (3) and (4), respectively, was presented in [11]. This study considers the HLITL distribution as presented in Definition 1.

Definition 1.
A random variable X is assumed to have the HLITL distribution if its PDF is represented as follows: where H = (1 + x) −2θ (1 + 2x) θ , λ and θ denote the shape parameters. A random variable with the PDF given by Equation (7) is denoted by ∼ (λ, θ). For λ = 1 in Equation (7), a new distribution called the Marshall-Olkin ITL distribution with δ = 0.5 is obtained. The PDF plots of the HLITL distribution for some specific values of λ and θ are displayed in Figure 1. The shape of the HLITL distribution is unimodal, positively skewed, and reverse J-shaped.
distribution parameters on the basis of SRS and RSS; and (4) provide significant improvement in the data modeling. The rest of the article is organized as follows. Section 2 defines the PDF, CDF, and hazard rate function (HRF) of the HLITL distribution. Section 3 derives the structural properties of the HLITL distribution. Section 4 presents the maximum likelihood (ML) parameter estimators as well as the simulation study of SRS and RSS. Section 5 describes the real data analysis. Section 6 concludes the paper.

HLITL Model Characterizations
The ITL distribution, having the CDF and PDF defined by Equations (3) and (4), respectively, was presented in [11]. This study considers the HLITL distribution as presented in Definition 1.

Definition 1.
A random variable X is assumed to have the HLITL distribution if its PDF is represented as follows: where 2 (1 ) ( 1 2 ) , H x x θ θ − = + + λ and θ denote the shape parameters. A random variable with the PDF given by Equation (7) is denoted by ~ , . For 1 λ = in Equation (7), a new distribution called the Marshall-Olkin ITL distribution with 0.5 δ = is obtained.
The PDF plots of the HLITL distribution for some specific values of λ and θ are displayed in Figure 1. The shape of the HLITL distribution is unimodal, positively skewed, and reverse J-shaped.  The associated CDF and HRF, respectively, are given by The HRF plots for specific values of parameters with various shapes are illustrated in Figure 2.  1 1 The HRF plots for specific values of parameters with various shapes are illustrated in Figure 2.

Basic Properties of the HLITL Distribution
The basic properties of the HLITL distribution, including moments, skewness, kurtosis, Rényi entropy, stochastic ordering, and Bonferroni and Lorenz curves, are derived in the following sections.

Useful Representation
The properties of the HLITL distribution are deduced through simple elicitation of its PDF. According to [19], the PDF given by Equation (7) can be represented as (1 ) Employing the binomial expansions, the PDF of the HLITL distribution becomes which further leads to , , 0 ( ; , ) ( ; ( 1)) , represents the ITL PDF with the shape parameter ( 1). d θ + Thus, the PDF of the HLITL distribution can be expressed as an infinite linear combination of ITL PDFs.

Basic Properties of the HLITL Distribution
The basic properties of the HLITL distribution, including moments, skewness, kurtosis, Rényi entropy, stochastic ordering, and Bonferroni and Lorenz curves, are derived in the following sections.

Useful Representation
The properties of the HLITL distribution are deduced through simple elicitation of its PDF. According to [19], the PDF given by Equation (7) can be represented as Employing the binomial expansions, the PDF of the HLITL distribution becomes which further leads to where The PDF g(x; θ(d + 1)) represents the ITL PDF with the shape parameter θ(d + 1). Thus, the PDF of the HLITL distribution can be expressed as an infinite linear combination of ITL PDFs.

HLITL Entropy
The Rényi entropy of a random variable X with the HLITL distribution for ϕ > 0 and ϕ = 1 is expressed as Substituting Equation (7) into Equation (13) leads to where By applying the binomial expansions to Equation (14), we get where B(.,.) stands for the beta function. The numerical results of I e of the HLITL distribution for some ϕ, λ, and θ values are listed in Table 1. Table 1. Numerical results of Rényi entropy for the HLITL distribution. Parameters We conclude from Table 1 that as the value of shape parameter θ increased, for the same value of parameter λ, the value of randomness decreased. In addition, as the value of ϕ increased, the value of randomness decreased for fixed values of θ and λ. Furthermore, the values in Table 1 are in the interval [−0.321, 0.533], meaning that λ and θ had pivotal effects on the information amount. Thus, these parameters had a strong effect on the Rényi entropy, showing different degrees of uncertainty.

Quantile Function
The quantile function of X ∼ HLITL(λ, θ) is given as q in (0, 1) as Setting q = 0.5 in Equation (16), we get the median M of the HLITL distribution. Furthermore, by setting q = 0.25 and 0.75 in Equation (16), the 25th and 75th percentiles are obtained, respectively. Based on the quantiles, Bowley's skewness is given by Moor's kurtosis is given by The Bo and Mo plots of the HLITL distribution were obtained based on the quantile function, shown in Figure 3.
in Equation (16), we get the median M of the HLITL distribution. Furthermore, by setting q = 0.25 and 0.75 in Equation (16), the 25th and 75th percentiles are obtained, respectively. Based on the quantiles, Bowley's skewness is given by Moor's kurtosis is given by The Bo and Mo plots of the HLITL distribution were obtained based on the quantile function, shown in Figure 3. Bowley's skewness was plotted for increasing values of and . From the plot of the skewness, it is clear that the distribution was positively skewed, and the skewness increased with the decrease in and increase in and vice versa.
Moor's kurtosis was plotted for increasing values of and . From the plot, it is clear that the kurtosis of the distribution increased with the increase in and .

Moments and Related Measures
If X has the PDF given by Equation (12), then its nth moment can be calculated as Bowley's skewness was plotted for increasing values of λ and θ. From the plot of the skewness, it is clear that the distribution was positively skewed, and the skewness increased with the decrease in θ and increase in λ and vice versa.
Moor's kurtosis was plotted for increasing values of λ and θ. From the plot, it is clear that the kurtosis of the distribution increased with the increase in λ and θ.

Moments and Related Measures
If X has the PDF given by Equation (12), then its nth moment can be calculated as Using the binomial expansion, the nth moment of the HLITL distribution can be obtained as follows: where Further, the nth central moment of X is defined as The numerical values for specific values of the parameters of the first four ordinary moments µ 1 , µ 2 , µ 3 , and µ 4 , the variance (σ 2 ), the coefficient of variation (CV = σ µ 1 ), the coefficient of skewness (SK = µ 3 (µ 2 ) 1.5 ), and the coefficient of kurtosis (KU = µ 4 (µ 2 ) 2 ) of the HLITL distribution are listed in Tables 2 and 3. Table 2. Some moments of the HLITL distribution.  Table 3. Some moments of the HLITL distribution.  As shown in Tables 2 and 3, for the same value of θ and for the increasing value of λ, the values of the moments increased, but the values of SK, KU, and CV decreased. At the same value of λ, the moment values increased with the value of θ, while the values of SK, KU, and CV decreased. As the value of θ increased, all measures decreased with the value of λ. Thus, the HLITL distribution was positively skewed and leptokurtic.

Bonferroni and Lorenz Curves
Bonferroni and Lorenz curves have been applied to several fields, such as economics and engineering, and they are respectively defined by where x 0 y f (y; λ, θ)dy, denotes the lower incomplete moment. For the HLITL distribution, the Bonferroni and Lorenz curves are respectively defined as follows: where ψ i,d,k is defined above and B(., ., x/(1 + x)) represents an incomplete beta function.

Parameters Estimation
In this section, the estimation of λ and θ of the HLITL distribution under the SRS and RSS are discussed. Additionally, we examine the behavior of the estimates, namely in the simulation study.

Parameter Estimation under SRS
In this section, the HLITL parameters are estimated by the ML method. Let X 1 , X 2 , . . . , X n be SRS obeying the HLITL distribution with observed values x 1 , x 2 , . . . , x n , respectively. The log-likelihood function of the HLITL distribution, denoted by ln , is obtained as follows: where H i = (1 + 2x i ) θ (1 + x i ) −2θ . The partial derivatives of the log-likelihood function associated with λ and θ are respectively given by where By solving the nonlinear equations ∂ln /∂λ = 0 and ∂ln /∂θ = 0 with respect to λ and θ, the ML estimators could be obtained.

ML Estimation under RSS
Suppose that X (i)ic , i = 1, 2, . . . , m, c = 1, 2, . . . , k denotes a selected RSS from the HLITL distribution, where the sample size is n = mk, m denotes the set size, and k is the number of cycles. For simplicity, assume that Y ic = X (i)ic . Then, the fixed c, Y ic is independent and has a PDF equal to the PDF of the ith order statistics. The likelihood function L R of the RSS of samples y 1c , y 2c , . . . , y mc can be expressed as where H ic = (1+2y ic ) θ (1+y ic ) 2θ . The log-likelihood function of the HLITL distribution is given by where The partial derivatives of lnL R associated with unknown parameters are expressed as follows: where h ic = 1+2y ic (1+y ic ) 2 θ ln 1+2y ic (1+y ic ) 2 . In solving the nonlinear equations ∂lnL R /∂λ = 0 and ∂lnL R /∂θ = 0 with respect to λ and θ numerically, the ML estimators can be obtained.

Simulation Procedures
This section presents the numerical method to obtain the ML estimates (MLEs) of λ and θ for the HLITL distribution under the RSS and SRS. A comparison study was conducted based on the mean square error (MSE) value, biases, and relative efficiency (RE). The following algorithm was performed using MATHEMATICA software [31] to obtain the MLEs and the suggested criteria measures.
Step 1: A random sample of size n = 30, 50, 100, 200, or 300 with a set size of m and number of cycles k where n = mk is generated from the HLITL distribution.
Step 3: For each n and specified parameter value, the MLE values under SRS and RSS are obtained.
Step 5: The outcomes of this study are given in Tables 4-8. Based on the numerical study, the following conclusions can be drawn. As presented in Tables 4 and 8, for a fixed value of θ, the MSE of the estimates of λ increased with the value of λ, while the MSE of the estimates of θ decreased under SRS and RSS with n. Furthermore, the MSE values of the estimates of λ were smaller than those of the estimates of θ for all values of n under both sampling schemes.
As λ increased from two to three and θ increased from five to eight, the MSE values of both estimates increased under both schemes. In addition, the MSE values of the estimates of θ were smaller than those of the estimates of λ for all sample sizes, as shown in Tables 4 and 8.
When the value of λ increased and the value of θ decreased, the MSE values of the estimates of λ increased, while the MSE values of the estimates of θ decreased under both sampling schemes (see Tables 4 and 6).
Generally, the MLE values of the HLITL distribution under RSS were smaller than those under the SRS. Additionally, when the MSE values of both estimates decreased, the RE increased. When n increased, the RE values increased, and the MSE values decreased.

Applications to Real Data
The experiments on two real datasets were conducted to assess the flexibility of the proposed HLITL model. The first dataset included the relief times of 20 patients receiving an analgesic [32]. The second dataset was taken from [33], which included the failure times of the air conditioning system of an airplane. The HLITL model was compared with four competing models: AP exponential (APE) [34], alpha power (AP) Lindley (APL) [35], Fréchet inverse exponential (OFIE) [36], and odd Fréchet inverse Rayleigh (OFIR) [37]. Some descriptive statistics of the two datasets are given in Table 9, and the boxplot is shown in Figure 4. competing models: AP exponential (APE) [34], alpha power (AP) Lindley ( chet inverse exponential (OFIE) [36], and odd Fréchet inverse Rayleigh (O descriptive statistics of the two datasets are given in Table 9, and the boxp Figure 4.    The values in Tables 12 and 13 show that the HLITL distribution had the smallest values among all fitted distributions. Consequently, this distribution can be considered more suitable than the others for the considered data. The estimated PDF, CDF, SF, and PP plots of the HLITL model are presented in Figures 5 and 6, where it can be seen that the HLITL model exhibits a close fit to both datasets.  The author of [38] mentioned that the total time test (TTT) represents a useful graphical approach to confirm whether the data can be applied to a specific distribution or not.  The author of [38] mentioned that the total time test (TTT) represents a useful graphical approach to confirm whether the data can be applied to a specific distribution or not. The empirical version of the TTT plot is expressed as The author of [38] mentioned that the total time test (TTT) represents a useful graphical approach to confirm whether the data can be applied to a specific distribution or not. The empirical version of the TTT plot is expressed as where r = 1, 2, . . . , n and y i:n denote the ith order statistics of the sample. In [38], it was shown that when the TTT plot represents a straight line, then the HRF is constant, and when the TTT plot is concave (or convex), the HRF increases (or decreases). Additionally, when the TTT plot is first convex and then concave, the HRF is U-shaped (bathtub); otherwise, the HRF is unimodal. The TTT plots presented in Figure 7 show that the estimated HRFs of the first and second datasets increased and decreased. This claim is also confirmed in Figure 8.

Conclusions
This paper proposes an HLITL distribution with two parameters. It is demonstrated that the HLITL distribution can be regarded as a mixture of ITL distributions. Its statistical properties, including the moments, quantile function, skewness, kurtosis, Bonferroni and Lorenz curves, and stochastic ordering, are defined. The maximum likelihood estimators

Conclusions
This paper proposes an HLITL distribution with two parameters. It is demonstrated that the HLITL distribution can be regarded as a mixture of ITL distributions. Its statistical properties, including the moments, quantile function, skewness, kurtosis, Bonferroni and Lorenz curves, and stochastic ordering, are defined. The maximum likelihood estimators of the HLITL distribution parameters are derived under both SRS and RSS schemes. It was shown empirically that the maximum likelihood estimates taken from the observed RSS were more efficient than the corresponding SRS. The merit of the HLITL distribution was For both real datasets, the MLE values of both parameters under RSS were obtained. The observed RSS of the first dataset with the set size of four were 1.1, 2.7, 3, and 4.1, and those of the second dataset with the set size of five were 1, 5, 7, 11, and 261. The MLE values of the HLITL distribution for the first dataset and their SE values wereλ = 6.653(5.558) andθ = 2.766(1.348), and those of the second dataset werê λ = 2.239(1.3448) andθ = 0.844(0.071).

Conclusions
This paper proposes an HLITL distribution with two parameters. It is demonstrated that the HLITL distribution can be regarded as a mixture of ITL distributions. Its statistical properties, including the moments, quantile function, skewness, kurtosis, Bonferroni and Lorenz curves, and stochastic ordering, are defined. The maximum likelihood estimators of the HLITL distribution parameters are derived under both SRS and RSS schemes. It was shown empirically that the maximum likelihood estimates taken from the observed RSS were more efficient than the corresponding SRS. The merit of the HLITL distribution was evaluated experimentally using two real datasets. The results show that it performed better than the other fitted distributions. Based on the obtained results, the proposed distribution can be useful for modeling data in various fields.