Statistical inference and data analysis of the record-based transmuted Burr X model

1 Introduction

The utilization of asymmetrical statistical distributions is widespread across nearly all disciplines, reflecting their fundamental role in understanding and interpreting uncertainty in various contexts, notably engineering, industrial, medical sciences, insurance, and environmental. As a result, it appears essential to obtain statistical models, which are a critical and challenging task. However, sometimes, there are cases in which these statistical distributions are not suitable for analyzing several data sets. For this, the author has worked to apply numerous methods for obtaining novel families of distributions that extend well-known models. These novel-generated family models have a crucial role in fitting skewed data sets. In relation to this, we refer several previous studies that have investigated the established probability distributions, specifically those conducted by Hamedani et al. [1], Cordeiro and Brito [2], Marshall and Olkin [3], Mahdavi and Kundu [4], Hassan et al. [5], Moakofi et al. [6], Eghwerido et al. [7], Sapkota et al. [8], Meraou and Raqab [9], Meraou et al. [10], and Thomas and Chacko [11].

In this context, Balakrishnan and He [12] proposed one of these procedures called the record-based transmuted mapping technique that is considered in numerous applied fields, such as insurance, medical science, biology, environment, and finance. Its cumulative distribution function (cdf) and corresponding probability distribution function (pdf) can be formulated as

and

where F ( x ; φ ) and f ( x ; φ ) represent the cdf and pdf of the parent distribution with parameter φ .

In the last few decades, the record-based transmuted mapping technique has been developed by different researchers in the literature. For example, Tanis and Saracoglu [13] introduced a record-based Weibull model by taking the Weibull distribution as the baseline model and establishing different properties of the proposed model. Arshad et al. [14] introduced a novel approach of generalization exponential distribution using record transmuted mapping procedure, and they studied different mathematical and distributional properties of the proposed model. Notably, the record-based transmuted model of Tanis proposes record-based transmuted Lindley distribution [15], and he applied the suggested model to COVID-19 patient data to demonstrate the potential of the proposed model among other new distributions. Many authors discussed digital transformation and employees with 4 years after COVID-19. In the same way, Sakthivel and Nandhini [16] provided the record transmuted power Lomax model with applications to the reliability area. Sobhi and Mashail [17] discussed moments of dual generalized order statistics and characterization for transmuted exponential model. Abu El Azm et al. discussed new transmuted generalized Lomax distribution. Mohamed et al. [18] introduced transmuted Topp-Leone length biased exponential model under competing risk model. A record-based transmuted Nadarajah-Haghighi model is defined by Kumar et al. [19].

As far as we know, the Burr X distribution (BXD) is a versatile statistical tool for modeling complex and asymmetric data and complementary risk scenarios. It has numerous applications in many practical cases, like fitting the lifetime record in the engineering field. One may refer to the studies of Usman and Ilyas [20], Al-Babtain et al. [21], Fayomi et al. [22], Raqab and Kundu [23], Yıldırım et al. [24], Korkmaz et al. [25], and Merovci et al. [26].

Surles and Padgett [27] provided the Burr X (BX) model. The associated probability density and cumulative density functions of the BX model are expressed respectively as follows:

and

In the present study, we take the BXD and apply the record-transmuted mapping technique to construct a new family of distributions that can be enhanced fitting capabilities in various practical applications when assessed against existing models. We referred to it as the record-transmuted Bur X (RT-BX) model; we sometimes called it record-transmuted power Bur X (RT-PBX) model. The proposed model can take a variety of shapes. As well as, we can obtain the basic distribution as a special case. The hazard function of the proposed distribution can exhibit various shapes, including increasing and decreasing. Further, various distributional and mathematical properties of the RT-BX model, like MGF, ordered statistics, and quantile function, are obtained as well and five entropy estimators for the RT-BX model are computed.

The rest of this study is outlined as follows: In Section 2, we construct the RT-BX model and thoroughly discuss its behavior of pdf and hazard rate function. Numerous mathematical and statistical properties are established in Section 3. In Section 4, several suggested entropy measures for the recommended distribution are defined, and its estimation parameters are developed in Section 5 by employing the maximum likelihood estimation (MLE) procedure. In Section 6, simulation experiment studies are explored to see the applicability of the MLE technique. Finally, three real-life applications are analyzed in Section 7 for validation purposes. Some important remarks are presented in Section 8.

2 Record-based transmuted BXD

2.1 Model description

In this subsection, we proposed certain distribution properties of the RT-BX model, such as probability density, cumulative density functions, survival, and hazard rate functions.

Let the BXD with parameters α and β be parent distribution. According to equations (1)–(4), the corresponding cumulative density function and probability density function of the proposed RT-BX model are, respectively, expressed as

(5) H y ( y ) = 1 − e − ( α y β ) 2 [ 1 + θ ( α y β ) 2 ]

and

(6) h Y ( y ) = 2 α 2 y 2 β − 1 e − ( α y β ) 2 [ 1 − θ + θ ( α y β ) 2 ] .

Figure 1 shows curves of the probability density function of the RT-BX model for different parameter values. From these plots, obviously the density is positively skewed and symmetric and decreasing when 0 < β ≤ 1 ⁄ 2 and uni-modal if β ≥ 1 ⁄ 2 . Theorem 1 ensures this conclusion.

Figure 1

Graphs of the RT-BX density for numerous parameter values.

Next the survival function with the associated hazard rate function of the RT-BX model can be formulated, respectively, by

S Y ( y ) = e − ( α y β ) 2 [ 1 + θ ( α y β ) 2 ]

and

(7) h r Y ( y ) = 2 α 2 y 2 β − 1 [ 1 − θ + θ ( α y β ) 2 ] 1 + θ ( α y β ) 2 .

The cumulative, survival, and hazard rate function plots of the RT-BX model are sketched in Figures 2 and 3, respectively. From Figure 3, it can be observed that the hazard rate function increases if β ≥ 1 ⁄ 2 and decreases if 0 < β ≤ 1 ⁄ 2 , which shows the flexibility of the proposed RT-BX model, and Theorem 2 confirms this conclusion.

Figure 2

Cumulative and survival plots for the RT-BX model using different values of the parameters.

Figure 3

Hazard rate function plots for the RT-BX model using different values of the parameters.

Next the cumulative hazard rate function of the RT-BX is

C H Y ( y ) = ( α y β ) 2 − ln [ 1 + θ ( α y β ) 2 ] .

The reversed hazard rate function can be formulated as

R Y ( y ) = 2 α 2 y 2 β − 1 e − ( α y β ) 2 [ 1 − θ + θ ( α y β ) 2 ] 1 − e − ( α y β ) 2 [ 1 + θ ( α y β ) 2 ] .

3 Mathematical properties

We developed here various statistical characteristics of the recommended RT-PBX distribution. From now on, let Y ∼ RT-BX( θ , α , β ).

3.2 Moments with related measures

The kth moment of Y is

(11) μ k ′ = 1 β α 2 + β ( k + 2 β ) Γ 1 2 β , 0 ( 1 − θ ) + θ Γ 1 2 β + 1 , 0 ,

where, Γ ( a , b ) denotes incomplete gamma function, and it is expressed as Γ ( a , b ) = ∫ b ∞ w a − 1 e − w d w .

Proof

(12) μ k ′ = ∫ 0 ∞ y k h Y ( y ) d y = 2 α 2 ∫ 0 ∞ y k + 2 β − 1 e − ( α y β ) 2 ( 1 − θ + θ ( α y β ) 2 ) d y .

Let t = ( α y β ) 2 , which implies that y = t 1 ⁄ 2 α 1 ⁄ β . Then, equation (12) can be rewritten as

μ k ′ = α β ( k + 2 β ) + 2 β ∫ 0 ∞ e − t t 1 − 2 β β ( 1 − θ + θ t ) d t = α β ( k + 2 β ) + 2 β ∫ 0 ∞ t 1 2 β − 1 e − t d t − θ ∫ 0 ∞ t 1 2 β − 1 e − t d t + θ ∫ 0 ∞ t 1 2 β e − t d t = 1 β α 2 + β ( k + 2 β ) Γ 1 2 β , 0 ( 1 − θ ) + θ Γ 1 2 β + 1 , 0 ,

which completes the proof. Consequently, from equation (11), the mean and second-ordered moment of Y are expressed, respectively, as

μ 1 ′ = 1 β α 2 + β ( 1 + 2 β ) Γ 1 2 β , 0 ( 1 − θ ) + θ Γ 1 2 β + 1 , 0

and

□ μ 2 ′ = 1 β α 2 + β ( 2 + 2 β ) Γ 1 2 β , 0 ( 1 − θ ) + θ Γ 1 2 β + 1 , 0 .

The variance and coefficient of variation ( V ) of Y are

Var ( Y ) = μ 2 ′ − μ 1 ′ 2 and V = μ 2 ′ − μ 1 ′ 2 μ 1 ′ .

The moment generating function (MGF) of Y is obtained as

(13) M Y ( t ) = E [ e t y ] = ∫ 0 ∞ e t y h Y ( y ) d y = 2 α 2 ∫ 0 ∞ e t y y 2 β − 1 e − ( α y β ) 2 ( 1 − θ + θ ( α y β ) 2 ) d y = 1 β ∫ j = 0 ∞ t j α 2 + β j + 2 β j ! Γ 1 2 β , 0 ( 1 − θ ) + θ Γ 1 2 β + 1 , 0 .

The proposed statistical property values of the RT-BX model are tabulated in Tables 1 and 2 by applying various choices of θ , α , and β . The same can easily be observed for these quantities from the plots presented in Figure 4. From these results of mathematical properties, we can see that

1. The findings indicate that μ 1 ′ and Var values decrease with the parameter values, whereas the values of V , S , and K are fixed.

2. Now, β increases and α and θ are fixed, the measures of μ 1 ′ , Var, V , and S decrease.

3. If θ tend to increase when β and θ are fixed, the records of μ 1 ′ and Var tend to increase, but the values of V , S , and K decrease.

Table 1

Distinct records of mathematical properties for the RT-BX model at θ = 0.5

	α	μ 1 ′	Var	V	S	K
β = 0.75	0.5	3.0334	3.4198	0.6096	0.8183	0.6445
	1	1.2038	0.5386	0.6096	0.8183	0.6445
	1.5	0.7011	0.1827	0.6096	0.8183	0.6445
	2	0.4777	0.0848	0.6096	0.8183	0.6445
β = 1.5	0.5	1.6533	0.3001	0.3313	‒ 0.0135	‒ 0.2945
	1	1.0415	0.1191	0.3313	‒ 0.0135	‒ 0.2945
	1.5	0.7948	0.0694	0.3313	‒ 0.0135	‒ 0.2945
	2	0.6561	0.0473	0.3313	‒ 0.0135	‒ 0.2945
β = 2.25	0.5	1.3794	0.1015	0.2310	‒ 0.3483	‒ 0.0323
	1	1.0136	0.0548	0.2310	‒ 0.3483	‒ 0.0323
	1.5	0.8465	0.0382	0.2310	‒ 0.3483	‒ 0.0323
	2	0.7449	0.0296	0.2310	‒ 0.3483	‒ 0.0323

Table 2

Distinct records of mathematical properties for the RT-BX model at θ = 0.75

	α	μ 1 ′	Var	V	S	K
β = 0.75	0.5	3.4122	3.4976	0.5481	0.7177	0.5191
	1	1.3541	0.5508	0.5481	0.7177	0.5191
	1.5	0.7886	0.1868	0.5481	0.7177	0.5191
	2	0.5374	0.0868	0.5481	0.7177	0.5191
β = 1.5	0.5	1.7713	0.2745	0.2958	‒ 0.0858	‒ 0.1425
	1	1.1159	0.1089	0.2958	‒ 0.0858	‒ 0.1425
	1.5	0.8516	0.0634	0.2958	‒ 0.0858	‒ 0.1425
	2	0.7030	0.0432	0.2958	‒ 0.0858	‒ 0.1425
β = 2.25	0.5	1.4484	0.0886	0.2055	‒ 0.4165	0.2257
	1	1.0643	0.0478	0.2055	‒ 0.4165	0.2257
	1.5	0.8888	0.0334	0.2055	‒ 0.4165	0.2257
	2	0.7822	0.0258	0.2055	‒ 0.4165	0.2257

Figure 4

3D curves of proposed statistical measures considering RT-BX with various selected parameter records.

4 Entropy information

In this section, numerous information entropies are established. First, the Rényi entropy [29] ( R 1 ( δ ) ) is the measure of variation in uncertainty. The associated Rényi entropy of RT-BX model is

Another uncertainty measure is the Shannon entropy [30] ( R 2 ). It is expressed as

A new Havrda and Charvat entropy [31] ( R 3 ( δ ) ) of the RT-BX can be considered in this study. It can be formulated as

Now, the Tsallis entropy [32] R 4 ( δ ) of the RT-BX distribution is defined by

Finally, the Arimoto entropy [33] ( R 5 ( δ ) ) associated with the RT-BX model is represented by

The numerical vales of the suggested entropy information are displayed in Tables 3 and 4 based on numerous selected parameters θ , α , β , and δ and Figures 5 and 6 show the sketches of the 3D plots of this information entropy.

Figure 5

Plots for μ 1 ′ , Var, CV, S , and K for α = 0.75 and ρ = 0.75 .

Figure 6

Plots for μ 1 ′ , Var, CV, S , and K for α = 1.5 and ρ = 0.5 .

5 ML estimator

Let { y 1 , y 2 , … , y n } be a random sample of the size n drawn from the RT-BX model. The associated likelihood function is now obtained as

On solving the below equations, we obtain the estimate of the given parameter of the RT-BX distribution under MLE method.

and

The solution cannot be found analytically and must be obtained using numerical methods. Here in this study, the Newton-Raphson method is commonly applied to obtain the final estimate of the unknown parameters for the RT-BX model numerically.

6 Simulation analysis

In this section, Monte Carlo (MC) simulation studies are conducted to assess the performance of the recommended ML estimator tool for the newly generated RT-BX model by applying numerous sample sizes n = { 300 , 500 , 700 , 900 , 1,000 } and various parameter set values of ( θ , α , β ) including Set 1 = ( 0.4 , 2.25 , 1.75 ) , Set 2 = ( 0.5 , 2.5 , 2 ) , Set 3 = ( 0.6 , 2.75 , 2.25 ) , and Set 4 = ( 0.75 , 3 , 2.5 ) . By using equation (10), we can generate a random sample of the RT-BX model. The computations were obtained employing the R program with a function optim for Newton-Raphson technique by taking the values of α , β , and θ as Set 1, Set 2, Set 3, and Set 4, respectively. The following algorithm describes the steps of random generating process from the suggested model:

1. Obtain q from the uniform distribution U [ 0 , 1 ] .

2. In the same way, obtain y with the formula

   y = α − 1 − 1 θ − W − 1 u − 1 θ e 1 θ 1 ⁄ 2 1 ⁄ β .

where ϱ = ( θ , α , β ) .

The results of these simulation experiments are reported in Tables 5, 6, 7, 8. Based on the findings presented in Tables 5–8, we can conclude that the final estimates are generally constant and tend to the initial parameters. Also, for all parameter sets, if we increase n , the ABs and MSEs decrease, which ensures that the suggested ML estimators are consistent and asymptotically unbiased, where (Est) is the estimated values.