A New Two-Parameter Burr-Hatke Distribution: Properties and Bayesian and Non-Bayesian Inference with Applications

Abstract

This article introduces a two-parameter flexible extension of the Burr-Hatke distribution using the inverse-power transformation. The failure rate of the new distribution can be an increasing shape, a decreasing shape, or an upside-down bathtub shape. Some of its mathematical properties are calculated. Ten estimation methods, including classical and Bayesian techniques, are discussed to estimate the model parameters. The Bayes estimators for the unknown parameters, based on the squared error, general entropy, and linear exponential loss functions, are provided. The ranking and behavior of these methods are assessed by simulation results with their partial and overall ranks. Finally, the flexibility of the proposed distribution is illustrated empirically using two real-life datasets. The analyzed data shows that the introduced distribution provides a superior fit than some important competing distributions such as the Weibull, Fréchet, gamma, exponential, inverse log-logistic, inverse weighted Lindley, inverse Pareto, inverse Nakagami-M, and Burr-Hatke distributions.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Survival and reliability analysis is an important area of statistics and it has various applications in several applied sciences such as engineering, economics, demography, medicine, actuarial science, and life testing. Different lifetime distributions have been introduced in the statistical literature to provide greater flexibility in modeling data in these applied sciences.

One of the important features of generalized distributions is their capability for providing superior fit for various life-time data encountered in the applied fields. Hence, the statisticians have been interested in constructing new families of distributions to model such data. Some recent notable families are the following: the exponential T-X [1], transmuted Burr-X [2], Marshall-Olkin Burr-III [3], Marshall-Olkin Burr [4], and log-logistic tan [5] families.

On the other hand, there are some useful techniques to add an additional parameter to extend and enhance the flexibility of the classical distributions such as the inverse-power (IP) transformation. Let $X$ and $Y$ be two random variables. The inverse transformation, say $X=Y^{-1}$, or the IP transformation, say $X=Y^{-1/\eta }$, has been adopted by many authors to construct generalized inverted distributions. For example, the generalized inverse gamma [6], the inverse Lindley with two parameters [7], the inverse Lindley [8], the inverse-power Maxwell [9], the inverse-power Lindley [10], and inverse-power Lomax [11].

In this paper, we are motivated to propose a more flexible version of the Burr-Hatke (BH) distribution to increase its flexibility in modeling real-life data. The BH model provides only a decreasing hazard rate (HR) shape; hence, its use will be limited to modeling the data that exhibits only increasing failure rate. The proposed distribution is called the inverse-power Burr-Hatke (IPBH) distribution. The IPBH model can accommodate right-skewed shape, symmetrical shape, reversed J shape, and left-skewed shape densities. Its HR can be an increasing shape, a unimodal shape, or a decreasing shape. The IPBH provides more accuracy and flexibility in fitting engineering and medicine data. The IPBH distribution was constructed using the inverse-power (IP) transformation.

Isaic-Maniu and Voda [12] proposed the BH distribution with shape parameter $\alpha$. Its cumulative distribution function (CDF) has the following form:$$\begin{array}{}\text{(1)}& Fx;\alpha =1-\frac{\exp -\alpha x}{x+1},\hspace{1em}\alpha >0,x>0.\end{array}$$

Its probability density function (PDF) takes the following form:$$\begin{array}{}\text{(2)}& fx;\alpha =\frac{\exp -\alpha x\alpha +\alpha x+1}{{x+1}^2},\hspace{1em}\alpha >0,x>0.\end{array}$$

We also considered ten various classical and Bayesian methods for estimating the IPBH parameters and provided detailed numerical simulations to explore their performances based on the mean square errors (MSE), mean relative estimates (MRE), and absolute biases (BIAS). The classical estimators proposed included the maximum product of spacing estimators (MPSE), Anderson-Darling estimators (ADE), Cramér-von Mises estimators (CVME), least-squares estimators (LSE), maximum likelihood estimators (MLE), right-tail Anderson-Darling estimators (RTADE), and weighted least-squares estimators (WLSEs). The Bayesian estimators of the IPBH parameters have been obtained under symmetric and asymmetric loss functions, namely, the square errors (SE), general entropy (GE), and linear exponential (LN) loss functions. We have compared the estimation methods by conducting extensive simulations study to explore their performances and to determine the best method of estimation, based on partial and overall ranks, which gives accurate estimates for the IPBH parameters.

It is shown empirically that the IPBH distribution can provide a more adequate fit than ten competing distributions, namely, the BH [12], Weibull (W), Fréchet (F), gamma (G), exponential (E), inverse log-logistic (ILL) [13], inverse weighted Lindley (IWL) [14], inverse Lindley (IL) [14], inverse Pareto (IP) [15], and inverse Nakagami-M (INM) [16] distributions.

This article is outlined in the following eight sections. The IPBH distribution is defined in Section 2. Some of its properties are discussed in Section 3. In Section 4, seven classical approaches of estimation are explored. The Bayesian estimators of the IPBH parameters under three loss functions are discussed in Section 5. In Section 6, the performances of classical and Bayesian approaches of estimation are explored via simulations. The applicability and flexibility of the IPBH distribution are illustrated in Section 7 using two real-life datasets. Some useful conclusions are presented in Section 8.

2. The IPBH Distribution

By applying the IP transformation to the BH CDF (1), the CDF of the IPBH distribution follows (for $x>0$) as$$\begin{array}{}\text{(3)}& Fx;\alpha ,\eta =\frac{\exp -\alpha x^{-\eta }}{x^{-\eta }+1},\hspace{1em}\alpha ,\eta >0.\end{array}$$

The corresponding PDF of the IPBH distribution reduces to$$\begin{array}{}\text{(4)}& fx;\alpha ,\eta =\frac{\eta \exp -\alpha x^{-\eta }\alpha +\alpha +1x^{\eta }}{x{x^{\eta }+1}^2},\hspace{1em}\alpha ,\eta >0,\end{array}$$where $\eta$ and $\alpha$ are shape parameters. The inverse BH (IBH) distribution follows simply as a special case by replacing $\eta =1$ in equation (4).

The survival function (SF) and HR function of the IPBH distribution take the following forms, respectively:$$\begin{array}{c}\text{(5)}& Sx;\alpha ,\eta =1-\frac{x^{\eta }\exp -\alpha x^{-\eta }}{x^{\eta }+1},hx;\alpha ,\eta =\frac{\eta \alpha +\alpha +1x^{\eta }}{x{x}^{\eta }+1x^{\eta }+1\exp -\alpha x^{-\eta }-x^{\eta }}.\end{array}$$

Possible shapes of the density and HR functions of the IPBH distribution are displayed in Figures 1 and 2, respectively.

[figure omitted; refer to PDF]

The ten estimation approaches are also adopted to estimate the IPBH parameters from the two datasets. Tables 8 and 9 report the estimates of $\alpha$ and $\eta$ along with the values of $-\widehat{\ell }$, $A$, $W$, K-S, and K-S $p$ value for both datasets, respectively. The proposed estimation approaches show a similar very well performance in estimating $\alpha$ and $\eta$ of the IPBH distribution. A visual comparison shows the close performance of these estimators as shown in Figure 6 which represents the P-P plots of the IPBH distribution for the ten methods.

Table 8

The estimates of $\alpha$, $\eta$, $-\widehat{\ell }$, A, $W$, K-S (stat), and K-S $p$ value of the IPBH distribution for dataset I.

	$\widehat{\alpha }$	$\widehat{\eta }$	$-\widehat{\ell }$	A	$W$	K-S (stat)	K-S $p$ value
MLE	5.7160	5.4950	20.0086	0.5175	0.0680	0.0762	0.8573
ADE	6.4795	5.7664	20.1976	0.4776	0.0568	0.0662	0.9452
CVME	7.1013	5.9308	20.5657	0.5025	0.0539	0.0767	0.8520
MPSE	4.9180	5.1487	20.5657	0.5025	0.0539	0.0767	0.8520
LSE	6.7066	5.8074	20.3068	0.4845	0.0548	0.0710	0.9084
RTADE	7.5598	6.0797	20.9502	0.5391	0.0552	0.0818	0.7922
WLSE	6.9539	5.9190	20.4733	0.4903	0.0548	0.0718	0.9013
BSE	5.8139	5.4722	20.0231	0.5246	0.0674	0.0808	0.8043
BLN	5.8148	5.4729	20.0230	0.5243	0.0673	0.0808	0.8050
BGE	5.8136	5.4719	20.0232	0.5247	0.0674	0.0808	0.8041

Table 9

The estimates of $\alpha$, $\eta$, $-\widehat{\ell }$, A, $W$, K-S (stat), and K-S $p$ value of the IPBH distribution for dataset II.

	$\widehat{\alpha }$	$\widehat{\eta }$	$-\widehat{\ell }$	A	$W$	K-S (stat)	K-S $p$ value
MLE	5.2423	4.0622	15.4046	0.1528	0.0261	0.1005	0.9875
ADE	5.1651	4.0264	15.406	0.1522	0.0265	0.0990	0.9895
CVME	6.0804	4.2881	15.4871	0.1770	0.0244	0.0924	0.9955
MPSE	6.3782	4.8187	15.4871	0.1770	0.0244	0.0924	0.9955
LSE	4.9267	3.9463	15.4211	0.1544	0.0279	0.0999	0.9882
RTADE	5.4326	4.1085	15.4092	0.1546	0.0254	0.0972	0.9915
WLSE	4.3142	3.7311	15.5535	0.1830	0.0342	0.1038	0.9822
BSE	5.4296	4.0412	15.4190	0.1627	0.0273	0.0977	0.9910
BLN	5.4366	4.0416	15.4198	0.1632	0.0273	0.0980	0.9906
BGE	5.4267	4.0410	15.4188	0.1625	0.0272	0.0976	0.9911

[figures omitted; refer to PDF]

8. Conclusions

In this article, we introduce a more flexible extension of the Burr-Hatke distribution called inverse-power Burr-Hatke (IPBH) distribution that provides more accuracy and flexibility in fitting engineering and medicine data. The new model was generated based on the inverse-power transformation technique. The hazard rate function of the IPBH distribution exhibits an increasing shape, a decreasing shape, or an upside-down bathtub shape. The IPBH model can accommodate right-skewed shape, symmetrical shape, reversed J shape, and left-skewed shape densities. Some of its basic mathematical properties are derived. The two parameters of the IPBH distribution are estimated using ten classical and Bayesian estimation approaches. The behavior and performance of these estimators are explored using simulation results. We also determined the best estimation approach using partial and overall ranks for all estimators. As expected, the Bayesian method outperforms other classical methods under the different loss functions. The flexibility and practical importance of the IPBH distribution are explored empirically using two real-life datasets. It is shown that the IPBH distribution has a superior fit compared to the Burr-Hatke distribution and other competing models.

For some possible directions for future studies, the IPBH model can be modified with “polynomial variable transfer” to introduce new model with several free parameters which makes it attractive for analysis and approximation of specific data from different areas such as growth theory, test theory, biostatistics, and computer viruses propagation. Furthermore, different approximation problems related to the “saturation” in Hausdorff sense can be explored for the new model along with some numerical examples using CAS Mathematica to validate the results. More details about these directions can be explored in [20, 21].

Moreover, the T-X family may be applied to define the new inverse-power Burr-Hatke-G family of distributions. Several properties of this new family may be established, its special sub-models may be explored, and their applications in different applied fields may also be addressed.

Acknowledgments

This study was funded by Taif University Researchers Supporting Project (no. TURSP-2020/279), Taif University, Taif, Saudi Arabia.

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