The exponentiated exponential burr XII distribution: Theory and application to lifetime data

Abstract

Lifetime data collected from reliability tests are among data that often exhibit significant heterogeneity caused by variations in manufacturing which make standard lifetime models inadequate. In this paper we introduce a new lifetime distribution derived from T-X family technique called exponentiated exponential Burr XII (EE-BXII) distribution. We establish various mathematical properties. The maximum likelihood estimates (MLE) for the EE-BXII parameters are derived. We estimate the precision of the maximum likelihood estimators via simulation study. Some numerical illustrations are performed to study the behavior of the obtained estimators. Finally the model is applied to a real dataset. We apply goodness of fit statistics and graphical tools to examine the adequacy of the EE-BXII distribution. The importance of this research lies in deriving a new distribution under the name EE-BXII, which is considered the best distributions in analyzing data of life times at present if compared to many distributions in analysis real data.

Keywords

EE-BXII distribution the maximum likelihood method Monte Carlo simulation variance covariance matrix

1 Introduction

Recently, many continuous distributions have been introduced in statistical literature. These distributions, however, are not suitable for the data sets from survival analysis, life testing, reliability, finance and environmental sciences. So, applications of the generalized models to these different sciences are clear requisite.

Generalization of the distribution is the only way to increase the applicability of the parent distribution. The generalized distributions are derived either by inserting a shape parameter or by transform to the parent distribution. So, the generalized distributions will be more suitable than the competing and sub models.

Burr XII distribution is the most commonly used distribution for non-monotonic hazard rates. Burr XII distribution has algebraic tails useful for modeling failures that occur with lesser frequency than those with corresponding models based on exponential tails. Burr XII distribution finds its application in flood frequency, software reliability, structural and wind engineering.

Studied [8] the general class of exponentiated distribution. Discussed [10] the exponentiated Weibull distribution. Many researchers utilized class of exponentiated distribution to create new distributions. For example, defined and studied [13] of exponentiated Frechet distribution, studied [14] the beta Gumbel distribution. Mathematical Problems in Engineering. Discussed [12] the exponentiated Gumbel distribution with climate application. Introduced [15] the beta exponentiated distribution. Studied [4] the beta generalized exponential distribution. Used [6] the Kumaraswamy distribution. Extended [2] the Weibull Pareto distribution. Recently, defined and studied [3], some the exponentiated T – X family of distribution with applications.

In this paper, we derive the EE-BXII distribution from the T-X family technique. The new distribution is obtained by connection between two distributions called T – X distribution. The random variable T is the generator of exponentiated exponential (EE) distribution and X is Burr XII (BXII) distribution.

The contents of the article are structured as follows. Section 2, derives the (EE-BXII) distribution, provides special models, we highlight the natures of density and failure rate functions. The plots are displayed for different parameter values of its probability density function (pdf), cumulative distribution function (cdf), reliability function (rf) and hazard rate function (hrf). We derived the r^th moment in Section 3. In Section 4, we address the maximum likelihood estimation for the EE-BXII parameters. We mention the variance covariance matrix in Section 5. In section 6, we evaluate the precision of the maximum likelihood estimators via simulation study. In section 7, we consider an application to explain the potentiality and utility of the EE-BXII model. We test the competency of the EE-BXII distribution using goodness of fit statistics. Finally, we discuss the results.

2 The (EE-BXII) distribution

For the EE – BXII distribution, the survival, failure rate, cumulative failure rate, density function, are given as follows:

Let r(t) be the pdf of anon – negative continuous random variable T defined on [0, ∞), is exponential distribution as to: $r (t) = β e^{- β t}, β > 0, t > 0 .$ (2-1)

Let F (x) and f (x) denote, respectively the cdf and pdf of a random variable X of Burr XII distribution as follows: $F (x) = 1 - {(1 + x^{α})}^{- k}, x > 0, α, k > 0$ (2-2) $f (x) = α {kx}^{α - 1} {(1 + x^{α})}^{- (k + 1)}, x > 0, α, k > 0 .$ (2-3)

We define the cdf for the EE – BXII distribution for a random variable X using the formula in [3] as follows:

$\begin{matrix} g (x) = c β f (x) {[F (x)]}^{c - 1} {[1 - {(F (x))}^{c}]}^{β - 1}, \\ x > 0, c, β > 0 . \end{matrix}$ (2-4)

Substitute from Equation (2-2) and Equation (2-3) in Equation (2-4), we get the EE-Burr XII, as to:

$\begin{matrix} g (x) = α kc β x^{α - 1} {(1 + x^{α})}^{- (k + 1)} {[1 - {(1 + x^{α})}^{- k}]}^{c - 1} \\ {1 - {[1 - {(1 + x^{α})}^{- k}]}^{c}}^{β - 1}, x > 0, α, c, β and k > 0 . \end{matrix}$ (2-5)

Where α, c, β and k are shape parameters. $\begin{matrix} g (x) = α kc β \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} {(- 1)}^{i + j} (\frac{β - 1}{i}) (\frac{c (i + 1) - 1}{j}) \\ x^{α - 1} {(1 + x^{α})}^{- k (j + 1) - 1}, \\ x > 0, α, c, β and k > 0 . \end{matrix}$ (2-6)

The cdf for EE-BXII distribution is: $\begin{matrix} G (x) = 1 - {1 - {[1 - {(1 + x^{α})}^{- k}]}^{c}}^{β}, \\ x > 0, α, c, β and k > 0 \end{matrix}$ (2-7)

The reliability function of EE- BXII distribution is $\begin{matrix} R (x) = {1 - {[1 - {(1 + x^{α})}^{- k}]}^{c}}^{β}, \\ x > 0, α, c, β andk > 0 . \end{matrix}$ (2-8)

The hazard rate function h (x) of the EE – BXII distribution is given by: $h (x) = \frac{α kc β x^{α - 1} {(1 + x^{α})}^{- (k + 1)} {[1 - {(1 + x^{α})}^{- k}]}^{c - 1}}{1 - {[1 - {(1 + x^{α})}^{- k}]}^{c}},$ (2-9)

Note that the EE-BXII reduces to the BXII distribution when β = c = 1. For β = 1, it becomes the exponentiated Pareto (EP) distribution. For β = c = k = 1 it reduce to the standard Log–Logistic (SLL) distribution. For α = 1, c = θa, we get the kumaraswamy – geralized Exponentiated Pareto (Kw – GEP) distribution. At β = c = α = 1, we get the Lomax (L) distribution or Pareto type II (PII) distribution. The sub models for EE-BXII are given in Table 1.

Table 1

The sub models of the EE-BXII

Model	α	c	K	β
EE-Burr XII	α	c	K	β
BXII	α	1	K	1
EP	α	c	K	1
SLL	α	1	1	1
Kw – GEP	1	θa	K	β
Lomax - PII	1	1	K	1

We plot pdf and failure rate function of the EE-BXII distribution for selected values of the parameters. Plots of the density function (2–5), reliability function (2–8) and hazard function (2–9), for some special values of α, c, β, k are given in Figs 1, 2 and 3, respectively.

Fig. 1

The pdf curves of EE-BXII with (α, c, k, β).

Fig. 2

The reliability curves of the EE – BXII distribution with (α, c, k, β).

Fig. 3

The hazard curves of the EE – BXII distribution with (α, c, k, β).

Figure 1: The pdf of EE-BXII distribution shows in Fig. 1 for various values of parameters α, c, βandk in equation (2–5). Also the pdf can be decreasing, symmetry and right skewness

The reliability function of the EE – BXII distribution shows in Fig. 2, which has decreasing behavior for selected values of parameters.

The hazard function of different parameters values for the EE – BXII distribution shows in Fig. 3 which has decreasing or J- shaped and up-side-down behavior.

3 Statistical properties of the EE–BXII distribution

Here, we present certain mathematical and statistical properties such as the ordinary moments, quantile, median and mode.

In this section, we present the statistical properties of the EE–BXII including the non-central moment, mean and variance of EE – BXII distribution. The r^th moment around zero of a EE – BXII distribution is given by: $\begin{matrix} μ_{r}^{'} = E (X^{r}) = = α ck β \int_{0}^{\infty} x^{α + r - 1} {(1 + x^{α})}^{- (k + 1)} {[1 - {(1 + x^{α})}^{- k}]}^{c - 1} {1 - {[1 - {(1 + x^{α})}^{- k}]}^{c}}^{β - 1} dx, r = 1, 2, \dots \\ = α ck β \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} {(- 1)}^{i + j} (\frac{β - 1}{i}) (\frac{c (i + 1)}{j}) \int_{0}^{\infty} \int_{0}^{\infty} x^{α + r - 1} {(1 + x^{α})}^{- k (j + 1) - 1} dx, \\ μ_{r}^{'} = α ck β \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} {(- 1)}^{i + j} (\frac{β - 1}{i}) (\frac{c (i + 1)}{j}) \frac{⌈ [1 + \frac{r}{α}] ⌈ [k (j + 1) - \frac{r}{α} - 2]}{⌈ [k (j + 1) - 1]} . \end{matrix}$ (3-1)

Subsisting r = 1 in Equation (3-1), we obtain the mean of EE – BXII distribution as follows: $μ^{'} = α kc β \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} {(- 1)}^{i + j} (\frac{β - 1}{i}) (\frac{c (i + 1)}{j}) \frac{⌈ [1 + \frac{1}{α}] ⌈ [k (j + 1) - \frac{1}{α} - 2]}{⌈ [k (j + 1) - 1]} .$ (3-2)

At r = 2 in Equation (3-1), we obtain the second moment for EE – BXII distribution as to: $μ_{2}^{'} = α kc β \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} {(- 1)}^{i + j} (\frac{β - 1}{i}) (\frac{c (i + 1)}{j}) \frac{⌈ [1 + \frac{2}{α}] ⌈ [k (j + 1) - \frac{2}{α} - 2]}{⌈ [k (j + 1) - 1]} .$ (3-3)

By using two Equation (3-2) and Equation (3-3), we get the variance: $\begin{matrix} var (x) = α ck β \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} {(- 1)}^{i + j} (\frac{β - 1}{i}) (\frac{c (i + 1)}{j}) \\ {\frac{⌈ [1 + \frac{2}{α}] ⌈ [k (j + 1) - \frac{2}{α} - 2]}{⌈ [k (j + 1) - 1]} - {(\frac{⌈ [1 + \frac{1}{α}] ⌈ [k (j + 1) - \frac{1}{α} - 2]}{⌈ [k (j + 1) - 1]})}^{2}} \end{matrix}$ (3-4)

If r = 3 in Equation (3-1), we obtain $μ_{3}^{'}$ as to $μ_{3}^{'} = α ck β \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} {(- 1)}^{i + j} (\frac{β - 1}{i}) (\frac{c (i + 1)}{j}) \frac{⌈ [1 + \frac{3}{α}] ⌈ [k (j + 1) - \frac{3}{α} - 2]}{⌈ [k (j + 1) - 1]} .$ (3-5)

And at r = 4 in Equation (3-1), we obtain $μ_{4}^{'}$ as to: $μ_{4}^{'} = α ck β \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} {(- 1)}^{i + j} (\frac{β - 1}{i}) (\frac{c (i + 1)}{j}) \frac{⌈ [1 + \frac{4}{α}] ⌈ [k (j + 1) - \frac{4}{α} - 2]}{⌈ [k (j + 1) - 1]} .$ (3-6)

The kurtosis for EE – BXII distribution is given by use two Equation (3-4) and Equation (3-6) as to: $\begin{matrix} kurt (x) = \frac{⌈ [1 + \frac{4}{α}] ⌈ [k (j + 1) - \frac{4}{α} - 2]}{⌈ [k (j + 1) - 1]} \div α ck β \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} {(- 1)}^{i + j} (\frac{β - 1}{i}) (\frac{c (i + 1)}{j}) \\ \times {\frac{⌈ [1 + \frac{2}{α}] ⌈ [k (j + 1) - \frac{2}{α} - 2]}{⌈ [k (j + 1) - 1]} - {(\frac{⌈ [1 + \frac{1}{α}] ⌈ [k (j + 1) - \frac{1}{α} - 2]}{⌈ [k (j + 1) - 1]})}^{2}} \end{matrix}$ (3-7)

The skewed for EE – BXII distribution is given by use two Equation (3-4) and Equation (3-5) as to: $\begin{matrix} SK (x) = ck β \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} {(- 1)}^{i + j} (\frac{β - 1}{i}) (\frac{c (i + 1)}{j}) \frac{⌈ [1 + \frac{3}{α}] ⌈ [k (j + 1) - \frac{3}{α} - 2]}{⌈ [k (j + 1) - 1]} \div \\ ck β \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} {(- 1)}^{i + j} (\frac{β - 1}{i}) (\frac{c (i + 1)}{j}) {\frac{⌈ [1 + \frac{2}{α}] ⌈ [k (j + 1) - \frac{2}{α} - 2]}{⌈ [k (j + 1) - 1]} \\ - {(\frac{⌈ [1 + \frac{1}{α}] ⌈ [k (j + 1) - \frac{1}{α} - 2]}{⌈ [k (j + 1) - 1]})}^{2}} \frac{3}{2} \end{matrix}$ (3-8)

3.1 The quantile, median and mode of EE – BXII distribution:

If T is variable for we called t_q is quantile EE – BXII distribution, by use (2–7) as to: $x_{q} = {[{(1 - {[1 - {1 - q}^{\frac{1}{β}}]}^{c})}^{- \frac{1}{k}} - 1]}^{\frac{1}{α}} .$ (3-9)

At q = 0.5 in Equation (4-1), we obtain the median of EE – BXII distribution as to: $x_{q} = {[{(1 - {[1 - {1 - 0.5}^{\frac{1}{β}}]}^{c})}^{- \frac{1}{k}} - 1]}^{\frac{1}{α}} .$ (3-10)

The mode of EE - BXII distribution is obtained from equation (2–5) as follows: $\begin{matrix} g^{'} (x) = α kc β \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} {(- 1)}^{i + j} (\frac{β - 1}{i}) (\frac{c (i + 1)}{j}) x^{α - 2} \\ \times {α (1 - k (j + 1) x^{α} + (α - 1) {(1 + x^{α})}^{- k (j + 1) - 1}} = 0 . \end{matrix}$ (3-11)

By equating equation (3–11) with and solving numerically, the mode for EE – BXII is obtained. The following are some results for different values of mean, median, mode Standard deviation (SD), skewness and kurtosis.

From Table 2:

The mean, SD, median and kurtosis are decreasing at α, c, k are constant but β varies, we can see the same result at c, k, β constant but α is various value.

The mean, median and kurtosis are increasing at α, k, β are constant but c is various value, but the SD is decreasing.

The mean, SD, median and kurtosis are decreasing at α, c, β are constant but k is various value.

The mean, median and kurtosis are decreasing at c, β are constant but α, k are various value, but SD is increasing.

The mean, SD, median and kurtosis are decreasing at α, c are constant but k, β various value.

The mean, SD, median and kurtosis are increasing at α, c, k and β are various value.

we note in general the distribution is unimodal and positively skewed.

Table 2

The mean, SD, Mode, Median, Skewness and Kurtosis

α	c	k	β	Mean	SD	Mode	Median	Skewness	Kurtosis
2	1	5	1	0.7761	0.3767	\|–3.002×10^-8	0.14867	17.5122	81.5019
2	1	5	2	0.2165	0.2599	–2.312×10^-8	0.0718	3.9569	10.4821
2	1	5	3	0.1143	0.1846	–2.133×10^-8	0.0473	3.4897	9.7860
4	1	5	1	0.3157	0.3419	7.009×10^-9	0.0718	3.8679	8.3785
5	1	5	1	0.3047	0.3284	6.213×10^-9	0.0087	3.6432	7.6053
2	0.5	5	1	0.4294	0.4549	–1.255×10^-8	0.0592	5.0754	19.2669
2	0.8	5	1	0.6447	0.4371	–3.303×10^-8	0.1153	9.0246	35.9706
2	1	5	1	0.7761	0.3765	–3.003×10^-8	0.1287	17.5122	81.5019
2	1	6	1	0.5615	0.3436	–6.925×10^-8	0.1225	10.1328	34.6881
2	1	7	1	0.4472	0.3114	5.697×10^-8	0.1041	7.8049	23.685
4	1	7	1	0.5891	0.3165	6.959×10^-8	0.1040	11.2318	29.5862
5	1	8	1	0.5613	0.3342	1.846×10^-8	0.0905	8.9515	21.5714
6	1	9	1	0.5441	0.3495	2.697×10^-8	0.0801	7.6818	17.5692
2	1	6	2	0.1794	0.2262	–1.615×10^-8	0.0595	3.8117	10.0123
2	1	7	3	0.0868	0.1468	–1.626×10^-8	0.0336	3.4538	9.7523
1	2	6	3	0.3409	0.2786	–1.745 × 10^-16	0.1062	5.2950	11.6339
4	3	7	4	0.4427	0.2886	5.79 × 10^-9	0.1179	7.4387	16.7381
5	4	8	5	0.5216	0.2889	1.803 × 10^-8	0.1213	10.2123	24.4879

Figures 4 –7 give some plots for the skewness and kurtosis of the EE–BXII distribution versus each parameter values.

Fig. 4

The Plot of Skewness and Kurtosis for selected values of the EE –BXII distribution versus the parameter c.

Fig. 5

The Plot of Skewness and Kurtosis for selected values of the EE –BXII distribution versus the parameter β.

Fig. 6

The Plot of Skewness and Kurtosis for selected values of the EE –BXII distribution versus the parameter α.

Fig. 7

The Plot of Skewness and Kurtosis for selected values of the EE –BXII distribution versus the parameter k.

4 Maximum likelihood estimates of the parameters

Let X₁, X₂, …, X_n be a random sample from EE–BXII distribution with parameters (α, c, k, β). The likelihood function is defined by: $L (θ | x) = \prod_{i = 0}^{n} f (x_{i})$ Where, f (.) is defined in (2–5) and $\underline{θ} = (α, c, k, β)$ . By taking the logarithm of likelihood function, see [3], we have: $\begin{matrix} l = lnlnL = nln [α kc β] + (α - 1) \sum_{i = 0}^{n} lnln [x_{i}] - (k + 1) \sum_{i = 0}^{n} \\ lnln [1 + x_{i}^{α}] + (c - 1) \sum_{i = 0}^{n} lnln [1 - {(1 + x_{i}^{α})}^{- k}] + (β - 1) \sum_{i = 0}^{n} \\ \ln [1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c}] \end{matrix}$ (4-1)

Differentiae Equation (4-1) with respect to α, c, β and k, respectively to have: $l_{j} = \frac{\partial lnL}{\partial θ_{j}} = \frac{1}{L} . \frac{\partial L}{\partial θ_{j}}, j = 1, 2, 3, 4 .$

From Equation (4-1), we have: $\begin{matrix} l_{1} = \frac{\partial l}{\partial α} = \frac{n}{α} + \sum_{i = 0}^{n} x_{i}] - (k + 1) \sum_{i = 0}^{n} \frac{{lnx}_{i}^{α} \ln [x_{i}]}{1 + x_{i}^{α}} \\ - (1 - c) \sum_{i = 0}^{n} \frac{{kx}_{i}^{α} lnln [x_{i}] (1 + x_{i}^{α})^{- (k + 1)}}{1 - {(1 + x_{i}^{α})}^{- k}} \\ + (1 - β) \sum_{i = 0}^{n} \frac{{ckx}_{i}^{α} lnln [x_{i}] (1 + x_{i}^{α})^{- (k + 1)} {(1 - {(1 + x_{i}^{α})}^{- k})}^{c - 1}}{1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c}} \end{matrix}$ (4-2)

$l_{2} = \frac{\partial l}{\partial c} = \frac{n}{c} + \sum_{i = 0}^{n} lnln [1 - {(1 + x_{i}^{α})}^{- k}] - (β - 1) \sum_{i = 0}^{n} \frac{\ln {(1 - {(1 + x_{i}^{α})}^{- k})}^{c} \ln [1 - {(1 + x_{i}^{α})}^{- k}]}{1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c}},$ (4-3)

$\begin{matrix} l_{3} = \frac{\partial l}{\partial k} = \frac{n}{k} - \sum_{i = 0}^{n} lnln [1 + x_{i}^{α}] + (c - 1) \sum_{i = 0}^{n} \frac{{(1 + x_{i}^{α})}^{- k} lnln [1 + x_{i}^{α}]}{1 - {(1 + x_{i}^{α})}^{- k}} \\ - (β - 1) \sum_{i = 0}^{n} \frac{c {(1 + x_{i}^{α})}^{- k} {(1 - {(1 + x_{i}^{α})}^{- k})}^{c - 1} lnln [1 + x_{i}^{α}]}{1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c}}, \end{matrix}$ (4-4) $l_{4} = \frac{\partial l}{\partial β} = \frac{n}{β} + \sum_{i = 0}^{n} lnln [1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c}] .$ (4-5)

Equating, the equations from Equation (4-2) to Equation (4-5) by zero and solving these equations simultaneously yields the maximum likelihood estimates (MLE_s) $(\hat{α}, \hat{c}, \hat{k}, \hat{β})$ of parameters (α, c, k, β). Since these nonlinear equations cannot be solved analytically. Clearly computer programs and numerical method are needed to solve these equations and obtain the parameter estimates. So that statistical software can be used to solve them numerically by means of iterative techniques such as the Newton –Raphson algorithm. The software package Mathematica 9.0 is used for this purpose.

5 The asymptotic variance - covariance matrix

The asymptotic variance –covariance matrix of the estimators $\hat{α}, \hat{c}, \hat{kand} \hat{β}$ are obtained using the second derivatives of the logarithm of likelihood function. In this section, we derive the asymptotic variance- covariance matrix by inverting, $I_{ij} (\underline{θ}$ ), that contains of the variances and covariance of estimates, By neglecting the expectation of the second derivative $I_{ij} (\underline{θ}) = E (- \frac{\partial^{2} lnlnL}{\partial θ_{i} \partial θ_{j}})$ , [see Cohen (1965)]. Unfortunately, the exact mathematical expressions for the above expectation are very difficult to obtained, an accordingly, we have the following approximate variance-covariance matrix.

$F = [I_{ij} (\underline{θ})]^{- 1} i, j = 1, 2, 3, 4, 5, 6$ and $\underline{θ} = (α, c, k, β)$ .

The second partial derivative of the parameters for the EE –BXII are given in the appendix:

5.1 The asymptotic confidence interval

For large sample size, the ML estimators under appropriate regularity conditions are consistent and asymptotically unbiased as well as asymptotically normally distribution. There for, the two-sided approximate 100 (1 - τ)% confidence intervals for the ML estimators say, $\hat{ω}$ of a population value ω can be obtained by $P (- z ⩽ \frac{\hat{ω} - ω}{σ_{\hat{ω}}} ⩽ z) = 1 - τ$ where z is the $100 (1 - \frac{τ}{2})$ th standard normal percentile. The two-sided approximate 100 (1 - τ)% confidence intervals for ω, will be given as follows: $L_{ω} = \hat{ω} - z_{\frac{τ}{2}} σ_{\hat{ω}} and U_{ω} = \hat{ω} + z_{\frac{τ}{2}} σ_{\hat{ω}},$

where $σ_{\hat{ω}}$ is the standard deviation, in this study $\hat{ω}$ is α, c, k, β, rf or hrf, respectively.

6 Simulation study

Simulation studies have performed using Mathematica 9.0 to illustrate the theoretical results of the estimation problem. The performance of the resulting estimators of the parameters has been considered in terms of their bias (Bias) and mean square error (MSE), where: $Bais = \frac{\sum_{i = 1}^{1000} \hat{θ}}{1000} - θ_{0}, MSE = \frac{\sum_{i = 1}^{1000} {(\hat{θ} - θ_{0})}^{2}}{1000}$

Step 1 1000 random sample of size n = 10, 20, 30, 40, 50, 100, 150 and 200 were generated from the EE-BXII distribution. The true parameter value are selected as (a = 1.2, c = 0.6, k = 3, β = 0.2).

Step 2 Newton- Raphson method is used for solving the Equation (4-2) –Equation (4-5) for α, c, k, β, respectively

Step 3 For each sample, the parameters of the distribution are estimated based on complete sample.

Step 4 The Bias and MSE of the estimators for the four parameters for all sample sizes are calculated and tabulated.

Step 5 According to large sample theory, for large sample size, n = 100, 200, 300, we obtained asymptotic variance and covariance of the estimators as described in Section 5.

Step 6 Finally, the approximate confidence limits at 95% for the parameters are computed.

From Table 3, we note that:

As the sample size increases, the MSE_S of the estimated parameters decreases.

The Bias estimator of the parameter α, β decreasing at the sample size is increasing.

Table 3
The Bias and MSE of the parameters ( $\underline{θ} = (α, c, k, β)$

n α c k β

Bias MSE Bias MSE Bias MSE Bias MSE

10 –0.2053 0.7879 15.363 23.024 4.9158 24.1648 0.6528 0.2244

20 –0.2103 0.4422 0.4025 0.1654 –1.1660 1.3596 0.4579 0.2098

30 –0.3442 0.0285 0.1826 0.0233 –1.2164 1.2795 0.0346 0.0198

40 –0.3456 0.0212 0.1744 0.0218 –1.3541 0.4161 0.0254 0.0128

50 –0.5337 0.0202 0.1623 0.0205 –2.1514 0.4086 0.0226 0.0124

100 –0.5642 0.0196 0.1596 0.0194 –3.2542 0.3962 0.0221 0.0120

150 –1.0681 0.0123 0.1324 0.0162 –4.5211 0.3054 0.0213 0.0115

200 –1.0542 0.0112 0.1301 0.0155 –4.8743 0.2981 0.0193 0.0102

n	α	c	k	β
10	–0.2053	0.7879	15.363	23.024	4.9158	24.1648	0.6528	0.2244
20	–0.2103	0.4422	0.4025	0.1654	–1.1660	1.3596	0.4579	0.2098
30	–0.3442	0.0285	0.1826	0.0233	–1.2164	1.2795	0.0346	0.0198
40	–0.3456	0.0212	0.1744	0.0218	–1.3541	0.4161	0.0254	0.0128
50	–0.5337	0.0202	0.1623	0.0205	–2.1514	0.4086	0.0226	0.0124
100	–0.5642	0.0196	0.1596	0.0194	–3.2542	0.3962	0.0221	0.0120
150	–1.0681	0.0123	0.1324	0.0162	–4.5211	0.3054	0.0213	0.0115
200	–1.0542	0.0112	0.1301	0.0155	–4.8743	0.2981	0.0193	0.0102

Fig. 8

Fitted (a) pdf, (b) cdf, (c) Survival and (d) PP plots of the EE-BXII distribution for time data set.

From Table 4, we note that the estimators and width for interval of estimators are decreases when the sample size is increasing.

Table 4

Confidence bounds of the estimates at a confidence level of 0.95

n	Parameters	Estimated mean	Lower boundary	Upper boundary	Width
100	α	1.07894	1.03668	2.19456	1.15788
	c	1.21323	1.06162	1.40324	0.29783
	k	1.14898	1.10541	1.32792	0.26628
	β	1.21986	1.19593	1.32283	0.12690
200	α	0.991207	0.843477	1.82589	0.98241
	c	1.19913	1.47256	1.73885	0.26629
	k	1.13091	1.08682	1.34863	0.26181
	β	1.21807	1.17309	1.23985	0.06676
300	α	0.984651	0.10079	0.99987	0.89908
	c	1.18572	1.05752	1.32385	0.26633
	k	1.20332	1.15126	1.25792	0.10666
	β	1.21016	1.16734	1.23214	0.06482

From Table 5, the variances of estimates are increasing when the sample size increasing for all parameters expect for the parameter β.

Table 5

Asymptotic variance and covariance of estimates matrix

n	Parameters	$\hat{α}$	$\hat{c}$	$\hat{k}$	$\hat{β}$
100	α	0.78486	0.41556	0.25342	0.20017
	c	0.41556	1.40460	0.22388	0.21671
	k	0.25342	0.22388	0.70550	–0.24612
	β	0.20017	0.21671	–0.24612	0.45092
200	α	0.76254	0.41365	0.24852	0.19871
	c	0.41365	1.40041	0.09231	0.07561
	k	0.24852	0.09231	1.70521	–0.24803
	β	0.19871	0.07561	–0.24003	0.45081
300	α	0.75873	0.40672	0.24793	0.19576
	c	0.40679	1.39981	0.03531	0.02573
	k	0.24793	0.03531	1.70451	–0.26146
	β	0.19576	0.02573	–0.26146	0.45021

7 Applications to real data

We consider an application to the failure times for a particular windshield device obtained by [11] for authentication of the flexibility, utility and potentiality of the EE-BXII model. We compare the EE-BXII distribution with models such as BXII, EP, SLL, Kw –GEP, Lomax –PII, McDonald Weibull (McW) in [7], beta Weibull (BW) in [9] and transmuted Marshall-Olkin Fréchet (TMOFr) in [1]. For selection of the optimum distribution, we compute “Cramer-von Mises (W*), Anderson Darling (A*), Akaike information criterion (AIC), corrected Akaike information criterion (CAIC) and Hannan-Quinn information criterion (HQIC)” for all competing and sub distributions. We compute the MLEs, their standard errors (in parentheses) and goodness of fit statistics (GOFs) values for the BXII, EP, SLL, Kw –GEP, Lomax –PII, McW, BW and TMOFr models.

7.1 Aircraft windshield data

Represents failure times of 84 Aircraft Windshield.

The data are mentioned in the Appendix.

Figures 8(a-d) infers that the proposed model is closely fitted to time data. Table 6, Shows the estimates of the parameters and SE for all competitive models.

Table 6
Estimates and SE (in parentheses) for Aircraft Windshield data

Distributions Estimates and SE

EE-Burr XII(α, c, k, β) 0.483 (0.245) 10.812 (8.34) 0.947 (0.75) 246.626 (189.193)

BXII(α, k) 3.171 (0.449) 0.352 (0.06)

EP(α, c, k) 1.563 (0.345) 3.411 (1.128) 1.182 (0.349)

SLL(α) 1.571 (0.137)

Kw –GEP(c, k, β) 4.373 (0.519) 0.26 (0.085) 220.976 (191.998)

Lomax - PII(k) 0.824 (0.0081)

McW (a, b, α, β, c) 17.686 (6.222) 33.639 (19.994) 1.940 (1.011) 0.306 (0.045) 16.721 (9.622)

BW (a, b, α, β) 34.180 (14.838) 11.496 (6.730) 1.360 (1.002) 0.298 (0.060)

TMOFr (α, β, σ, λ) 200.747 (87.275) 1.952 (0.125) 0.102 (0.017) –0.869 (0.101)

Distributions	Estimates and SE
EE-Burr XII(α, c, k, β)	0.483 (0.245)	10.812 (8.34)	0.947 (0.75)	246.626 (189.193)
BXII(α, k)	3.171 (0.449)	0.352 (0.06)
EP(α, c, k)	1.563 (0.345)	3.411 (1.128)	1.182 (0.349)
SLL(α)	1.571 (0.137)
Kw –GEP(c, k, β)	4.373 (0.519)	0.26 (0.085)	220.976 (191.998)
Lomax - PII(k)	0.824 (0.0081)
McW (a, b, α, β, c)	17.686 (6.222)	33.639 (19.994)	1.940 (1.011)	0.306 (0.045)	16.721 (9.622)
BW (a, b, α, β)	34.180 (14.838)	11.496 (6.730)	1.360 (1.002)	0.298 (0.060)
TMOFr (α, β, σ, λ)	200.747 (87.275)	1.952 (0.125)	0.102 (0.017)	–0.869 (0.101)

From the Table 7, it is clear that our proposed model is best fitted, with smallest values for all statistics.

Table 7

Goodness of fit tests for Aircraft Windshield data

Distribution	AIC	CAIC	HQIC	A^*	W^*
EE-Burr XII	281.937	282.443	285.845	1.54655	0.19141
BXII	342.854	343.003	344.809	4.51619	0.65524
EP	324.777	325.077	327.709	4.25024	0.63884
SLL	387.548	387.597	388.525	5.01924	0.53438
Kw –GEP	283.12	283.42	286.052	1.87653	0.24333
Lomax - PII	406.442	406.491	407.419	21.27859	1.44182
McW	283.899	284.669	296.053	1.591	0.199
BW	305.028	305.534	314.751	3.220	0.465
TMOFr	309.472	309.978	319.195	2.404	0.320

8 Conclusion

We derive the EE-BXII distribution from the T-X family procedure. The EE-BXII density can be symmetrical, right-skewed, left-skewed and exponential shapes. The EE-BXII failure rate can take decreasing shapes. We derived certain mathematical and statistical properties such as quantiles, sub-models, moments, and reliability function. We address the maximum likelihood estimation for the EE-BXII parameters. We evaluate the precision of the maximum likelihood estimators via simulation study. We consider an application to aircraft windshield data to elucidate the flexibility, utility and potentiality of the for a repairable items model. We compute goodness of fit for testing the acceptability and competency of the EE-BXII distribution. We ascertain empirically that the proposed model is suitable for aircraft windshield data analysis and has applications in engineering, artificial intelligence engineering, natural science, medicine, and etc.

Data availability

In order to obtain the numerical dataset used to carry out analysis reported in the manuscript, please contact the author (Majdah Badr).

Conflicts of interest

The authors declare no conflicts of interest.

Funds

This project has no funds.

Footnotes

Appendix

(5-1) $\begin{matrix} \frac{\partial^{2} l}{\partial α^{2}} = - \frac{n}{α^{2}} - (1 + k) \sum_{i = 1}^{n} (- \frac{Ln [x_{i}]^{2} x_{i}^{2 α}}{{(1 + x_{i}^{α})}^{2}} + \frac{Ln [x_{i}]^{2} x_{i}^{α}}{1 + x_{i}^{α}}) + \\ (- 1 + c) \sum_{i = 1}^{n} (- \frac{k^{2} Ln [x_{i}]^{2} x_{i}^{2 α} (1 + x_{i}^{α})^{- 2 - 2 k}}{{(1 - {(1 + x_{i}^{α})}^{- k})}^{2}} + \frac{(- 1 - k) kLn [x_{i}]^{2} x_{i}^{2 α} (1 + x_{i}^{α})^{- 2 - 2 k}}{1 - {(1 + x_{i}^{α})}^{- k}} + \frac{kLn [x_{i}]^{2} x_{i}^{2 α} (1 + x_{i}^{α})^{- 1 - k}}{1 - {(1 + x_{i}^{α})}^{- k}}) \\ + (- 1 + β) \sum_{i = 1}^{n} (- \frac{c^{2} k^{2} Ln [x_{i}]^{2} x_{i}^{2 α} (1 + x_{i}^{α})^{- 2 - 2 k} {(1 - {(1 + x_{i}^{α})}^{- k})}^{- 2 + 2 c}}{{(1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c})}^{2}} - \frac{(- 1 + c) {ck}^{2} Ln [x_{i}]^{2} x_{i}^{2 α} (1 + x_{i}^{α})^{- 2 - 2 k} {(1 - {(1 + x_{i}^{α})}^{- k})}^{- 2 + c}}{{(1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c})}^{2}} \\ - \frac{c (- 1 + k) kLn [x_{i}]^{2} x_{i}^{2 α} (1 + x_{i}^{α})^{- 2 - k} {(1 - {(1 + x_{i}^{α})}^{- k})}^{- 1 + c}}{1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c}} - \frac{ckLn [x_{i}]^{2} x_{i}^{α} (1 + x_{i}^{α})^{- 1 - k} {(1 - {(1 + x_{i}^{α})}^{- k})}^{- 1 + c}}{1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c}}) . \end{matrix}$

Similarly, we can see (5-2) $\begin{matrix} \frac{\partial^{2} l}{\partial α \partial c} = \sum_{i = 1}^{n} \frac{kLn [x_{i}] x_{i}^{α} {(1 + x_{i}^{α})}^{- 1 - k}}{1 - {(1 + x_{i}^{α})}^{- k}} + \\ (- 1 + β) \sum_{i = 1}^{n} - (\frac{ckLn [x_{i}] Log [1 - (1 + x_{i}^{α})^{- k}] x_{i}^{α} (1 + x_{i}^{α})^{- 1 - k} {(1 - {(1 + x_{i}^{α})}^{- k})}^{- 1 + 2 c}}{{(1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c})}^{2}} \\ - \frac{kLn [x_{i}] x_{i}^{α} (1 + x_{i}^{α})^{- 1 - k} {(1 - {(1 + x_{i}^{α})}^{- k})}^{- 1 + c}}{1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c}} - \frac{ckLn [x_{i}] Ln [1 - {(1 + x_{i}^{α})}^{- k}] x_{i}^{α} (1 + x_{i}^{α})^{- 1 - k} {(1 - {(1 + x_{i}^{α})}^{- k})}^{- 1 + c}}{1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c}}) . \end{matrix}$ (5-3) $\begin{matrix} \frac{\partial^{2} l}{\partial α \partial k} = \frac{\partial^{2} l}{\partial k \partial α} = - \sum_{i = 1}^{n} \frac{Ln [x_{i}] x_{i}^{α}}{1 + x_{i}^{α}} + (- 1 + c) \sum_{i = 1}^{n} (- \frac{kLn [x_{i}] Ln [1 + x_{i}^{α}] x_{i}^{α} (1 + x_{i}^{α})^{- 1 - 2 k}}{{(1 - {(1 + x_{i}^{α})}^{- k})}^{2}} + \\ \frac{Ln [x_{i}] x_{i}^{α} (1 + x_{i}^{α})^{- 1 - k}}{1 - {(1 + x_{i}^{α})}^{- k}} - \frac{kLn [x_{i}] Ln [1 + x_{i}^{α}] x_{i}^{α} (1 + x_{i}^{α})^{- 1 - k}}{1 - {(1 + x_{i}^{α})}^{- k}}) + \\ (- 1 + β) \sum_{i = 1}^{n} (- \frac{c^{2} kLn [x_{i}] Ln [1 + x_{i}^{α}] x_{i}^{α} (1 + x_{i}^{α})^{- 1 - 2 k}}{{(1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c})}^{2}} - \\ \frac{(- 1 + c) ckLn [x_{i}] Ln [1 + x_{i}^{α}] x_{i}^{α} (1 + x_{i}^{α})^{- 1 - 2 k} {(1 - {(1 + x_{i}^{α})}^{- k})}^{- 2 + c}}{1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c}} - \\ \frac{cLn [x_{i}] x_{i}^{α} (1 + x_{i}^{α})^{- 1 - k} {(1 - {(1 + x_{i}^{α})}^{- k})}^{- 1 + c}}{1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c}} + \frac{ckLn [x_{i}] Ln [1 + x_{i}^{α}] x_{i}^{α} (1 + x_{i}^{α})^{- 1 - k} {(1 - {(1 + x_{i}^{α})}^{- k})}^{- 1 + c}}{1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c}}) . \end{matrix}$ (5-4) $\frac{\partial^{2} l}{\partial α \partial β} = \frac{\partial^{2} l}{\partial β \partial α} = \sum_{i = 1}^{n} - \frac{ckLog [x_{i}] x_{i}^{α} (1 + x_{i}^{α})^{- 1 - k} (1 - (1 + x_{i}^{α})^{- k})^{- 1 + c}}{1 - (1 - (1 + x_{i}^{α})^{- k})^{c}} .$ (5-5) $\frac{\partial^{2} l}{\partial c^{2}} = \frac{n}{c^{2}} + (- 1 + β) \sum_{i = 1}^{n} (- \frac{Ln {[1 - (1 + x_{i}^{α})^{- k}]}^{2} (1 - (1 + x_{i}^{α})^{- k})^{2 c}}{{(1 - (1 - (1 + x_{i}^{α})^{- k})^{c})}^{2}} - \frac{Ln {[1 - (1 + x_{i}^{α})^{- k}]}^{2} (1 - (1 + x_{i}^{α})^{- k})^{c}}{1 - (1 - (1 + x_{i}^{α})^{- k})^{c}}) .$ (5-6) $\begin{matrix} \frac{\partial^{2} l}{\partial c \partial k} = \frac{\partial^{2} l}{\partial k \partial c} = \sum_{i = 1}^{n} \frac{kLog [x_{i}] x_{i}^{α} (1 + x_{i}^{α})^{- 1 - k}}{1 - {(1 + x_{i}^{α})}^{- k}} + (- 1 + β) \sum_{i = 1}^{n} (- \frac{ckLog [x_{i}] Log [1 - {(1 + x_{i}^{α})}^{- k}] x_{i}^{α} (1 + x_{i}^{α})^{- 1 - k} {(1 - {(1 + x_{i}^{α})}^{- k})}^{- 1 + 2 c}}{{(1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c})}^{2}} - \\ \frac{kLog [x_{i}] x_{i}^{α} (1 + x_{i}^{α})^{- 1 - k} {(1 - {(1 + x_{i}^{α})}^{- k})}^{- 1 + c}}{1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c}} - \frac{ckLog [x_{i}] Log [1 - {(1 + x_{i}^{α})}^{- k}] x_{i}^{α} (1 + x_{i}^{α})^{- 1 - k} {(1 - {(1 + x_{i}^{α})}^{- k})}^{- 1 + c}}{1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c}}) . \end{matrix}$ (5-7) $\begin{matrix} \frac{\partial^{2} l}{\partial c \partial β} = \frac{\partial^{2} l}{\partial β \partial c} = \sum_{i = 1}^{n} \frac{Ln [1 + x_{i}^{α}] (1 + x_{i}^{α})^{- k}}{1 - {(1 + x_{i}^{α})}^{- k}} + (- 1 + β) \sum_{i = 1}^{n} (- \frac{cLn [1 + x_{i}^{α}] Ln [1 - {(1 + x_{i}^{α})}^{- k}] (1 + x_{i}^{α})^{- k} {(1 - {(1 + x_{i}^{α})}^{- k})}^{- 1 + 2 c}}{{(1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c})}^{2}} - \\ \frac{Ln [1 + x_{i}^{α}] (1 + x_{i}^{α})^{- k} {(1 - {(1 + x_{i}^{α})}^{- k})}^{- 1 + c}}{1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c}} - \frac{cLn [1 + x_{i}^{α}] Ln [1 - {(1 + x_{i}^{α})}^{- k}] (1 + x_{i}^{α})^{- k} {(1 - {(1 + x_{i}^{α})}^{- k})}^{- 1 + c}}{1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c}}) . \end{matrix}$ (5-8) $\frac{\partial^{2} l}{\partial k^{2}} = \sum_{i = 1}^{n} - \frac{Ln [1 - {(1 + x_{i}^{α})}^{- k}] {(1 - {(1 + x_{i}^{α})}^{- k})}^{c}}{1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c}}$ (5-9) $\frac{\partial^{2} l}{\partial k \partial β} = \frac{\partial^{2} l}{\partial β \partial k} = \sum_{i = 1}^{n} - \frac{ckLn [x_{i}] x_{i}^{α} {(1 + x_{i}^{α})}^{- 1 - k} {(1 - {(1 + x_{i}^{α})}^{- k})}^{- 1 + c}}{1 - {(1 - {(1 + x_{i}^{α})}^{- k})}^{c}}$ (5-10) $\frac{\partial^{2} l}{\partial β^{2}} = - \frac{n}{β^{2}}$

Aircraft Windshield Data: represents failure times of 84 Aircraft Windshield. The data are:

0.040, 1.866, 2.385, 3.443, 0.301, 1.876, 2.481, 3.467, 0.309, 1.899, 2.610, 3.478, 0.557, 1.911, 2.625, 3.578, 0.943, 1.912, 2.632, 3.595, 1.070, 1.914, 2.646, 3.699, 1.124, 1.981, 2.661, 3.779, 1.248, 2.010, 2.688, 3.924, 1.281, 2.038, 2.82,3, 4.035, 1.281, 2.085, 2.890, 4.121, 1.303, 2.089, 2.902, 4.167, 1.432, 2.097, 2.934, 4.240, 1.480, 2.135, 2.962, 4.255, 1.505, 2.154, 2.964, 4.278, 1.506, 2.190, 3.000, 4.305, 1.568, 2.194, 3.103, 4.376, 1.615, 2.223, 3.114, 4.449, 1.619, 2.224, 3.117, 4.485, 1.652, 2.229, 3.166, 4.570, 1.652, 2.300, 3.344, 4.602, 1.757, 2.324, 3.376, 4.663.

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n	α		c		k		β
	Bias	MSE	Bias	MSE	Bias	MSE	Bias	MSE
10	–0.2053	0.7879	15.363	23.024	4.9158	24.1648	0.6528	0.2244
20	–0.2103	0.4422	0.4025	0.1654	–1.1660	1.3596	0.4579	0.2098
30	–0.3442	0.0285	0.1826	0.0233	–1.2164	1.2795	0.0346	0.0198
40	–0.3456	0.0212	0.1744	0.0218	–1.3541	0.4161	0.0254	0.0128
50	–0.5337	0.0202	0.1623	0.0205	–2.1514	0.4086	0.0226	0.0124
100	–0.5642	0.0196	0.1596	0.0194	–3.2542	0.3962	0.0221	0.0120
150	–1.0681	0.0123	0.1324	0.0162	–4.5211	0.3054	0.0213	0.0115
200	–1.0542	0.0112	0.1301	0.0155	–4.8743	0.2981	0.0193	0.0102