#### Abstract

In this paper, a new three-parameter lifetime distribution, alpha power transformed inverse Lomax (APTIL) distribution, is proposed. The APTIL distribution is more flexible than inverse Lomax distribution. We derived some mathematical properties including moments, moment generating function, quantile function, mode, stress strength reliability, and order statistics. Characterization related to hazard rate function is also derived. The model parameters are estimated using eight estimation methods including maximum likelihood, least squares, weighted least squares, percentile, Cramer–von Mises, maximum product of spacing, Anderson–Darling, and right-tail Anderson–Darling. Numerical results are calculated to compare the performance of these estimation methods. Finally, we used three real-life datasets to show the flexibility of the APTIL distribution.

#### 1. Introduction

The inverse Lomax (IL) is originally developed as a lifetime distribution. The inverse Lomax is member of family of generalized beta distribution. Kleiber and Kotz [1] showed that IL distribution can be used in economics. The IL distribution has many applications in modeling of different trends of hazard rate function (hrf), i.e., decreasing or upside-down bathtub failure rate of life testing of components. The IL distribution is used in [2] to get Lorenz ordering relationship among ordered statistics. McKenzie et al. [3] used the IL model and applied it to geophysical databases. Singh et al. [4] investigated the reliability estimators of IL distribution under Type II censoring. Bayesian estimation of mixture of the IL model under the censoring scheme was studied in [5]. Inverse power Lomax distribution was studied in [6] and Weibull IL distribution in [7].

The probability density (pdf) and cumulative distribution function (cdf) of the IL distribution are as follows:

Mahdavi and Kundu [8] introduced the alpha power transformation (APT) method to add an additional parameter to a family of distributions to increase flexibility in given family. The cdf and pdf of APT-G family areand the corresponding pdf is

In the literature, many probability distributions are generalized using this approach; for example, alpha power transformed Weibull (APTW) distribution in [9], APT generalized exponential distribution in [10], APT Lindley distribution in [11], APT extended exponential distribution in [12], alpha power inverted exponential distribution in [13], alpha power Inverse-Weibull distribution in [14], APT inverse-Lindley distribution in [15], APT power Lindley studied in [16], and APT Pareto distribution proposed in [17].

The main goal of this research article is to introduce a simpler and more flexible model called APT inverse Lomax (APTIL) distribution. Furthermore, the key motivations of using APTIL distribution in the practice are follows:(i)To improve the flexibility of the existing distributions by using APT-G(ii)To introduce the extended version of the IL distribution whose closed form of cdf exist. (iii)To provide better fits than the competing modified models

The rest of the paper is arranged as follows. In Section 2, we define the new model called AP inverse Lomax (APTIL) distribution. Various statistical properties of the APTIL distribution are derived in Section 3 along with more attractive expressions for quantile function, median, mode, moments, order statistics, and stress strength parameter. Lemmas 1 and 2 contain expressions for stochastic ordering and characterization related to hazard function, and the asymptotic behavior of density is derived. In Section 4, we estimate the parameters using eight methods of parameters estimation as follows: maximum likelihood (ML), least squares (LS), weighted least squares (WLS), percentile (PC), Cramer–von Mises (CV), maximum product of spacing (MPS), Anderson–Darling (AD), and right-tail Anderson–Darling (RTAD). Section 5 deals with three applications to show the efficiency of the proposed model. Final remarks are mentioned in Section 6.

#### 2. APTIL Distribution

The random variable (rv) *X* is said to have APTIL distribution denoted by APTIL(*α*, *a*, and *b*) with two shape parameters and one scale as *α*, *a*, and *b*, respectively. The pdf of *X* for is

The survival function (sf) and the hrf for APTIL distribution for *x* > 0 are in the following forms:

Figure 1 demonstrates the graphs of pdf and hazard function of APTIL distribution for different values of *α*, *a*, and *b*. Clearly, the pdf of APTIL distribution is the function for *α* ≠ 1 and *a* < 1 and unimodal for *α* ≠ 1 and *a* > 1. The hrf of the APTIL model can be decreasing or upside-down bathtub for *α* ≠ 1 and *a* < 1 and *a* > 1, respectively.

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##### 2.1. Useful Expansions

Here, an explicit expression for APTIL pdf is given. By using the series representation of exponential function, equation (5) can be written aswhere denotes the pdf of IL distribution given in equation (1) with parameters and *b*.

##### 2.2. Quantile Function and Median

The generation from APTIL distribution is obtained by inverting equation (6):

If *q* uniform (0, 1), then *X* APTIL(*α*, *a*, *b*), the *q*th quantile function of APTIL(*α*, *a*, *b*) is given byand the median can be obtained asand when *α* = 1, the median of the APTIL distribution is equal to the median of the IL distribution. The analysis of shape of distribution can be performed by study of skewness and kurtosis. Using Bowley’s coefficient of skewness as follows:

Moor’s coefficient of kurtosis is given aswhere is the quantile function.

#### 3. Properties of APTIL Distribution

This section deals with some statistical properties of APTIL distribution.

##### 3.1. Mode

The mode of APTIL distribution is derived by :

The mode of APTIL distribution cannot be expressed in the closed form. Computer software, e.g., Mathematica or *R*, can be used to compute the mode of APTIL distribution for specific values of parameters.

For *α* = 1, the mode of IL distribution can be easily calculated from the following equation:

##### 3.2. Asymptotic Behavior

The behavior of APTIL distribution is investigated here as

Lemma 1. *It can be shown that, for APTIL, and *

*Proof. *

##### 3.3. Moments

Theorem 1. *Let X be a rv from APTIL distribution, then its r^{th} moment is*

*Proof. *Let *X* be an r.v. with pdf given in equation (5). For any real number , the *r*^{th} moments of APTIL distribution are obtained asUsing expression from equation (10), we getwhere denotes Euler’s constant and [18]. The mean of *X* can be obtained using equation (20) by putting *r *=* *1:where denotes Euler’s constant. The *r*^{th} central moment of *X* is derived asThe variance of APTIL distribution is easily obtained as

Lemma 2. *Let X be a r. v. with pdf (equation (5)). For any real number the integral,is calculated as*

*Proof of Lemma 2 is given in Appendix.*

##### 3.4. Moment Generating Function

The m.g.f. of *X* is obtained as

*Proof. *The moment generating function can be derived byUsing Lemma 2, the moments, moment generating function, characteristic generating function, and raw moment can be easily obtained by

##### 3.5. Order Statistics

Let be r.sample from the APTIL(*α a*, b) distribution with order statistics . The pdf of rv *X*_{(r)} (*r* = 1, 2, …, *n*) is obtained as

The pdf of can be expressed aswhere . Particularly, pdf of the first and *n*^{th} order statistics can be easily derived from equation (32) asrespectively.

##### 3.6. Stress-Strength Model

Let *X*_{1} and *X*_{2} be two independent random variables with and distributions, respectively. If *X*_{1} represents “stress” and *X*_{2} represents “strength,” the reliability is defined by Then, we can write

Using Lemma 2 from equation (27), we have

The effect of parameters *a*, b, and *α* on mean, variance, skewness, and kurtosis is displayed in Figures 2 and 3, respectively.

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*Remark 1. *(i)The mean and variance of APTIL distribution increase as “*a*” or “*b*” increase for fixed value of *α*(ii)For increasing *α*, the mean of distribution decreases as “*b*” decreases for higher value of “*a*” and increases for lower value of “*a*”(iii)For increasing *α*, the variance of distribution increases as “*b*” increases for lower value of “*a*”(iv)The skewness and kurtosis of APTIL distribution decrease as “*a*” increases or “*b*” decreases for fixed value of *α*

##### 3.7. Characterization Based on Hazard Function

In this section, characterizations of APTIL distribution based on the hazard function are presented. It is known that hazard function *h*(*x*) satisfies the following differential equation:

Theorem 2. *Let X: (0, ∞) be a continuous r. v. with pdf (5) iff its hrf h(x) satisfies the differential equation:under the boundary conditions h(0) ≥ 0.*

*Proof. *If rv *X* has the hrf given in (8), thenNow the result follows.

Conversely if equation (38) holds, thenwhich implies *C* = 0.

#### 4. Estimation of Parameters

The parameters of the APTIL distribution are estimated using various methods including maximum likelihood (ML), least squares (LS), weighted least squares (WLS), percentile (PC), Cramer–von Mises (CV), maximum product of spacing (MPS), Anderson–Darling (AD), and right-tail Anderson–Darling (RTAD) methods of estimation.

##### 4.1. ML Estimation

Let *X*_{1}, …, *X*_{n} have the observed values from APTIL distribution. The MLEs of the proposed model parameters *α*, *a*, and *b* are derived using the log-likelihood function say which is given by

The ML equations of the APTIL distribution are given by

Equating , , and with zeros and solving simultaneously, we obtain the ML estimators of *α*, *a*, and *b*.

##### 4.2. Ordinary and Weighted LS Estimators

Suppose is a random sample from APTIL distribution with corresponding ordered sample of . The mean and variance of APTIL are independent of unknown parameter and are as follows:where is the cdf of APTIL distribution with being the *i*^{th} order statistic. Then, LS estimators ([19]) are obtained by minimizing the SSE:with respect to *α*, *a*, and *b*. So, the LS estimators (LSEs) of the parameters *α*, *a*, and *b* of the APTIL are obtained by minimizing the following:with respect to *α*, *a*, and *b*.

The WLS estimators [19] of *α*, *a*, and *b* can be obtained by minimizing the following expression:with respect to *α*, *a*, and *b*.

##### 4.3. PC Estimator (PCE)

Let be order statistics. Using on PC method of estimation ([20, 21]), the estimators of *α*, *a*, and *b* are derived by minimizing the following:with respect to *α*, *a*, and *b*.

##### 4.4. The Cramer-von Mises Minimum Distance Estimators

The CV method is based on the difference between the estimated cdf and the empirical cdf ([22, 23]). The CV estimators are obtained by minimizing

Macdonald [24] stated about the CV method that it depends on minimum distance estimators providing empirical evidence that the bias of the estimator is smaller than the other minimum distance estimators.

##### 4.5. Maximum Product of Spacing Estimators

The MPS method is a powerful alternative to the ML method for estimating the population parameters of continuous distributions ([25]). Letbe the uniform spacings of a random sample from the APTL distribution, where

The MPS estimator is obtained by maximizing the *geometric mean* (GM) of the spacings:w.r.t. *α*, *a*, and *b*. The MPS estimator of *α*, *a*, and *b* can be obtained by maximizing the logarithm of the GM of sample spacing’s equation (51). No closed solution exists, so the numerical method is used to find the estimates.

##### 4.6. Anderson-Darling and Right-Tail Anderson-Darling Estimators

The method of Anderson–Darling estimation was introduced by [26] in the context of statistical tests. By adapting it to the APTIL model, the Anderson–Darling estimates (ADEs) of *α*, *a*, and *b* can be obtained by minimizing, with respect to *α*, *a*, and *b*, the function given by

Similarly, the right-tail Anderson–Darling estimates (RTADEs) of *α*, *a*, and *b* can be obtained by minimizing, with respect to *α*, *a*, and *b*, the function given by

#### 5. Simulation Study

Here, we come up with a numerical study to compare the behavior of different estimates. We generate 1000 random samples of size *n* = 50, 100, and 200 from the APTIL distribution. Four sets of the parameters are assigned as follows: set1, set2, set3, and set4. The MLE, LSE, WLSE, CVE, PCE, MPSE, ADE, and RTADE of , and are determined. Then, the estimates of all methods and their mean square errors (MSEs) are documented in Tables 1–4.

Form Table 5, for the parameter combinations, we can conclude that the ML estimation method outperforms all the other estimation methods (overall score of 20.5). Therefore, depending on our study, we can consider the ML estimation method is the best for APTIL distribution.

#### 6. Applications

In this section, we utilized three data sets to show that APTIL can be a better life testing distribution compared with some known probability distributions such as APT Weibull (APW) distribution [9], alpha power transformed inverse exponential (APTIE) distribution [10], Marshall Olkin length biased exponential (MOLBE) distribution [27], APT inverted Weibull (APIW) distribution [14], APT Pareto (APTP) distribution [17], and inverse Lomax (IL) distribution [4]. The corresponding probability density functions of competitor models are given below:

The first data set was originally reported by [28] which represents the maximum annual flood discharges of the North Saskachevan in unit of 1000 cubic feet per second, of the North Saskachevan River at Edmonton, over a period of 47 years. The data are as follows: 19.885, 20.940, 21.820, 23.700, 24.888, 25.460, 25.760, 26.720, 27.500, 28.100, 28.600, 30.200, 30.380, 31.500, 32.600, 32.680, 34.400, 35.347, 35.700, 38.100, 39.020, 39.200, 40.000, 40.400, 40.400, 42.250, 44.020, 44.730, 44.900, 46.300, 50.330, 51.442, 57.220, 58.700, 58.800, 61.200, 61.740, 65.440, 65.597, 66.000, 74.100, 75.800, 84.100, 106.600, 109.700, 121.970, 121.970, and 185.560.

The second data set represents the marks in Mathematics for 48 students in the slow pace programme in the year 2013 [29]. The data are as follows: 29, 25, 50, 15, 13, 27, 15, 18, 7, 7, 8, 19, 12, 18, 5, 21, 15, 86, 21, 15, 14, 39, 15, 14, 70, 44, 6, 23, 58, 19, 50, 23, 11, 6, 34, 18, 28, 34, 12, 37, 4, 60, 20, 23, 40, 65, 19, and 31.

The third data consists of a sample of 30 failure times of air-conditioned system of an airplane [30]. The data are as follows: 23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5, 12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, 52, and 95.

For the selection of best fit model, we used the following criteria: Akaike information criterion (AIC), Bayesian information criterion (BIC), Anderson–darling (A∗), and Cramer–von Mises (W∗) test. The maximum likelihood estimates are presented in Table 6, and the goodness of fit measures are presented in Table 7.

We can use the likelihood ratio (LR) test to compare the fit of the ALTIL distribution with other models for given data sets. The form of the test is suggested its namewhere the LR is the ratio of two likelihood functions; the simpler model (*s*) has fewer parameters than the general () model. The LR test rejects the null hypothesis if where denotes the upper 100% point of the distribution.

The shape hazard function for modeling can be analyzed using graphical technique called total time in the test (TTT) plot (for more details, see [31]). From Figure 4, for the first and second data, the TTT plot is concave and provides evidence of the monotonic hazard rate. For the third data set, The TTT plot is convex and according to [31], it provides evidence that the hazard rate is decreasing.

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The APTIL distribution gives the lowest values of AIC, BIC, , and tests among all the fitted models to these data sets. So, it could be selected as the best model among them. The fitted pdf and estimated cumulative distribution function of the APTIL are displayed in Figures 5–7 for the three data sets, respectively.

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The empirical data and estimated density plot show closeness which depict that the APTIL model fits all three data sets well. The APTIL model is compared with other competitive models. The estimated cdf curve of APTIL model also confirms the above results.

The likelihood ratio test is performed to compare APTIL distribution with other fitted models to test *H*_{o} against *H*_{1} discussed above, and results are shown in Table 8. Low values of the likelihood ratio mean that the observed result was much less likely to occur under the null hypothesis as compared with the alternative. We conclude that APTIL distribution provides better fit than all other competitive models at the level of significance ≤0.05.

#### 7. Conclusions

In this research, we proposed and studied the APTIL distribution. Some structural characteristics of the APTIL distribution are derived. The asymptotic behavior of its density function is studied. Characterization related to hazard rate function is also obtained. Estimation of the population parameters is achieved using eight various procedures. Simulation results are carried out to assess the performance of estimators. Real data sets are used for the applications to show the flexibility of the APTIL model.

#### Appendix

#### A. Proof of Lemma 2

By using the series representation of exponential function and Taylor’s series expansion of the function in equation (27), we have

Using the following relation given by [30],

#### B. Code for Simulation

For MLE and MPSE,

For other methods,

#### Data Availability

Data are given within the manuscript, and they are available in Section 6.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was funded by the Deanship of Scientific Research (DSR), King Abdul Aziz University, Jeddah, under grant no. (DF-285-305-1441). The authors gratefully acknowledge the DSR technical and financial support.