Lomax gamble type two distributions with applications to lifetime data

Ijaz, Asim and Alamgir (2019) [11] introduced Lomax exponential distribution and Lomax inverse exponential by Abdulkadir, Jerry and leren (2020) [1] . The Gumbel Type-Two (GTT) distribution has received some further studies in literatures and some of the recent studies are presented in Table1.

Table 1: Extensions of Gumbel Type-two (GTT) Distribution and Authors

Acronyms

Probability Distributions Authors Year LGTT Lomax Gumbel Type-two Proposed 2022 GGTT Gompertz Gumbel Type-two Ogunde, Olalude, Omosigho, 2021 [14] ETGTT Exponentiated Transformation of Gumbel Type-II Sindhu, Shafiq and Al-M dalla 2021 EGTT Extended Gumbel Type-II Ogunde, Fayose, Ajayi, Omosigho 2020 [13] ExGTT Exponentiated Gumbel Type-two Okorie Akpanta and Ohakwe 2016

This present study is motivated by the expected benefits inherent in Lomax-G family of distribution revealed by recent studies of some researchers that have generalized flexible distributions with the aid of the Lomax-Generator. This research will contribute to knowledge from a further extension of Gumbel Type-Two (GTT) distribution and enhance its wide applications and popularity among researchers. The remaining part of the study is organized as follows; section 2 contains the resources for constructing the

LGTT distribution, the definitions and presentations of the lifetime distribution. Some of the properties of the distribution were investigated in section 3 with various graphical description of the distribution while section 4 was devoted for estimating the parameters using the method of maximum likelihood estimation. Section 5 contains several applications for testing the usefuln ess of the distribution and the study was concluded in section 6.

Materials and Methods

The Gumbel Type-two distribution is to be extended using the Lomax-G technique as follows;

Let 𝑋 be a random variable from the Gumbel Type-2 distribution; then the cdf is defined as;

And the associated density function is the derivative of equation ( 1) given as;

Where 𝑘 is scale and 𝜃 the shape parameters of the distribution. The CDF of Lomax-G distribution as defined and studied by Cordeiro et al. (2014) [3] is as follows;

The pdf associated with equation ( 3) is given by;

G ( x ) is the cumulative distribution function (cdf) and g(x) is the probability density function (pdf) of the baseline distribution while 𝛼, 𝛽 are shape parameters.

The Lomax Gumbel Type -Two (LGTT) Distribution

The new distribution is derived by substituting the cdf in equation ( 1) into Lomax-G cdf in equation (3) .The new distribution has cdf derived and presented as;

The density function corresponding to the cdf is given as;

Sub models of LGTT distributions are obtained as follows; i. The Lomax Inverse Exponential distribution (LOMINEXD) defined and studied by Abdulkadir et al. (2020) is a special case of (LGTT) when parameter 𝑘 = 1 Plots of the density function and cdf for various parameter values is presented in Figure 1.

Some Properties of Lomax Gumbel Type-Two Distributions 2.2.1 Reliability Analysis

The Survival Function is given as;

The survival function for the LomGumT2 is derived and given as;

The Hazard Function (Failure Rate) is given as;

Plots of the survival and hazard rate functions are as shown in figure 2 for some values of parameters of the distribution. LGTT for some values of parameters.

Asymptotic Behavior of Lomax Gumbel Type-Two Distribution

The Asymptotic properties of the LGTT distribution is investigated by taking limits of the density function, and hazard rate function as 𝑥 → ∞ 𝑎𝑛𝑑 𝑎𝑠 𝑥 → 0 using theorems (1 to 2)

Theorem 1: Limit of Lomax Gumbel Type -2 density function as 𝑥 → ∞ 𝑖𝑠 0 and as → 0 𝑖𝑠 ∞ Proof:

= ∞ Theorem 2: The limit of LGTT hazard rate function as 𝑥 → ∞ 𝑖𝑠 0 and as 𝑥 → 0 𝑖𝑠 ∞ Proof:

The hazard rate function is given as,

Asymptotes of hazard rate function as

Asymptotes of hazard rate function as 𝑥 → 0 lim 𝑥→0 ℎ ( 𝑥 ) = ∞ Theorem 3: Let 𝑓 ( 𝑥 ) 𝑎𝑛𝑑 ℎ(𝑥) be the probability density and the hazard function of Lomax Gumbel Type -2 distribution respectively, then the asymptotes for the functions 𝑎𝑠 𝑥 → 0 and 𝑥 → ∞ implies 𝑓 ( . ) = ℎ ( . )

Proof:

Results follow from proofs of theorems ( 1) and (2).

The Asymptotes of Lomax Gumbel Type-2 distribution hazard rates h(x) is constant and the distribution also exhibits decreasing failure rates as shown in figure 3. The reliability function revealed a decreasing and upside bath tub shapes. The plots of the hazard also revealed constant and decreasing failure rates.

The Quantile Function and Median

Let 𝑋 be a random variable from the Lomax-Gumbel Type-two distribution having cdf 𝐹 ( 𝑥; 𝛼, 𝛽, 𝑘, 𝜃 ) ; the quantile function is defined by; 𝑢 = 𝑃 ( 𝑋 ≤ 𝑥 𝑢 ) = 𝐹(𝑥), where U is a uniform random variable on U(0,1). The Quantile for the proposed Lomax-Gumbel Type -2 distribution is obtained as;

Simulation is achievable by generating random samples from the Lomax-Gumbel Type-Two distribution using the random variable 𝑋 as follows;

The median is obtained by substituting 𝑢 = 0.5 into 𝑄 ( 𝑢 ) , the median of Lomax-Gu mbel Type-Two distribution is given by

Order Statistics:

Mathematical expression for the order statistics is presented in this section. Suppose we have 𝑋 1 , 𝑋 2 , … , 𝑋 𝑛 as random sample of size 𝑛 from the Lomax Gumbel Type-two distribution with the CDF and PDF given as 𝐹 ( 𝑥 ) 𝑎𝑛𝑑 𝑓 ( 𝑥 ) respectively, let 𝑋 (1) , 𝑋 (2) , … , 𝑋 (𝑛) be the associated order statistics from the random variables in order of magnitude, then the probability density function of the order statistics 𝑋 (𝑠) of observation in the 𝑠 𝑡ℎ position is given by;

Using the pdf 𝑓(𝑥) in equation ( 6) and the corresponding cdf; the distribution of the 𝑠 𝑡ℎ order statistics for Lomax Gumbel Typetwo distribution is obtained as;

The order statistics for smallest observation is derived when 𝑠 = 1 and is given as;

The largest order statistics is derived when 𝑠 = 𝑛

Estimation of Parameters of LGTT Distribution

The method of maximum likelihood estimation (MLE) is utilized for the estimation of parameters of the distribution. Let 𝑋 1 , 𝑋 2 , … , 𝑋 𝑛 be random sample of size 𝑛 from the Lomax Gumbel Type-Two distribution with density function 𝑓(𝑥) as given in equation ( 6) with set of parameters 𝜑 = (𝛼, 𝛽, 𝑘, 𝜃).

𝑓 ( 𝑥

The likelihood function of the distribution is obtained as;

The log Likelihood function 𝑙𝑜𝑔𝐿𝑖𝑘 [ 𝑓 ( 𝑥, 𝜑 )] denoted as 𝐿𝑜𝑔𝐿 is

The next step is to generate derivatives of 𝐿𝑜𝑔𝐿 [ 𝑋/𝛼, 𝛽, 𝑘, 𝜃 ] with respect to the parameters; The parameters can be obtained but the complexity of the functions will require solutions to the above equations to be estima ted through statistical software. Fisher information matrix for generating variances for the confidence intervals for estimated parameters 𝜑 ̂= (𝛼 ̂ 𝛽 ̂ 𝜃 ̂ 𝑘 ̂) can be obtained after second derivates of the normal equations

The 100 (1 -𝜖)% confidence intervals for parameters of Lomax Gumbel Type-2 distribution are provided as follows;

𝛼 ̂± 𝑍 𝜖/2 √𝛪 -1 𝛼𝛼 (𝛩 ̂), 𝛽 ̂± 𝑍 𝜖/2 √𝛪 -1 𝛽𝛽 (𝛩 ̂), 𝜃 ̂± 𝑍 𝜖/2 √𝛪 -1 𝜃𝜃 (𝛩 ̂) and 𝑘 ̂± 𝑍 𝜖/2 √𝛪 -1 𝑘𝑘 (𝛩 ̂)

Where 𝑍 𝜖 is used as the 100 (1 -𝜖)% upper percentile of the standard normal distribution.

Results and Discussions from Application to Real Dataset

Five real life data sets were analyzed in this section to investigate the performance of the LGTT distribution; descriptive statistics of the five datasets are presented in Table 1. LGTT distribution will be compared with GGTT, ExGTT, EGTT and ETGTT which are some notable generalizations of the GTT distribution.

LGTT distribution will also be compared with some related distributions developed using the Lomax generalized families of distribution. MLE of the model parameters are obtained using R statistical software. The goodness of fit measures for selection of the best model to the data are reported for the Akaike Information Criteria (AIC), Bayesian Information Criteria (BIC), the P-values, Kolgomorov Smirnoff (K-S) statistics and the Log-Likelihhod (𝐿𝐿) estimates.

Data 1: Relief Times of Patients on Analgesic

This dataset represents the lifetime's data relating to relief times (in minutes) of 20 patients receiving an analgesic and reported by Gross and Clark (1975). The data was recently analyzed by leren et al. (2019) [9] using LOMINLIND distribution with Log Likelihood (LL) estimates LL=16.3968. The flexibility of LGTT distribution will be assessed by comparing with LOMINLIND, Lomax-Exponential (LomExp) and Lomax-Go mpertz (LomGo) distributions.

. Results of analysis on Table 3 and graphical plots of density function with the estimated cdfs displayed in Figure 3 shows that

LGTT distribution can be considered as better model for the data set than the Lomax Gompertz, Lomax Exponential and LOMINLIND distributions. [16] . Results of analysis on Table 4 and graphical plots of density function with the estimated cdfs displayed in Figure 4 shows th at

LGTT distribution can be considered as better model for the data set than the Lomax Gompertz distribution.

Data 3: Survival times in month's of 20 acute myeloid leukemia patients

The third data represents survival times in months of 20 acute myeloid leukemia patients which was used by Okor ie et al. (2016) [15] for testing the performance of Exponentiated Gumbel (ExGTT) distribution. Results of analysis on Table 5 and graphical plots of density function with the estimated cdfs displayed in Figure 5 shows th at

LGTT distribution can be considered as better model for the data set than other extensions of the GTT distributions.

Data 4: Guinea pigs data

The fourth data represents survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli, reported by Bjerkedal (1960) [2] . It has received several applications and recently by Ogunde et al. (2020) [13] for testing the flexibility of Extended Gumbel Type-two (EGTT) and Ogunde et al. (2021) [14] in Gompertz Gumbel Type-two (GGTT) distributions. Results of analysis on Table 6 and graphical plots of density function with the estimated cdfs displayed in Figure 6 shows th at

LGTT distribution can be considered as better model for the data set than all the competitive distributions derived by extensions of the GTT distribution.

Data 5: Bladder cancer data

This dataset is the remission times (in months) of a random sample of 128 patients with bladder cancer. It has been widely ap plied by notable researchers and recently by leren et al. (2019) [9] , Ogunde et al. (2020) [13] using EGTT and Ogunde et al. (2021) [14] in the study of GGTT distributions. The LGTT distribution is applied to the data and compared with some notable generalizations of the Gumbel Type-two distribution GGTT, EGTT, ExGTT, ETGTT, and GTT distributions. Results of analysis on Table 7 and graphical plots of density function with the estimated cdfs displayed in Figure 7 shows that

LGTT distribution can be considered as better model for the bladder cancer data than other extensions of the GTT distribution s already used for the dataset.

Conclusion of the Study

The Gumbel Type-Two (GTT) was generalized in this study to address the monotonic failure rates and enhance the potential for modeling various types of positively skewed data. The new distribution called Lomax Gumbel Type -Two (LGTT) is positively skewed and close to a bell shape depending on values of parameters. The reliability and hazard rate functions were derived while expressions were obtained for the Quantile function and the order statistics. The new distribution has the Lomax Inverse Exponential distribution as a sub-model. Extensive applications to diverse lifetime datasets were carried out to test the potentials of LGTT. It was compared with (GGTT, EGTT, ExGTT and ETGTT) some popular distributions existing in literatures that were also developed by extension of the GTT as baseline distribution using different methodologies. Results have proved that extension of GTT using the Lomax-G method produced superior performance than the competitive distributions. In addition, the LGTT also shows greater flexibility than some other distribution developed using the Lomax-G.

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