Transmuted Lower Record Type Fréchet Distribution with Lifetime Regression Analysis Based on Type I-Censored Data

Caner Tanış; Buğra Saraçoğlu; Coşkun Kuş; Ahmet Pekgör; Kadir Karakaya

doi:10.2991/jsta.d.210115.001

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Volume 20, Issue 1, March 2021, Pages 86 - 96

Transmuted Lower Record Type Fréchet Distribution with Lifetime Regression Analysis Based on Type I-Censored Data

Authors

Caner Tanış¹^{, *}, Buğra Saraçoğlu², Coşkun Kuş², Ahmet Pekgör³, Kadir Karakaya²

¹Department of Statistics, Faculty of Science, Cankiri Karatekin University, Cankiri, 18100, Turkey

²Department of Statistics, Faculty of Science, Selçuk University, Konya, 42250, Turkey

³Department of Statistics, Faculty of Science, Necmettin Erbakan University, Konya, 42090, Turkey

^*Corresponding author. Email: caner.tanis@gmail.com

Corresponding Author

Caner Tanış

Received 15 April 2019, Accepted 20 November 2020, Available Online 25 January 2021.

DOI: 10.2991/jsta.d.210115.001 How to use a DOI?
Keywords: Fréchet distribution; Lower records; Point estimates; Lifetime regression
Abstract: This paper introduces a new lifetime distribution by mixing the first two lower record values and various distributional properties are examined. Statistical inference on distribution parameters are discussed with five estimators. A Monte Carlo simulation study is carried out to evaluate the risk behavior of these estimators for different sample of sizes. The distribution modeling analysis is provided based on real data to demonstrate the fitting ability of the proposed model. In addition, a lifetime regression model is described by re-parameterization on the log lifetimes. The superiority of proposed regression model is revealed in well-known models.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

In two decades, many statistical distributions have been brought into the literature. These distributions are obtained as a member of the family of distributions. Transmuted families can be given as an example to the family of distributions. In general, transmuted families are given based on order statistics. The transmuted distributions has been introduced by Shaw and Buckley [1,2] using a quadratic transmutation map. Order statistics, transmuted distributions can be sorted as [3–5]. Recently, Balakrishnan [6] proposed a new record based transmuted family of distributions. Tanış and Saraçoğlu [7] examined a special model based on Weibull distribution of record based transmuted family of distributions. In terms of both distributional properties and statistical inference, in parallel with the study of [6].

In our study, we obtained the transmuted version of the Fréchet distribution based on lower record values. In Section 2, a transmutation lower record type map of order 2 is introduced. In Section 3, we have suggested a sub-model called transmuted lower record type Fréchet (TLRT-F) distribution and the some distributional properties are obtained such as moments, incomplete moments, Bonferroni and Lorenz curves. In Section 4, the unknown parameters are estimated by five methods including maximum likelihood estimators, least squares estimators, weighted least squares estimators, Anderson-Darling estimators, and Cramer von-Mises estimators. A simulation study is performed in order to compare the performance of these estimators in terms of mean squared errors (MSEs) and bias. A lifetime regression is introduced based on TLRT-F distribution in Section 5. In Sections 6 and 7, two applications with real data are presented to show the applicability of introduced distribution.

2. TRANSMUTATION LOWER RECORD TYPE MAP OF ORDER 2

Let XL1 and XL2 be the lower record values from a population with cumulative distribution function (cdf) Gx. Let us define a new random variable based on these records:

Y=XL1,U<p XL2,U>p,(2.1)

where U is standard uniform random variable and p∈0,1. The cdf of Y is written by

Fx=pPXL1≤x+1−pPXL2≤x =pGx+1−pGx1−logGx =Gx1−plogGx. (2.2)

It is noticed that the distribution with cdf (2.2) is called “transmuted lower record type distribution (TLRT).” Using (2.2), the probability density function (pdf) and hazard function(hf) of TLRT distribution are given by

fx=gx1−p1+logGx,(2.3)

and

hx=gx1−p1+logGx1−Gx1−plogGx,(2.4)

respectively.

3. TLRT-F DISTRIBUTION AND DISTRIBUTIONAL PROPERTIES

In this section, we investigate the sub-model of TLRT family. Let X be a random variable having Fréchet distribution. The cdf and pdf of X are defined by

Gx=exp−βx α,(3.1)

and

gx=αβ αx−α+1exp−βx α, x>0(3.2)

respectively, where α>0 is a shape parameter and β>0 is scale parameter. Substituting the cdf (3.1) and pdf (3.2) into (2.2) and (2.3), the following cdf and pdf are obtained as

Fx;θ=exp−βx α1+pβx α,(3.3)

and

fx;θ=αβ αx−α+1exp−βx α1−p+pβx α, x>0(3.4)

where α,β>0,p∈0,1, and θ=α,β,p. The distribution with cdf (3.3) is called TLRT-F θ distribution.

The hf of TLRT-F distribution is given by

hx;θ=αβ αx−α+1exp−βx α1−p+pβx α1−exp−βx α1+pβx α.(3.5)

Figures 1–2 illustrates the possible shapes of pdf and hf for selected parameters. The quantile function of the TLRT-F is given by

Qq;θ=β−pWqexp−1∕pp+1p −1α,(3.6)

where q∈0,1 and W⋅ is a Lambert function. Using (3.6), the median can be easily written by

Q0.5;θ=β−pWexp−1∕p2p+1p −1α.

For r∈ℕ+, the rth moment of TLRT-F distribution is given by

EXr=1−pβ rΓ1−rα+pβ rΓ2−rα,(3.7)

where Γ⋅ is a gamma function. Using (3.7), the mean and variance of TLRT-F distribution are obtained by

EX=1−pβΓ1−1α+pβΓ2−1α,

and

VarX=1−pβ 2Γ1−2α+pβ 2Γ2−2α −1−pβΓ1−1α+pβΓ2−1α 2,

respectively. The moment generating function of TLRT-F distribution is also given by

Mt=1−p∑r=0∞trr!β rΓ1−rα+p∑r=0∞trr!β rΓ2−rα.

The incomplete moments of TLRT-F distribution is obtained by

mry=1−pβ rΓ1−rα,βy α+pβ rΓ2−rα,βy α,(3.8)

where Γa,x is incomplete gamma function defined by

Γa,x=∫x∞ta−1e−tdt.

The Bonferroni and Lorenz curves for TLRT-F are given, respectively, by

Bπ=1−pβ rΓ1−rα,βν α+pβ rΓ2−rα,βν απ1−pβΓ1−1α+pβΓ2−1α,(3.9)

and

Lπ=1−pβ rΓ1−rα,βν α+pβ rΓ2−rα,βν α1−pβΓ1−1α+pβΓ2−1α,(3.10)

where ν=Qπ;θ and Q is quantile function defined in (3.6).

Stochastic and the other ordering are important means for evaluating the comparative properties for a positive continuous random variable. The following theorem shows that the TLRT-F random variables can be ordered with respect to the likelihood ratio.

Theorem 3.1.

Let X~ TLRT-F α,β,p1 and Y~TLRT−Fα,β,p2. If p1<p2 then X is smaller than Y in the likelihood ratio order, i.e., the ratio function of the corresponding pdfs is decreasing in x.

Proof.

For any x>0, the ratio of the densities is given by

gx=1−p1+p1βx α1−p2+p2βx α.

Consider the derivative of loggx in x

dloggxdx=αβ αp1−p2xα−11−p1+p1βx α1−p2+p2βx α<0

for p1<p2 and hence proof is completed.

Corollary 3.1.

It follows from [8] that X is also smaller than Y in the hazard ratio, mean residual life and stochastic orders under the conditions given in Theorem 3.1.

In order to generate the data from TLRT-F distribution, an acceptance-rejection (AR) sampling method is given in the following algorithm. In this algorithm, the Weibull distribution is chosen as a proposal distribution. The AR algorithm is given as follows:

Algorithm 1:

Generate data on random variable Y from Weibull distribution with pdf g given as follows:
gz;α,β=αβyβzα−1exp−yβzα.
Generate U from standard uniform distribution(independent of Y).
If
U<fY;θk×gY;α,β
then set X=Y (“accept”); otherwise go back to A1 (“reject”), where pdf f is given as in (3.4) and
k=maxz∈ℝ +fz;θgz;α,β.

The output of this algorithm suggests a random data on X from TLRT-F θ. It is noticed that the Algorithm 1 is used for all simulations in the paper.

4. STATISTICAL INFERENCE ON DISTRIBUTION PARAMETERS

In this section, we propose five estimators to estimate the unknown parameters of the TLRT-F θ distribution. They are the maximum likelihood, least squares, weighted least squares, Cramér-von Mises type and Anderson–Darling type estimates. A simulation study is performed to observe the performances of the methods discussed here.

4.1. Point Estimation

Let x1,x2,…,xn be a realization of a random sample from the TLRT-F θ distribution and x1<x2<⋯<xn denotes the corresponding observed order statistics. Then, the log-likelihood function is written by

ℓθ=nlogα+nαlogβ−α+1∑i=1nlogxi −∑i=1nβxi α+∑i=1nlog1−p+pβxi α. (4.1)

Hence, the maximum likelihood estimate (MLE) of θ is given by

θ^1=argmaxθℓθ,

where θ=α,β,p is the parameter vector and θ^1=α^,β^,p^ is MLE of θ. Let us define the following functions which are used to define the different types of estimators:

QLSθ=∑i=1nFxi−in+12, QWLSθ=∑i=1n(n+2)(n+1)2i(n−i+1)Fxi−in+12, QCvMθ=112n+∑i=1nFxi−2i−12n2,

and

QADθ=−n−1n∑i=1n2i−1logFxi+1n∑i=1nlog1−Fxi,

where F⋅ is cdf of TLRT-F θ distribution given in (3.3). Then, the least squares estimator (LSE), weighted least squares estimator (WLSE), Anderson–Darling estimator (ADE) and the Cramér-von Mises estimator (CvME) of θ are given, respectively, by

θ^2=argminθQLSθ,(4.2)

θ^3=argminθQWLSθ,(4.3)

θ^4=argminθQADθ,(4.4)

θ^5=argminθQCvMθ. (4.5)

The optimization problems can be solved by some numerical methods such as Nelder-Mead, BFGS, L-BFGS-B or CG. These methods can be easily employed by optim function in R.

4.2. Simulation Study for Point Estimates

In the simulation study, 5000 trials are used to estimates the bias and MSE of the MLEs, LSEs, WLSEs, ADEs and CVMEs. The sample sizes are considered as n = 50, 100, 250, 500, 750 and 1000. Two parameter settings are considered. The results are given in the Tables 1 and 2. In the simulation, the samples are generated from TLRT-F θ distribution by using AR sampling given in Algorithm 1. The numerical methods BFGS, Nelder-Mead, CG and L-BFGS-B are used to achieve the values of estimates by optimizing the objective functions (4.1)–(4.5). The values obtained from the numerical method that optimizes the objective functions are selected as parameter estimates. The bias and MSEs of estimators are presented in Tables 1 and 2. From Tables 1 and 2, it is observed that the bias and MSEs of the all estimators decrease to zero as expected. The WLSEs and ADEs have similar MSEs for all sample of size and they are better than the others. The MSE of MLEs tends to MSE of WLSEs and ADEs for large sample size.

		Bias			MSE
n		α	β	p	α	β	p
MLEs	50	0.0986	−0.0534	−0.1802	0.0367	0.0183	0.0995
	100	0.0587	−0.0299	−0.1155	0.0234	0.0155	0.0803
	250	0.0333	−0.0155	−0.0761	0.0158	0.0126	0.0648
	500	0.0235	−0.0113	−0.0585	0.0115	0.0097	0.0503
	750	0.0243	−0.0139	−0.0573	0.0095	0.0078	0.0421
	1000	0.0209	−0.0123	−0.0517	0.0084	0.0069	0.0385

LSEs	50	−0.0028	0.0326	−0.0241	0.0318	0.0442	0.0645
	100	0.0049	0.0139	−0.0259	0.0214	0.0223	0.0614
	250	0.0044	0.0120	−0.0165	0.0155	0.0152	0.0455
	500	0.0129	−0.0014	−0.0334	0.0112	0.0103	0.0402
	750	0.0140	−0.0034	−0.0311	0.0094	0.0080	0.0314
	1000	0.0126	−0.0030	−0.0293	0.0087	0.0071	0.0294

WLSEs	50	0.0150	0.0031	−0.0598	0.0251	0.0231	0.0590
	100	0.0142	0.0024	−0.0410	0.0179	0.0161	0.0503
	250	0.0102	0.0029	−0.0284	0.0125	0.0113	0.0404
	500	0.0122	−0.0022	−0.0314	0.0094	0.0083	0.0333
	750	0.0146	−0.0060	−0.0336	0.0077	0.0065	0.0280
	1000	0.0135	−0.0057	−0.0322	0.0072	0.0059	0.0264

ADEs	50	0.0205	0.0078	−0.0548	0.0254	0.0267	0.0562
	100	0.0183	−0.0027	−0.0467	0.0162	0.0139	0.0533
	250	0.0114	0.0021	−0.0308	0.0127	0.0113	0.0436
	500	0.0104	−0.0007	−0.0277	0.0092	0.0082	0.0332
	750	0.0121	−0.0036	−0.0282	0.0075	0.0064	0.0268
	1000	0.0118	−0.0040	−0.0284	0.0070	0.0058	0.0258

CvMEs	50	0.0261	0.0320	−0.0211	0.0371	0.0469	0.0738
	100	0.0195	0.0134	−0.0254	0.0240	0.0237	0.0669
	250	0.0105	0.0115	−0.0172	0.0165	0.0157	0.0482
	500	0.0163	−0.0018	−0.0343	0.0118	0.0105	0.0417
	750	0.0164	−0.0038	−0.0321	0.0098	0.0082	0.0325
	1000	0.0145	−0.0034	−0.0302	0.0090	0.0073	0.0302

MSE, mean squared error; MLE, maximum likelihood estimate; LSE, least squares estimator; WLSE, weighted least squares estimator; ADE, Anderson Darling estimator; CvME, Cramér-von Mises estimator.

Table 1

Average bias and MSEs of the estimates for the true parameters θ=1,0.5,0.7.

		Bias			MSE
n		α	β	p	α	β	p
MLEs	50	0.5643	−0.1473	−0.2707	0.6119	0.0448	0.1357
	100	0.3529	−0.0996	−0.1791	0.3561	0.0305	0.0899
	250	0.1590	−0.0445	−0.0820	0.1522	0.0151	0.0440
	500	0.0748	−0.0203	−0.0400	0.0759	0.0081	0.0230
	750	0.0483	−0.0126	−0.0242	0.0485	0.0053	0.0144
	1000	0.0317	−0.0076	−0.0139	0.0325	0.0036	0.0087

LSEs	50	0.3154	−0.0816	−0.1645	0.4937	0.0476	0.1010
	100	0.2540	−0.0614	−0.1239	0.3698	0.0361	0.0813
	250	0.1667	−0.0378	−0.0787	0.2271	0.0219	0.0553
	500	0.0999	−0.0229	−0.0496	0.1338	0.0132	0.0340
	750	0.0720	−0.0164	−0.0347	0.0940	0.0093	0.0242
	1000	0.0474	−0.0100	−0.0217	0.0672	0.0068	0.0169

WLSEs	50	0.3550	−0.1027	−0.1956	0.4285	0.0402	0.0995
	100	0.2705	−0.0747	−0.1384	0.3017	0.0284	0.0684
	250	0.1533	−0.0398	−0.0753	0.1610	0.0157	0.0404
	500	0.0813	−0.0208	−0.0414	0.0869	0.0088	0.0226
	750	0.0528	−0.0130	−0.0251	0.0546	0.0055	0.0136
	1000	0.0327	−0.0074	−0.0141	0.0360	0.0038	0.0083

ADEs	50	0.3749	−0.1044	−0.1930	0.4107	0.0361	0.1043
	100	0.2808	−0.0775	−0.1377	0.2808	0.0254	0.0620
	250	0.1515	−0.0391	−0.0721	0.1495	0.0143	0.0367
	500	0.0796	−0.0202	−0.0394	0.0809	0.0082	0.0208
	750	0.0525	−0.0129	−0.0245	0.0528	0.0054	0.0133
	1000	0.0330	−0.0074	−0.0138	0.0352	0.0037	0.0082

CvMEs	50	0.4034	−0.0813	−0.1560	0.6166	0.0502	0.1073
	100	0.2947	−0.0611	−0.1193	0.4201	0.0375	0.0853
	250	0.1814	−0.0376	−0.0767	0.2425	0.0225	0.0574
	500	0.1064	−0.0227	−0.0483	0.1395	0.0135	0.0350
	750	0.0757	−0.0161	−0.0335	0.0965	0.0094	0.0247
	1000	0.0498	−0.0097	−0.0206	0.0684	0.0069	0.0171

MSE, mean squared error; MLE, maximum likelihood estimate; LSE, least squares estimator; WLSE, weighted least squares estimator; ADE, Anderson Darling estimator; CvME, Cramér-von Mises estimator.

Table 2

Average bias and MSEs of the estimates for the true parameters θ=3,2,0.9.

5. TLRT-F REGRESSION ANALYSIS

The log-location-scale regression models are studied by several authors such as [9–12]. In this section, we describe the usage of log-location-scale TLRT-F regression analysis.

Let X be a TLRT-F θ random variable. Let us consider a reparameterization by β=expμ and α=1∕σ. Thus, the log-lifetime Y=logX is a random variable with the pdf and cdf

hy;τ=exp−y−μσexp−exp−y−μσ1−p+pexp−y−μσσIℝy,(5.1)

and

Hy;τ=exp−exp−y−μσ1+pexp−y−μσ,(5.2)

where τ=μ,σ,p,μ and σ are location and scale parameters, respectively.

The distribution of Y with cdf (5.2) is denoted log-TLRT-F μ,σ,p. Let us consider the regression model

Y=μ+σε,(5.3)

where Y1,Y2,…,Yn are random sample from log-TLRT-F μi=ZiTβ,σ,p distribution, and Y=Y1,Y2,…,YnT. Furthermore, β=β1,…,βpT,μ=μ1,…,μnT,ε =ε1,…,εnT,μi=Zi Tβ and Zi=Zi1,…,ZiqT are ith values of covariates for i=1,2,…,n. In addition, ε i=Yi−μ i∕σ for i=1,2,…,n is a random error from log-TLRT-F μ=0,σ=1,p. In this model, the location parameter is assumed as a linear function of covariates Zi.

Let us discuss the MLEs of parameters η=β,σ,p in the model (5.3) under Type-I right censoring. Suppose that the log lifetimes Yii=1,2,…,n are Type-I right censored (at logci) from log-TLRT-F μ i,σ,p, where ci is censoring time for lifetime Xi. Let us define

Ti=minYi,logci, i=1,2,…,n.

Hence, the log-likelihood function based on the Type-I right censored sample T1,T2,…,Tn is written by

ℓη=∑i=1nω iloghti;ZiTβ,σ,p+1−ω ilog1−Hti;ZiTβ,σ,p,(5.4)

where

ω i=0,Ti>logci 1,Ti≤logci

is an indicator function and ti denotes the observed value of Ti,i=1,2,…,n.

The MLE of η can be obtained by maximizing the log-likelihood given in (5.4). Some numerical methods such as Nelder-Mead, BFGS, CG or L-BFGS-B can be used for the maximization problem. These methods are available in R.

6. REAL DATA APPLICATION

In this section, we report a data modeling analysis for the glass fibers data. For the comparison issue, we consider TLRT-F, Fréchet (Fr), transmuted Log-logistic (TLL) [13], Weibull (W), exponentiated Exponential (EE) [14], transmuted Weibull (TW) [15], Lindley (L), Gompertz (G) distributions. The pdf of these distributions are given in Table 3. Table 4 presents the MLEs (standard errors) and Table 5 contains −2×log-likelihood value, Akaike's information criteria (AIC), Kolmogorov–Smirnov test statistic (KS), Anderson–Darling statistic (A*), Cramer von–Mises statistic (CVM) and p values based on these statistics for the all distributions given in Table 3. Figure 3 shows that the fitted cdfs for glass fibers data. According to results in Table 5, it is observed that the TLRT-F has minimum KS, A* and CVM values. Hence, it can be said that the TLRT-F can be a good alternative to modeling real data.

fFrα,β=αβαx−α+1exp−βxα,		α,β>0
fTLLα,β,λ=eαxβ−11+eαxβ−λeαxβ−11+eαxβ3,		α,β>0,λ∈−1,1
fWα,β=αβxβα−1exp−xβα,		α,β>0
fEEθ,α=θαe−αx1−e−αxθ−1,		α,θ>0
fTWα,β,λ=αβxβα−1exp−xβα1+λ−2λexp−xβα,		α,β>0,λ∈−1,1
fLθ=θ2θ+11+xe−θx,		θ>0
fGa,b=aebxexp−abebx−1,		a,b>0

Table 3

List of the lifetime distribution to modeling real data.

Distribution	MLEs
TLRT-F	α̂=4.09080.5563,β̂=1.67640.0879,p̂=0.86090.1689
Fr	α̂=5.43780.5192,β̂=1.41080.0344
TLL	α̂=1.61030.1265,β̂=8.07420.0704,λ̂=0.40560.6049
W	α̂=3.06200.2403,β̂=1.78750.0784
EE	θ̂=165.996884.3295,α̂=3.55570.3665
TW	α̂=3.45050.2645,β̂=2.07880.0868,λ̂=0.85120.1146
L	θ̂=0.94590.0931
G	â=0.17910.0407,b̂=1.05080.1173

MLE, maximum likelihood estimate; TLRT-F, transmuted lower record type Fréchet; TLL, transmuted Log-logistic; W, Weibull, EE, exponentiated exponential; TW, transmuted Weibull; L, Lindley; G, Gompertz.

Table 4

MLEs (standard errors) for glass fibers data.

Distribution	−2LogL	AIC	KS	p-value (KS)	A*	p-value (A*)	CVM	p-value (CVM)
TLRT-F	38.9698	44.9698	0.0662	0.9283	0.4285	0.8195	0.0563	0.8389
Fr	40.1277	44.1277	0.0772	0.8187	0.5291	0.7167	0.0699	0.7540
TLL	44.5394	50.5394	0.0755	0.8383	0.5586	0.6873	0.0460	0.9018
W	92.7338	96.7338	0.2051	0.0084	5.2609	0.0022	0.8853	0.0044
L	170.9594	172.9594	0.4387	1.50E-11	15.8383	9.52E-06	3.3144	5.11E-09
TW	83.4567	89.4567	0.1921	0.0165	4.2427	0.0067	0.6739	0.0145
G	128.7679	132.7679	0.2964	2.11E-05	8.9041	4.70E-05	1.6596	6.33E-05
EE	45.1408	49.1408	0.0717	0.8794	0.6638	0.5892	0.0724	0.7385

MLE, maximum likelihood estimate; TLRT-F, transmuted lower record type Fréchet; TLL, transmuted Log-logistic; W, Weibull, EE, exponentiated exponential; TW, transmuted Weibull; L, Lindley; G, Gompertz; AIC, Akaike's information criteria; KS, Kolmogorov–Smirnov test; A*, Anderson Darling statistic; CVM, Cramer von–Mises.

Table 5

Selection criteria statistics for glass fibers data.

Glass fibers data

This data set is generated data to simulate the strengths of glass fibers in Mahmoud and Mandouh [16]. The data are as follows: 1.014, 1.081, 1.082, 1.185, 1.223, 1.248, 1.267, 1.271, 1.272, 1.275, 1.276, 1.278, 1.286, 1.288, 1.292, 1.304, 1.306, 1.355, 1.361, 1.364, 1.379, 1.409, 1.426, 1.459, 1.46, 1.476, 1.481, 1.484, 1.501, 1.506, 1.524, 1.526, 1.535, 1.541, 1.568, 1.579, 1.581, 1.591, 1.593, 1.602, 1.666, 1.67, 1.684, 1.691, 1.704, 1.731, 1.735, 1.747, 1.748, 1.757, 1.800, 1.806, 1.867, 1.876, 1.878, 1.91, 1.916, 1.972, 2.012, 2.456, 2.592, 3.197, 4.121.

7. LIFETIME REGRESSION ANALYSIS WITH REAL DATA

Lawless [17] reported the failure times for epoxy insulation specimens in an accelerated voltage life test. The sample size is n=60 and there are three levels of voltage 52.5, 55.0 and 57.5. Let yi be failure times for epoxy insulation specimens (in min) and vi1 are voltages (kV). The following regression model can be written by

yi=β 0+β 1vi1+σzi,

where the realization yi is assumed an observation from LTLRT-F distribution given in (5.1). For comparison, the LW (see, Lawless [17]) and LTLGBXII (see, Yousof et al. [9]) lifetime regression models are considered. The results of analyses are presented in Table 6. To compare LTLRT-F with LW and LTLGBXII regression models, the MLEs of the model parameters, the asymptotic standard errors of these estimates and the values of the minus log-likelihood and AIC measures are given in Table 6. It is noticed that the results for LW and LTLGBXII are reproduced from Yousof et al. [9]. Figure 4 exhibit, the fitted and empirical survival function plots. From Figure 4 and Table 6, one can conclude that the fitted LTLRT-F regression model can be strongly recommended for modeling survival data given in the literature.

Model	β 0	β 1	σ	p			−ℓ	AIC
LTLRT-F	15.6527	−0.1564	1.2378	0.9880			78.278	164.556
	3.5915	0.0625	0.2637	0.3261
	[0.0000]	[0.0123]

	β 0	β 1	σ	γ	θ	β	−ℓ	AIC

LTLGBXII	14.4513	−0.1790	0.8024	0.6860	7.7247	0.7089	78.2	168.4
	4.876	0.074	0.827	0.685	7.027	0.984

	β 0	β 1	σ				−ℓ	AIC

LW	22.0313	−0.2745	0.8453				83.699	173.398
	3.0454	0.0553	0.0904

MLE, maximum likelihood estimate; AIC, Akaike's information criteria.

Table 6

MLEs of the parameters, standard errors in second line, p-values in [⋅] and the AIC statistics.

8. CONCLUSION

In this study, a new extension to generate a new family of distribution is introduced by a transmuted lower record type map of order 2. This method will be lead to obtain new families of distribution called TLRTs. A sub-model of this family is considered and lifetime regression analysis is introduced based on this sub-model. The introduced regression model is applied to a real data set and the results show that our model is a good alternative to modeling real data.

CONFLICTS OF INTEREST

There is no conflict of interest.

AUTHORS' CONTRIBUTIONS

The authors thank the relevant editor and reviewer for their valuable suggestions, which were very useful in improving the study.

Funding Statement

This study is not supported by any funding council.

ACKNOWLEDGMENTS

We thank to Dr. Emrah Altun for his constructive comments on regression part of the study.

REFERENCES

1.W.T. Shaw and I.R. Buckley, The Alchemy of Probability Distributions: Beyond Gram Charlier & Cornish Fisher Expansions, and Skew-Normalor Kurtotic-Normal Distributions, King's College, London, U.K., 2007, pp. 64. Technical report, Financial MathematicsGroup

2.W.T. Shaw and I.R. Buckley, The alchemy of probability distributions: Beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation, map, 2009. http://arxiv.org/abs/0901.0434

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 20 - 1
Pages: 86 - 96
Publication Date: 2021/01/25
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.d.210115.001 How to use a DOI?
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TY  - JOUR
AU  - Caner Tanış
AU  - Buğra Saraçoğlu
AU  - Coşkun Kuş
AU  - Ahmet Pekgör
AU  - Kadir Karakaya
PY  - 2021
DA  - 2021/01/25
TI  - Transmuted Lower Record Type Fréchet Distribution with Lifetime Regression Analysis Based on Type I-Censored Data
JO  - Journal of Statistical Theory and Applications
SP  - 86
EP  - 96
VL  - 20
IS  - 1
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.210115.001
DO  - 10.2991/jsta.d.210115.001
ID  - Tanış2021
ER  -

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