NOVEL DISTRIBUTION FOR MODELING UNCENSORED AND CENSORED SURVIVAL TIME DATA AND REGRESSION MODEL Текст научной статьи по специальности «Математика»

NOVEL DISTRIBUTION FOR MODELING UNCENSORED AND CENSORED SURVIVAL TIME DATA AND REGRESSION MODEL

Adubisi O. D.1

Adubisi C. E.2 •

department of Mathematics and Statistics, Federal University Wukari, Nigeria.

2Department of Physics, University of Ilorin, Nigeria.

[email protected] [email protected]

Abstract

This work proposes a new one-parameter model titled the type II Topp Leone half logistic (TIITLhl) model which is characterized by an increasing and decreasing hazard rate function quite dependent on the shape parameter. Some structural properties and basic functions used in reliability analysis are derived. Simulations are carried out for both uncensored and censored samples. The uncensored simulation results indicated that the estimators perform quite well in producing good parameter estimates at finite sample sizes. However, the Anderson Darling estimator (ADE) average estimate tend to the true parameter value faster than other methods with minimum bias. More so, simulation based on censored samples using different censoring proportions showed that the bias, MSE and MRE values decrease as the sample size increases for the different censoring proportions. Two uncensored and censored datasets from the medical and environmental sciences were analysed to show the relevance, flexibility and adaptability of the TIITLHL model, and the new model achieved the best performance when compared with six other competing lifetime models. In addition, the log-TIITLHL regression model constructed and compared with two existing models showed that this model will be a useful option in survival investigation.

Keywords: Half-logistic distribution, Classical estimation methods, Monte-Carlo simulation, type II censoring, Type-II-Topp-Leone-G class

1. Introduction

The standard half-Logistic (HL) model pioneered by [1] has gained a lot of popularity as a significant model given its extensive applicability in lifetime modeling and reliability analysis. The cumulative density function (CDF) of the standard HL model is

1- e-k

G(k) = -—e—r, k > 0, (1)

v ' 1 + e-k

and the corresponding probability density function (PDF) to (??) is

2e-k

= (ire-p,k >a (2)

Several authors have pioneered various extensions of the HL model such as the type I HL family of distributions by [2], inverse HL model by [3], Poisson HL model by [4], type II HL family of

distributions by [5], Kumaraswamy HL model by [6], extended HL model by [7], Transmuted HL model by [8], new type I HL model by [9], odd Lindley HL model by [10], weighted HL model by [11], modified HL model by [12], Poisson-logarithmic HL model by [13], extended type I HL family of distributions by [14] and Gamma power HL model by [15].

The type II Topp Leone-G (TIITL-G) class of distributions was pioneered by [16]. The TIITL-G class has just one parameter which imply that the proposed extended HL model in this study will have a single shape parameter. The CDF and PDF of the TIITL-G class are

r t 1 T

F (k) = 1 - 1 - G2 (k) , (3)

and

f (k) = 2tg (k) G (k) 1 - G2 (k) , (4)

where t > 0 is a shape parameter, G(k) and g(k) are considered as the CDF and PDF of the baseline model. The novelty of this study is the creation of a new one-parameter lifetime model titled the TIITLHL model, investigation of six different estimation methods for the new model with applicability to uncensored and censored survival time datasets, and introduction of a new log-TIITLHL regression model for analysing censored response variable.

The remaining parts are outlined like this: Part 2 introduces the CDF and PDF of the TIITLhl model. Part 3 presents reliability analysis and several important structural properties of the TIITLHL model. Six classical estimation approaches are discussed in Part 4 to appreciate the parameters of the TIITLHL model for uncensored sample. The maximum likelihood estimator based on the type-II right censored scheme is presented in Part 5. The finite sample performance of the TIITLhl estimators is presented in Part 6 using Monte Carlo experiments. Part 7 deals with a new TIITLhl regression model. The applications and empirical results are presented in Part 8. Finally, Part 9 presents the conclusion.

2. Model Genesis

This part introduces a new one-parameter model called the TIITLHL model by inserting Eqs (1) and (2) into Eqs (3) and (4), then the CDF, PDF, survival function and hazard rate function (HRF) of the TIITLHL model are

F(k, t) = 1 -

1

1- e

-k

1 + e-

, k > 0, t > 0,

f (k, t)

4Te

k

1e

k

(1 + e-k)2 I 1 + e-k

1

1e

k

1 + e

-k

T-1

, k > 0, T > 0,

(5)

(6)

Figure 1 depicts the graphical shapes of the TIITLHL density function (PDF) with selected values for t. The density function (PDF) is uni-modal, right-skewness, and heavy-tailed. The survival (Reliability) function (SF) and hazard (failure) rate (HRF) of the TIITLHL model, take the forms

S(k, t)

1

1 - e-1 + e-

and

h(k, t )

4Te

-k

1e

-k

(1 + e-k)2 + e-k J More so, the reversed HRF of the TIITLHL model is

1

1e

-k

1 + e-

(7)

(8)

T

2

k

2

T

2

k

k

1

2

k

r(k, t) =

4re-

1 - e-

(1 + e-k)2 I 1 + e-k

1

1 e-

1 + e

k

t- 1

1

1

1 e-

1 + e

k

^ -1

(9)

and the cumulative HRF takes the form

H(k, t) = -t log

1

1e

k

1 + e

-k

(10)

The graphical shapes of the HRF for TIITLhl model with various selected values of t are depicted in Figure 2. The model is characterized by an increasing-decreasing HRF.

3. Structural Properties This part describes the statistical properties of the TIITLHL model..

3.1. Quantile function, Bowley's skewness and Moor's kurtosis

If the random variable (r.v) K ~ TIITLHL(t), then the quantile function by inverting Eq (5) takes the form

k = - log •

1 - [1 - (1 - u) t]1 1 + [1 - (1 - u)T] 2

(11)

where u ~ uniform(0,1). By setting u = 0.5 in Eq (11), the median (M) of the TIITLHL model takes the form

2

2

k

k

k

k

2

Figure 2: Survival function plot (a), hazard rate function plot (b), reversed HRF plot (c) and cumulative HRF plot (d) of the TIITLhl model.

k=- log!1 -11 - (05);':}. (12)

U + [1 - (0.5)T ]1 J

The Bowley's skewness [17] and Moor's kurtosis [18] are found using the following expressions, respectively.

s Q (3; t) - 2Q (j; t) + q( j; t) (13)

Sk = -^-)-a-)-. (13)

Q (4; t) - Q (4; t) _ Q (7; t) - Q (5; T) - Q (8; t) + Q (1; t)

Ku = Q (6; t) - Q (2; t) . (14)

where Q(.) is the quantile function.

3.2. Dispersion index and Coefficient of variation

The dispersion index (DI) tells when a model is suitable for modeling equi-dispersed (DI = 1), under-dispersed (DI < 1) and over-dispersed (DI > 1). The coefficient of variation (CV) is a relative measure of variability and a high CV value shows higher variability. The expressions for the DI and CV functions are

DI

and

CV

Var(X)

(Var(X)) 2

E(X)

Q( 4;t)-q( 1;t)

_135_

Q(4;t)+q( j;T)+Q(4;t) •

3

( Q( 3;t)-q( 4;t) ) 2

( 1.35 ) 2

Q(4;t)+q( 1;t)+q(4;t) •

3

(15)

(16)

where Q(.) is the quantile function.

Table 1 reports the numerical values of the mean (ME), variance (VAR), standard deviation (STD), median (M), skewness (Sk), kurtosis (Ku), Dispersion index (DI) and Coefficient of variation (CV) for the TIITLhl model using selected values of t.

Table 1: The numerical values of ME, VAR, STD, M, Sk, Ku, DI and CV

T ME VAR STD M Sk Ku DI CV

0.2 5.282 17.356 4.166 4.836 0.238 0.629 3.286 0.789

0.5 2.784 3.531 1.879 2.634 0.177 0.484 1.268 0.675

1.0 1.832 1.293 1.137 1.763 0.135 0.370 0.706 0.621

1.5 1.459 0.769 0.877 1.412 0.117 0.321 0.527 0.601

2.0 1.247 0.543 0.737 1.212 0.108 0.294 0.436 0.591

2.5 1.107 0.419 0.647 1.078 0.102 0.277 0.378 0.584

3.0 1.006 0.341 0.584 0.980 0.097 0.266 0.339 0.581

3.5 0.928 0.287 0.536 0.905 0.095 0.257 0.310 0.578

4.0 0.865 0.248 0.498 0.845 0.092 0.251 0.287 0.576

4.5 0.814 0.218 0.467 0.795 0.091 0.246 0.268 0.574

5.0 0.771 0.194 0.441 0.753 0.089 0.242 0.252 0.572

The ME, STD, Sk and K u values of the TIITLhl model decrease as the selected values of t increase. The TIITLhl model is positively skewed and beneficial for over-and-under dispersed datasets. Figure 3 depict the plots of the ME, VAR, Sk and Ku of the TIITLhl for selected values of t and support the conclusion reached using Table 1.

3.3. Moments and Moment generating function

The rth raw moment of the TIITLhl model is given as

T — 1 œ

F = EE (b + c + 1)-r—1 r (r + 1) •

a=0 b,c=0

(17)

Proof. The rth raw moment of the TIITLhl model is found using y!r = r krf (k; t) dk,

4t/o°° k

e k f 1-e k

1

(1+e—k)2 \ 1+e—k

By utilising Taylor series expansions in Eq (??), we have

T — 1 œ

1—e— 1+e—

t—1

(18)

dk,

t —± w f<œ

Fr = E E Va,b,c kre-k(b+c+1)dk,

a=0 b,c=0 0

where ßa,b,c = 4t (-1)a+b+c ( T - 1 ) (

2a +1 b

-2 (1 + a) - 1 c

Let

z = k (b + c + 1) ^ k

(b+c+1) '

2

Figure 3: The Mean, Variance, skewness and kurtosis plots of the TIITLhl model.

dk - 1 ^dk- dz

dz (b+c+1) ^ (b+c+1)'

Hence,

T-1 M .M / _ \ r

i'M ' z z dz

= «5 & l(bT7TT)J e"(b+T+T), (20)

T-1 m .. M

ti = EE V*,b,c (b + c + 1)-r-W zre-zdz, (21)

«=0 b,c=0

;r<

TIITLhl model takes the form

By utilising the gamma integral function r (a + 1) = f0 zae zdz. The rth raw moments of the

T-1 TO

F'r = EE #a,b,c (b + c + 1)-r-1 r (r + 1). (22)

a=0 b,c=0

The first four moments are found by inserting r = 1,2,3,4 into Eq (22), respectively. ■

The moment generating function (MGF) of the TIITLHL model is given as

to t-1 to tr A

Mk (t) = E EE E -AT* (b + c + 1)-r-1 r (r + 1), (23)

r=0 a=0 b,c=0 ■

Proof.

The MGF of the TIITLhl model, say Mk (t) is found using

/ n m tr i'm

Mk (t) = E (etk) = £ M Krf (k;t)

v 7 r=0 1/0

dk:

M tr £ ^,

r=0

By inserting Eq (7) into Eq (24), the MGF takes the form

M T —1 M ±r

Mk (t) = ££ £

tr ft

a,b,c

(b + c + 1)—r—1 r (r + 1).

r=0 «=0 b,c=0

(24)

(25) ■

3.4. Order statistics

If ki, k2,..., kn be a random sample from the TIITLHL model with ki:n < k2:n < ... < kn:n as the order statistics (O.S). The pdf of the pth O.S of the TIITLHL model is

fr.n (k)=4Te—k (1 + e—k)—2 (p) n! u—

(p — 1)! (n — p)!

( r T ^ p —1 r

{1 — 1 — (p)2 } 1 — (p)

T[(n — p) + 1]—1

Proof. The pdf of the pth O.S can be found using

fp:n (k)

n!

— 1)! (n — p)!

g (k) [G (k)]p—1 [1 — G (k)]n—p,

(26)

(27)

where B (.,.) is the beta function. By inserting Eqs (5) and (6) into Eq (27), the pdf of the pth O.S of the TIITLhl model after some simplification takes the form

fp, (k)=4Te—k (1 + e—k)—2 (9) n! U —

(p — 1)! (n — p)!

( r T ^ p —1 r

{1 — 1 — (p)2 } 1 — (p)

T[(n—p)+1]—1

(28)

(1-e—k

where p = (^1+e+k J. By substituting p = 1 and p = n into Eq (28), the lowest and highest order statistics are obtained. ■

4. Methods of estimation for uncensored sample

In this part, the parameter of the TIITLHL model is estimated via the maximum likelihood estimation (MLE), maximum product spacing estimation (MPSE), Anderson Darling estimation (ADE), least square estimation (LSE), weighted least square estimation (WLSE), and Cramer Von Mises estimation (CVME).

4.1. The MLE

If k1, k2,..., kn be the random observed values from TIITLHL model. Then, the MLE function L(t) takes the form

L (T) = (4T)n n

e—ki r 21 T—1

( 9i) 1 — ( 9i)

i=1 (1 + e—ki)

\2

(29)

where pi = Q+^j?. ). The log-likelihood function of the TIITLhl model takes the form

log (L (T)) = n log(4T) — £ ki — 2 £ log (1 + e—kM + £ log (pt) + (t — 1) £ log 1 — (pi)

nl

i=1 i=1

The first derivative of Eq (30) with respect to t is

i=1

i=1

r

2

2

2

d log (L (t)) n " r 2

8( ( )) = - + Elog 1 -()2 T ¡=1

dT

(31)

The R (optim function) is employed to estimate the TIITLHL parameter using numerical approaches.

4.2. The LSE and WLSE

Minimizing with respect to t, the LS estimate TLS can be found using

ls (t) = e

i=1

1

1

1 e-

1+e

k

n+1

Likewise, minimizing with respect to t, the WLS estimates TWLS can be found using

WLS (t) = e

(n + 2)(n + 1)2

!=- i(n - i +1)

1

1

1e

k

1 + e

k

n+1

(32)

(33)

4.3. The MPSE

The MPS for the TIITLHL model with ordered sample k(1:n), k(2:n), . . . , k(n:n) is given as follows

GM (T|kn:n)

n+1

n Di (ki, T) i=1

1

n+1

(34)

where Di (ki, t) = F (k(i:n) |t j - F (k(i-1:n) |t j ; i = 1,2,...,n + 1. and F (k, t) is given in Eq (5).

4.4. The ADE

Minimizing with respect t, the AD estimate Tad can be found using

AD (T) = -n - n E (2i - 1) [logF (k(i:n) |t) + logF (^+1-^) |t)

i=1

where F (k, t) = 1 - F (k, t) and F (k, t) is given in Eq (5).

4.5. The CVME

Minimizing with respect t, the CVM estimates TCVM can be found using

1

CVM (t) = - + E

i=1

1

1

1e

-k

1 + e-

2(i - 1) + 1 2n

(35)

(36)

5. MLE for type II right censoring

Given that a fixed number of failed units have been observed, a life testing experiment is concluded. Then the remaining units are designated as type-II-censored. Let k^), k(2),..., k(p), p < n denote the ordered values of a random sample (r.s) k\, k2,..., kn (failure times) and observations cease after the pth unsuccessful unit occurs, then the likelihood function is given

L (t; k)

n!

(n - p)! [R(kp;"-pnf (ki;t) .

(37)

2

T

2

k

2

T

2

2

T

2

k

p

If ki, k2,..., kn be r.s from the TIITLhl (t), then the likelihood function is

L (t; k)

(n - p)!

1 -

1 - e-kp 1 + e-kp

T(n-p) p Í 4Teki

n { ~1 + e-ki)2

i=1

1

1 - e

-ki

1 + e-

t- 1

The log-likelihood function without the constant term is

(38)

I (t; k) a p [log (4) + log (t)] + t (n - p) log

1

Í 1-e-kp \1+e-kp

-yp k y¿=1 ki

-2yp=1 log(1 + e-ki) + yf=! Jog (g-k-) + (t - 1) y=1 log

^ ( 1-LkLx 2

1 1 1+e-k¿

(39)

Setting JTl (t; k) = 0. The MLE (T) can be found as solution of

T + (n - p) log

1

1 - e-kp 1 + e-kp

+ y log

i=1

1

1e

ki

1 + e

-ki

(40)

Using the R (optim function), the non-linear equation in Eq (40) is solved numerically to obtain the MLE t.

2

2

n

k

2

2

6. Simulation

The Monte Carlo simulations for uncensored and censored samples are executed for the TIITLhl parameter (Pa.).

6.1. Simulation based on uncensored sample

The simulations using MLE, LSE, WLSE, MPSE, ADE, and CVME approaches for the TIITLHL parameter are presented in this subpart. The simulation is carried out as follows:

• Set the parameter value t = 0.5, 2.5 for the Monte Carlo simulation process.

• Random samples of sizes n = 20,70,150,250,350 with replicates N = 5000 generated using Eq (11).

• The MLE, LSE, WLSE, MPSE, ADE, and CVME processes are executed to find the estimates of parameter (t).

• Compute the average estimate (AVEs), absolute biases (ABs), mean square errors (MSEs) and mean relative error (MREs) using the information in the preceding step.

Tables 2 and 3 reports the AVEs, ABs, MSEs and MREs for the MLE, LSE, WLSE, MPSE, ADE, and CVME methods with different sample sizes. The results are graphically summarized in Figures 4 and 5. As seen from these graphs, the ABs, MSEs and MREs tend to zero as n increases for the six estimation methods. However, the ADE average estimate tend to the true parameter value faster than other estimation methods with minimum AB.

Table 2: The six estimators AVE for t = 0.5 based on uncensored sample.

n Measures MLE MPSE ADE LSE WLSE CVME

20 AVE 0.527 0.489 0.518 0.522 0.520 0.526

70 AVE 0.507 0.492 0.504 0.505 0.505 0.506

150 AVE 0.503 0.495 0.502 0.502 0.502 0.503

250 AVE 0.502 0.496 0.501 0.501 0.501 0.501

350 AVE 0.501 0.497 0.500 0.500 0.500 0.500

Figure 4: AB, MSE and MRE of Estimators in Table 2.

Table 3: The six estimators AVE for t = 2.5 based on uncensored sample.

n Measures MLE MPSE ADE LSE WLSE CVME

20 AVE 2.637 2.444 2.591 2.610 2.598 2.632

70 AVE 2.536 2.460 2.521 2.525 2.524 2.531

150 AVE 2.517 2.475 2.509 2.511 2.511 2.514

250 AVE 2.508 2.480 2.503 2.504 2.504 2.505

350 AVE 2.504 2.483 2.500 2.501 2.501 2.502

Figure 5: AB, MSE and MRE of Estimators in Table 3.

6.2. Simulation based on type-II-right censored sample

The simulation is executed for the MLE using different random sample sizes n = (20,70,150,250,350) generated with Eq (11). The length of censored sample test is given by p = nm, where (m) is the censoring proportion (0 < m < 1). Table 4 reports that the bias, MSE and MRE values decrease as the sample size increases for the different censoring proportions considered.

Table 4: The MLE, Bias, MSE and MRE based on censored sample.

n m t t Bias MSE MRE

20 0.3 0.5 0.881 0.381 0.145 0.762

2.5 4.405 1.905 3.629 0.762

0.5 0.5 0.752 0.252 0.063 0.503

2.5 3.758 1.258 1.583 0.503

0.7 0.5 0.781 0.281 0.079 0.562

2.5 3.906 1.406 1.976 0.562

70 0.3 0.5 0.611 0.111 0.012 0.223

2.5 3.057 0.557 0.310 0.223

0.5 0.5 0.650 0.150 0.022 0.299

2.5 3.248 0.748 0.560 0.299

0.7 0.5 0.657 0.157 0.025 0.315

2.5 3.287 0.787 0.620 0.315

150 0.3 0.5 0.551 0.051 0.003 0.101

2.5 2.753 0.253 0.064 0.101

0.5 0.5 0.554 0.054 0.003 0.108

2.5 2.771 0.271 0.073 0.108

0.7 0.5 0.573 0.073 0.005 0.145

2.5 2.863 0.363 0.132 0.145

250 0.3 0.5 0.567 0.066 0.004 0.132

2.5 2.831 0.331 0.110 0.132

0.5 0.5 0.587 0.087 0.008 0.175

2.5 2.936 0.436 0.190 0.175

0.7 0.5 0.579 0.079 0.006 0.158

2.5 2.896 0.396 0.157 0.158

350 0.3 0.5 0.485 -0.015 2E-04 0.030

2.5 2.424 -0.076 0.006 0.030

0.5 0.5 0.453 -0.047 0.002 0.095

2.5 2.264 -0.236 0.056 0.095

0.7 0.5 0.424 -0.076 0.006 0.153

2.5 2.118 -0.382 0.146 0.153

7. The log-TIITLhl regression model

Let K denotes a random variable which follows the TIITLhl model with parameter t. Utilizing the transformation Y = log (K) with location and scale parameters added, the density of Y is

fTIITLhl (У, т, -, a) = 4t exp [(- exp (i1 + exp [- exp (

y--

J 1—exp[— exp( ^ 1+exp[- exp(

( 1—exp[— exp( y--- )] 1 \ 1+exp[-exp( y-- )]

t-1

(41)

where t, a > 0, y, - £S. The random variable Y has the log-TIITLHL (LTIITLHL) model with location - and scale a parameters, respectively. The survival function to Eq (41) is

/

stiitlhl (y, ^ a)

1

\

1 — exp — exp (y/)

1 + exp — exp (

2\ t

(42)

By inserting z = (y - -) /a, z £ S into Eq (41). The standardized log-TIITLHL density takes the form

ftiitlhl t) = a exp [(z) - exp(z)] {1 + exp [- exp(z)]}

2

{1-exp[- exp(z)] 1+exp[- exp(z)]

1 _ \ i-exp[-exp(z)n2 ]

11+exp[-exp(z)] J J

t—1

(43)

x

Let Ki = (ki1,...,kim)T be the explanatory vector associated with the ith response variable yi for i = 1,..., n. A regression model based on the TIITLhl density function is given by

yi = KTß + azi, i = 1,..., n,

(44)

where Z{ is the random error which follows the density function Eq (43), ß = (ßi,..., ßm) , a > 0, t > 0 are unknown parameters and Ki is modeling ^i = KTß. The density and survival functions of yi are

/tiitIhl (yi; T, a, ßT) = 4т exp [zi - exp (zi)] {1 + exp [- exp (zi)]}-

f i-exp[-exp(zi)П (i _ f 1-exp[-exp(zi)] 1

\ 1+exp[-exp(zi)] J I \ 1+exp[-exp(zi)] J

^ 1

and

Stiitlhl (yi; ^ a, ßT)

1

1 - exp [-exp (zi)]

1 + exp [-exp (zi)]

(45)

(46)

Let F and C denote the sets of units for which yi is the log-lifetime or log-censoring, respectively. The log-likelihood for & = ^r, a, from Eq (44) is

I (*) = r log (4T) + Eier [zi - exp (zi)] - 2 Eier log {1 + exp [- exp (zi)]}

+EieF log i 1+exp[-:xp(z;)]}+(t -1) e^ log (1 - {1+exp[-exp(z;)] }2)

(47)

+t Eiec log (1 - {1+exp[-exp(z;)] }2).

where zi = (yi - KTß) /a and r is the number of uncensored observations (failures). The MLE Ь of Ь can be obtained by maximizing Eq (47) using the R (optim function).

8. Applications

The potentiality of the introduced model is illustrated by means of five applications.

8.1. Applications to uncensored data

The first dataset consist of survival time of 72 Guinea pigs infected with virulent tubercle bacilli. The data was initially reported by [19], and analysed by [6] and [20]. The second dataset, discussed by [21] consists of 30 observations of March precipitation (in inches) in Minneapolis/St Paul.

Figure 6: Box, TTT, kernel-density, strip, and Violin plots of first uncensored data.

Figures 6 and 7 depict the box plot, TTT plot, kernel density plot, strip plot, and Violin plot of the first and second datasets to check for outliers and symmetric nature. As depicted, both

2

2

T

2

Figure 7: Box, TTT, kernel-density, strip, and Violin plots of second uncensored data.

datasets have certain outliers and are asymmetrical in nature. More so, the total test time (TTT) plot depicts that the utilized datasets have increasing HRF which means that the TIITLhl model can be used to model these datasets.

We fit the datasets with the TIITLhl, odd Lindley HL (OLIHL), Generalized HL (GHL), Inverse Lindley (ILIN), Rayleigh (R), Inverse Rayleigh (IR), and Muth (M) models. The MLEs for all the models are computed in R-software via the BFGS method (optim function ). The W*, A* and KS measures (we used abbreviations) for model comparisons. The MLEs and their standard errors (SEs) in parentheses, log-likelihood (LL), and the information criteria are presented in Tables 5 and 6 for both datasets, respectively. The measures reveal that the TIITLHL model provides an appropriate fit to both datasets (with lowest values of the goodness-of-fit statistics).

Table 5: The MLEs and SEs of the fitted models with goodness-of-fit measures for first uncensored data.

Model MLEs (SEs) LL W* A* CAIC AIC BIC HQIC KS(P-Value)

TIITLHL 1.257(0.148) -96.83 0.074 0.498 195.7 195.7 197.9 196.6 0.088(0.636)

OLIHL 0.289()0.024 -157.28 0.891 4.940 316.6 316.6 318.8 317.5 0.393(0.000)

GHL 0.786(0.093) -104.94 0.080 0.531 211.9 211.9 214.2 212.8 0.218(0.002)

ILIN 1.359(0.124) -129.20 0.929 5.634 260.5 260.4 262.7 261.3 0.232(0.001)

R 1.442(0.085) -98.96 0.133 0.814 200.0 199.9 202.2 200.8 0.107(0.376)

IR 0.480(0.028) -204.82 1.632 9.214 411.7 411.6 413.9 412.5 0.623(0.000)

M 0.480(0.028) -123.46 0.118 0.735 249.0 248.9 251.2 249.8 0.434(0.000)

Figure 8: The fitted pdfs (left panel) and cdfs (top-left,right panel) of competing models, fitted HRF (top-right,right panel), P-P plot (bottom-left,right panel) and Q-Q plot (bottom-right,right panel) of the TIITLhl model for first uncensored data.

Table 6: The MLEs and SEs of the fitted models with goodness-of-fit measures for second uncensored data.

Model MLEs (SEs) LL W* A* CAIC AIC BIC HQIC KS(P-Value)

TIITLHL 1.351(0.247) -38.34 0.01483 0.1155 78.83 78.68 80.08 79.13 0.059(0.999)

OLIHL 0.364 (0.048) -56.22 0.25812 1.6009 114.58 114.44 115.84 114.89 0.307(0.007)

GHL 0.828(0.151) -42.43 0.01530 0.1195 87.01 86.87 88.27 87.32 0.182(0.274)

ILIN 1.583(0.227) -45.22 0.09750 0.6014 92.59 92.44 93.84 92.89 0.228(0.089)

R 1.376 (0.125) -38.92 0.02607 0.1979 79.99 79.85 81.25 80.30 0.084(0.985)

IR -0.927(0.085) -44.14 0.16293 0.9881 90.42 90.27 91.67 90.72 0.240(0.064)

M 0.185(0.091) -48.72 0.02471 0.1898 99.59 99.45 100.85 99.89 0.367(0.001)

Figures 8 and 9 display the histogram and Kaplan-Meier empirical cdf in conjunction with fitted pdfs and cdfs of the TIITLHL and competing models for both datasets. Figures 8 and 9, also depict the fitted HRF of the TIITLhl model with corresponding probability-probability (P-P) and quantile-quantile (Q-Q) plots. The superiority of the TIITLHL model is supported by these figures.

Figure 9: The fitted pdfs (left panel) and cdfs (top-left,right panel) of competing models, fitted HRF (top-right,right panel), P-P plot (bottom-left/right panel) and Q-Q plot (bottom-right,right panel) of the TIITLhl model for second uncensored data.

Tables 7 and 8 report the estimation of the TIITLhl parameter based on six estimation methods for the both uncensored datasets. From the results in Tables 7, the MPSE and WLSE are considered as the appropriate methods with smaller KS and larger P-values than the other methods. Likewise, the results in Table 8 attest that the MLE, ADE and WLSE are the appropriate methods with smaller KS and larger P-values than the other methods.

Table 7: The results of the six estimation methods for first uncensored data.

Estimate^Method^ MLE MPSE CVME ADE LSE

WLSE

T

KS P-Value

1.257 0.088 0.600

1.215 0.081 0.700

1.279 0.091 0.600

1.262

0.089 0.600

1.266 1.229 0.089 0.083 0.610 0.700

Table 8: The results of the six estimation methods for second uncensored data.

Estimate^Method^ MLE MPSE CVME ADE LSE WLSE

T 1.351 1.266 1.354 1.347 1.011 1.351

KS 0.059 0.073 0.060 0.059 0.155 0.059

P-Value 0.999 0.999 0.999 0.999 0.470 0.999

8.2. Applications to censored data

The first dataset represents the relief times (in minutes) of twenty patients receiving an analgesic. The data was initially reported by [22] and the complete sample analysed by [23] and [24]. The censored sample (number of failures) p, is chosen as 50% (censoring scheme). The MLE, KS and P-Value for the TIITLhl model are reported in Table 9.

Table 9: The MLE and performance measure for the first censored data.

Models MLE KS P-Value

TIITLhl T = 0.877 0.230 0.240

The second dataset represents the survival time of 72 Guinea pigs infected with virulent tubercle bacilli. The complete sample was analysed by [6] and [20]. The censored sample (number of failures) p, is chosen as 70% (censoring scheme). The MLE, KS and P-Value for the TIITLhl model are reported in Table 10. It is evident that the TIITLHL appropriately fits the two survival time censored datasets.

Table 10: The MLE and performance measure for the second censored data.

Models MLE KS P-Value

TIITLhl T = 1.174 0.077 0.790

It is evident that the TIITLHL appropriately fits the two censored samples.

8.3. Regression model application

The usefulness of log-TIITLHL regression model is demonstrated by means of a real data analysis. The log-TIITLHL regression model is compared with log-exponential (LE) and log-Burr-Hatke-exponential (LBHE) regression models [25]. The utilized dataset contains 100 individuals having HIV+ obtained from the Bolstad2 package in R-software. The observed survival times (yi), in months, with censoring indicator (0 = alive and 1 = death) is analysed with two explanatory variables: ki1, (0 = no and 1 = yes) represent the history of drug usage and ki2 represent the ages of patients . The proposed regression model is

yi = J00 + № + 02 ki2 + azi (48)

where z, has density Eq (43). The MLE method is utilized in estimating the unknown parameters of log-TIITLHL, LE and LBHE regression models. Table 11 reports the regression models estimated parameters, -LL and performance measures (AIC, BIC, AICc and HIQC values). The results provided in Table 11 indicates that the LTIITLhl regression model has the lowest value of -LL and performance measures values, respectively. Hence, it is concluded that log-TIITLHL regression model provides appropriate fit than LE and LBHE regression models. More so, the estimated regression parameters j80, and j82 are statistically significant at 5% level of significance.

Table 11: The regression models estimated parameters and performance measures.

LBHE L-E Log-TIITLHL

Parameters Estimates SE P-Value Estimates SE P-Value Estimates SE P-Value

T 1.508 13.659 - 1.599 13.783 - 26.968 48.251 -

a 0.778 0.067 - 0.839 0.072 - 1.684 0.143 -

6.883 7.064 0.330 6.542 7.256 0.367 2.303 0.130 <0.001

h -0.091 0.014 <0.001 -0.091 0.014 <0.001 -0.023 0.009 0.020

fa -1.021 0.193 <0.001 -1.049 0.189 <0.001 -0.261 0.109 0.017

-LL 128.059 128.502 128.051

AIC 266.12 267.00 266.10

BIC 279.14 280.03 279.13

AICc 266.76 267.64 266.74

HQIC 271.39 272.28 271.37

9. Conclusion

This work introduced a new one-parameter model titled the TIITLhl model and provided some of its properties. The consistency of the maximum likelihood estimator and five other estimators are proven by uncensored and censored simulation studies. Applications to real medical and environmental sciences datasets revealed its flexibility and adaptability. The log-TIITLhl regression model constructed and fitted to HIV+ data, and compared with other existing models showed that the model will be a useful choice in survival investigation for practitioners. Overall, the five applications showed the usefulness of the new model for asymmetric, uncensored and censored data. In future works, the Bayesian analysis of TIITLhl accelerated failure time model, the TIITLhl-G family of distributions and the discrete case of the TIITLhl model will be addressed.

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