Bayesian estimation of the parameters of exponentiated exponential ...

Bayesian estimation of the parameters of exponentiated exponential distribution under progressive interval type-I censoring scheme with binomial removals Arun Kaushik Banaras Hindu University, Varanasi, India December 20, 2015 This research is funded by Council of Scientific and Industrial Research (CSIR) - New Delhi

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1

Background

2

Estimating Model

3

EM method of estimation

4

Bayesian Estimation Metropolis Hasting Procedure

5

Application

6

Comparison of the Estimators

7

Conclusions

Arun Kaushik

Introduction I Unless you are lucky, its difficult to get a complete data in medical field. In numerous life testing experiments, situations do arise when the units under study are lost or removed from the experiments while they are still alive. Sometimes, test units are often inspected at specified times only and thus we can only record the number of failed units in place of their exact time of failure. Aggarwala (2001)[1] proposed the progressive type-I interval censoring, which is combination of these two situations.

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Introduction II To have a clear visualization of this censoring scheme, let us consider an experiment with n bladder cancer patients for whom remission times are to be recorded.

Remission time A period during which symptoms of disease are reduced (partial remission) or disappear (complete remission) with regard to the cancer. Remission means absence of sign of cancer on scans or on the patient at the time of examination by the doctor. Doctors use the word ‘remission’ instead of cure when talking about cancer because they cannot be sure about the disappearance of cancer cells in the body and there is a possibility that cancer cell may reappear in future.

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Data arising setup Experimenter may decide to call the patients for examinations at some specified interval of time. At the initial time T0 = 0 experimenter start experiment with taking n number of patients. At the end point of every specified interval (follow up date), the patients are expected to visit the doctor who examines whether the patient’s, cystoscopy/biopsy test is positive or not. a

Number of recurrence Time

0

t1

t2

tm−1

tm

The patients are called for regular check up at scheduled times and those who turn up are checked. 1 Arun Kaushik (BHU)

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Data arising setup At the first follow-up date, there is a possibility that some patients may not turn up. At the first visit scheduled at time T1 , only n − R1 patients out of the total n patients report i.e. R1 patients leave the experiment during the time interval (0, T1 ]. Experimenter examines these n − R1 patients and finds that cancer has reoccurred in D1 patients. Number of droppings

D1


R1

0

t1

t2

tm−1

tm

It may be noted here that the exact time of recurrence is not known to the experimenter; he only has the information about the number of 2 recurrences during this time period. Arun Kaushik (BHU)

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Data arising setup At second visit scheduled at time T2 , n − R1 − D1 − R2 out of the remaining n − R1 − D1 patients in the experiment after their first visit report i.e. R2 patients leave the experiment at this stage (during the time interval (T1 , T2 ]). Number of droppings


0

R1

R2

D1

D2

t1

t2

tm−1

tm

Experimenter examines these patients and finds that cancer has 3 reoccurred in D2 patients out of remaining n − R1 − D1 − R2 patients, Arun Kaushik (BHU)

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Data arising setup

The experiment continues till the mth visit in the same way. At this stage (mth visit) Number of droppings


0

R1

R2

D1

D2

t1

Rm

Dm−1

t2

tm−1

tm

4

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Data arising setup

Remaining Rm = n − D1 − D2 · · · − Dm − R1 − R2 · · · Rm−1 units are removed i.e. the experiment is terminated at this stage.

Number of droppings


0

R1

R2

D1

D2

t1

Rm

Experiment terminate, all remaining units removed

Dm−1

t2

tm−1

tm

“Data: it happen or didn’t.” 5

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1

Background

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Estimating Model

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Application

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Conclusions

Arun Kaushik

Literature Review I Chen and Lio (2010)[6] have discussed the estimation problem for generalized exponential distribution under progressive type-i interval censoring scheme assuming the number of removals at each stage being fixed. This assumption that the proportions (pi ) of the patients leaving the experiment during (Ti−1 , Ti ] is known in advance i.e. they prefixed the proportion p1 , p2 , . . . , pm and considered that at i th stage, bni ∗ pi c patients leave the experiment. Patients dropping out from the clinical trial at any stage is beyond the control of the experimenter.

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Literature Review II Perhaps, keeping a similar thought in mind, Yuen and Tse (1996)[13] discussed progressive censoring scheme with binomial removal. Ashour and Afify (2007)[2] and Tse, Yang and Yuen (2000)[12] have used PTII censoring scheme with binomial removals assuming that the exact value of the life times of the units are observable. In general, the number of units dropped atP i th stage, Ri follows the binomial distribution with parameter (n − il=1 Dl + Rl , pi ) for i = 1, 2, 3, · · · , m − 1.

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Maximum Likelihood Estimation

Likelihood for the above discussed problem can be written as L(α, θ|R, D, T ) ∝ L1 (α, θ|R, D, T )L2 (P|R, D, T ), where, L1 (α, θ|R, D, T ) =

m h Y

1 − e −θTi

α

α idi h α iri − 1 − e −θTi−1 1 − 1 − e −θTi

i=1

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Maximum Likelihood Estimation The corresponding log likelihood function can be written as logL1 (T , α, θ) = C

m X

di ln

1 − e −θTi

α

α − 1 − e −θTi−1

i=1

α +ri ln 1 − 1 − e −θTi . Schnabel et al.(1985) A modular system of algorithms for unconstrained minimization. ACM Trans.Math. Software, 11, 419-440.

Corresponding Fisher Information matrix −Lαα −Lαθ I (α, θ|data) = , −Lθα −Lθθ

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Background

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Estimating Model

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Application

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Conclusions

Arun Kaushik


The expectation-maximization (EM) algorithm is broadly applicable approach to the iterative computation of maximum likelihood estimates and it is useful in a variety of incomplete data problems where algorithms such as the Newton-Raphson method may turn out to be more complicated and often has less convergence rate. On each iteration of the EM algorithm, there are two steps called the expectation step or the E-step and the maximization step or the M-step. Therefore, the algorithm is called as EM algorithm. The details about the development of EM algorithm can be found in Dempster, Laird and Rubin (1977)[7]

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EM method of estimation For the present study, the EM algorithm for finding the MLEs of parameters in the two-parameter EE distribution is developed as follows. We assume that τi,j , j = 1, 2, . . . , di , denote the exact survival times of those di units which fail within subinterval (Ti−1 , Ti ] and ∗ , j = 1, 2, . . . , r would have been the exact survival times for τi,j i those ri units which drops within time interval (Ti−1 , Ti ]. The log likelihood, ln(LC ), for the lifetimes of n items following the EE distribution can be written as:   ! di ri m m X X X X  ln(LC ) = [ln (α) + ln (θ)] (di + ri ) − θ τij + τij∗  i=1

i=1

j=1

j=1

  di ri m X X X ∗  + (α − 1) ln 1 − e −θ τij + ln 1 − e −θ τij (1) i=1 Arun Kaushik (BHU)

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EM Algorithm I 1 Choose, starting values of α and θ , say α

ˆ (0) and θˆ(0) and set k = 0. 2 In the k + 1th iteration,

X The E-step considers expectations of the terms involving τij and τij∗ in above equations, and finally reduces to the following: m X n = − (di E2i + ri E4i ) α(k) i=1

n θ(k)

Arun Kaushik (BHU)

=

m X

(di E1i + ri E3i ) − (α − 1)

i=1

m X

(2)

(di E5i + ri E6i ) ,

(3)

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EM Algorithm II X The M-step requires solving (2) and (3) to obtain the next values, α(k+1) and θ(k+1) , of α and θ, respectively, as follows: n i=1 (di E2i + ri E4i )

α ˆ (k+1)

=

− Pm

θˆ(k+1)

=

Pm

(4)

n (k+1) − 1) (d E + r E ) (d E + r E ) − (ˆ α i 1i i 3i i 5i i 6i i=1 (5)

3 Checking convergence, if the convergence occurs then the current

α ˆ (k) and θˆ(k) are the approximated maximum likelihood estimates α 2 of θ via EM algorithm; otherwise, set k = k + 1 and go to Step

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1

Background

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Estimating Model

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Application

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Conclusions

Arun Kaushik

Bayesian Estimation Before proceeding further, we make the selection for the prior distributions of the parameters. Following Berger and Sun (1993)[4], it is assumed that both α and θ are independently distributed gamma variates, g1 (α) =

λν11 −(λ1 α) (ν1 −1) e α Γ(ν1 )

;

0 < α < ∞, λ1 > 0, ν1 > 0 (6)

and λ2ν2 −(λ2 θ) (ν2 −1) e θ g2 (θ) = Γ(ν2 )

;

0 < θ < ∞, λ2 > 0, ν2 > 0,

(7) Here, all the hyper parameters λ1 , ν1 , λ2 and ν2 are assumed to be known and can be evaluated following the same method as discussed by Singh. et. al. (2013)[11]. We compute the Bayes estimate of the unknown parameters under squared error loss function. Arun Kaushik (BHU)

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Bayesian Estimation The joint posterior density of α and θ for the given data can be written as π(α, θ|R, D, T ) = =

L(α, θ|R, D, T ) g1 (α) g2 (θ) R∞R∞ 0 0 L(α, θ|R, D, T ) g1 (α) g2 (θ) dα dθ J RR ∞ , (6) 0 Jdαdθ

where, J = J(α, θ)

=

e −(λ1 α+λ2 θ) α(ν1 −1) θ(ν2 −1)

k h Y

(1 − e −θTi )α − (1 − e −θTi−1 )α

idi

i=1

h

× 1 − (1 − e

Arun Kaushik (BHU)

−θTi α

)

iri

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.

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Metropolis Hasting Procedure

We have considered a bivariate normal distribution as the proposal density i.e. N2 (µ, Σ) where Σ is the variance-covariance matrix. We take absolute value of generated observation. The Metropolis-Hastings algorithm associated with the target density π(·) and the proposal density N2 (µ, Σ) produces a Markov chain Θi through the following steps.

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0

1 Set initial values Θ0 = [α0 , θ0 ] .

0 2 Generate new candidate parameter values Θ∗ = [α∗ , θ∗ ] from

N2 (µ, Σ). 3 Calculate the ratio

o n π(Θ∗ ) , 1 . ρ Θ∗ , Θi−1 = min π(Θ i−1 ) 4 Accept candidate Θ∗ as

( Θ∗ with probability ρ Θ∗ , Θi−1 i Θ = Θi−1 with probability 1 − ρ Θ∗ , Θi−1

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Here, we propose the use of MLEs of (α, θ), as initial value for MCMC process. One choice for correlation structure Σ for the target posterior distribution, would be the asymptotic variance-covariance matrix ˆ see Natzoufras (2009) [10] for details. I −1 (ˆ α, θ), Acceptance rate for such a choice of Σ is about 15%. Gelmen et. al. (1995)[8], suggested that the correlation structure of proposal density can be taken as c 2 Σ with c 2 = 5.8/d, where d is the number of parameters.

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Bayesian point estimation Let h(·) be a function of α and θ. Then, the Bayes estimate of h(·) under squared error loss function is given by hˆB (α, θ) = Eπ (h(α, θ)) RR ∞ 0 RRh(α, θ) J dα dθ . = ∞ 0 J dα dθ

(7)

Generate MCMC samples  [E (Θ|data)] = 

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1 N − N0

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N X

 Θi  ,

i=N0 +1

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HPD interval estimation

For computation of the higher posterior density (HPD) credible interval of Θ, order the MCMC sample of Θ as Θ(1) , Θ(2) , Θ(3) , · · · , Θ(N) . Construct all the 100(1-γ)% credible intervals of Θ say (Θ(1) , Θ(Nb1−γc+1) ), (Θ(2) , Θ(Nb1−γc+2) ) · · · ,(Θ(bNγc) , Θ(N) ). The HPD credible interval of α and β is that interval which has the shortest length.

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1

Background

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Estimating Model

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Application

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Conclusions

Arun Kaushik

Application

In the present study We have studied two data sets. The first data set considered by us, represent the survival times for patients with plasma cell myeloma, already reported in Carbone et. al. (1967)[5]. Data contains the response time to therapy of 112 patients with plasma cell myeloma (a tumor of the bone marrow composed of cells normally found in bone marrow) treated at the National Cancer Institute, Bethesda, Maryland.

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θ

0.10

0.15

Application

0.05

ML

0.00

Initial

0

1

2

3

4

5

α Figure: Contour plot for plasma cell myeloma data

Maximum likelihood estimates for parameters are α ˆ ML = 1.4325, θˆML = 0.0571 and EM estimates are α ˆ EM = 1.4247, ˆ θEM = 0.05783. Arun Kaushik (BHU)

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Application

0.5

chain 3

chain

1

1 0.4

2

2

3

3

2

q

a

0.3

0.2 1 0.1

0.0

0 4000

5000

6000

7000

8000 0.0 0.2 0.4 0.6 0.8

4000

5000

6000

7000

8000 0 2

4

6

Figure: Iteration and density plot of MCMC samples for plasma cell myeloma data

α ˆ B = 1.4301, θˆB = 0.0581 under non-informative independent priors. The 95% HPD interval estimates for α is obtained as (1.0001,1.6109) and for θ as (0.0424, 0.0719). Arun Kaushik (BHU)

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The second data set, considered here, arose in the tests on endurance of deep groove ball bearings. Reported by Lawless (2002)[9]. This data contains the number of million revolutions before failure for each of the 23 ball bearings. The data points are exact observations. For the illustration of our methodology, we have generated censored data for prefixed number of inspections by specifying the inspection times and dropping probabilities.

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We fixed the experimentation time as 140 unit of time and considered to have 8 inspections during this period. We have considered four different inspection plans. The first plan consists of equally spaced inspection time i.e. at 20, 40, . . . , 140 units of time. 1,

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2,

8,

4,

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3,

2,

2

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0.07 0.05 0.03 0.01

0.01

0.03

0.05

0.07

Contour plot for censored data with Inspection scheme A when dropping scheme 1 and complete data set

5

10

15

20

Censored

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5

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15

20

Complete

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Censoring scheme

Name 1 2 3 4 5

Dropping Probabilities 0*7,1 0.2*7,1 0.2*3,0.1*4,1 0.1*4,0.2*3,1 0.25,0*6,1

Arun Kaushik (BHU)

Name A C B D

Inspection time 20 40 60 80 100 120 140 33.00 45.60 51.96 67.80 68.88 98.64 1 36.96 46.85 55.88 65.55 77.31 94.54 1 40 50 60 70 100 120 140

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ML, EM and Bayes estimates of α and θ along with their AC and HPD intervals from simulated data by applying different dropping and inspection scheme on ball bearing dataset Inspection Dropping Scheme scheme A 1 A 2 A 3 A 4 A 5 B 1 B 2 B 3 B 4 B 5 C 1 C 2 C 3 C 4 C 5 D 1 D 2 D 3 D 4 D 5 Complete

αˆML 5.7256 5.6571 5.6128 5.7293 5.6273 5.6181 7.8160 7.9877 7.4394 6.6682 5.868 8.678 7.4335 7.0232 6.4041 7.0402 10.7939 13.6243 9.4270 8.4050 5.2525

AC Interval of α HPD Interval of α αÊM αˆB Lower Upper Lower Upper 1.0579 9.3032 5.7265 5.7256 1.259 8.203 0.0000 10.1801 5.6573 5.6574 0.0000 9.0799 0.0000 9.5055 5.6137 5.614 0.0673 8.4045 0.0000 9.4265 5.7297 5.7299 0.0791 8.3262 0.2748 9.3497 5.6282 5.6278 0.4651 8.2486 0.0000 9.1908 5.6190 5.619 0.0000 8.0897 0.0000 33.0908 7.8169 7.8162 0.0000 31.9897 0.0000 28.4876 7.9883 7.989 0.0000 27.3872 0.0000 17.7743 7.4401 7.4404 0.0000 16.6734 0.0000 11.3201 6.6683 6.6687 0.0000 9.0188 0.0000 9.4336 5.8687 5.8693 0.0000 8.3327 0.0000 46.1005 8.6781 8.6786 0.0000 45.0000 0.0000 38.6954 7.4337 7.4343 0.0000 37.5953 0.0000 22.1020 7.0233 7.0242 0.0000 21.0012 0.0000 12.6524 6.4044 6.4051 0.0000 9.5510 0.0000 21.8707 7.0411 7.0415 0.0000 20.7702 0.0000 64.2913 10.7939 10.7942 0.0000 63.1908 0.0000 47.2660 13.6250 13.6248 0.0000 46.1648 0.0000 23.0812 9.4277 9.4276 0.0000 19.9800 0.0000 13.7972 8.4050 8.4063 0.0000 12.6967 1.2716 9.2933 — 5.2428 1.572 9.3819

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θˆML 0.039 0.0385 0.0385 0.0389 0.039 0.0393 0.0395 0.0395 0.0394 0.0394 0.0391 0.0395 0.0390 0.0393 0.0405 0.0398 0.0409 0.0408 0.0404 0.0401 0.0322

AC Interval of θ Lower Upper 0.0184 0.0454 0.0179 0.0449 0.018 0.045 0.0183 0.0454 0.0184 0.0455 0.0187 0.0459 0.0159 0.0489 0.0164 0.0484 0.0173 0.0474 0.0182 0.0466 0.0186 0.046 0.0166 0.0483 0.0166 0.0463 0.0178 0.0469 0.0201 0.0473 0.019 0.0465 0.0165 0.0511 0.0169 0.0506 0.0184 0.0483 0.0187 0.0474 0.0198 0.0449

θÊM

θˆB

0.0394 0.0387 0.0389 0.0393 0.0394 0.0394 0.0401 0.0396 0.0399 0.0394 0.0393 0.0397 0.0391 0.0394 0.0408 0.0403 0.0411 0.0413 0.0409 0.0406 —

0.039 0.0389 0.0388 0.0389 0.0394 0.0391 0.0392 0.0391 0.0393 0.0396 0.0392 0.0400 0.0392 0.0394 0.0401 0.0398 0.0404 0.0403 0.0395 0.0402 0.0319

HPD Interval of θ Lower Upper 0.0245 0.0412 0.0240 0.0415 0.0241 0.0413 0.0242 0.0411 0.0248 0.0415 0.0244 0.0415 0.0235 0.04291 0.0231 0.0428 0.0241 0.0422 0.0247 0.0424 0.0246 0.0420 0.0230 0.0439 0.0229 0.0435 0.0239 0.0425 0.0247 0.0427 0.0251 0.0423 0.0229 0.0443 0.0225 0.0437 0.0238 0.0431 0.0243 0.0426 0.0256 0.0427

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MLEs and 95 % Interval estimates for model parameters under different choice of censoring scheme

MLEs and ACIs of α

MLEs and ACIs of θ 0.05

60

Inspection Scheme 40

Inspection Scheme 0.04 A

A

B

B C 20

0.03

C

D

D 0.02

0 1

2

3

4

1

5

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2

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Dropping Scheme

Dropping Scheme

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1

Background

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Estimating Model

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5

Application

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Conclusions

Arun Kaushik

In this section we have studied the performance of the various estimators on the basis of their simulated mean square error (MSE). The MSE’s of the estimators are estimated on the basis of Monte-Carlo simulation study of 2000 samples. It is noted that Newton-Raphson algorithm has convergence rate of 85%-90%, whereas the EM algorithm converges most of the times. We report the results omitting those cases where the algorithms do not converge.

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To simulate progressive Type-I interval censored sample from the considered distribution, we propose the use of the algorithm given by 4 as : Determine the number of [3, pp.200] after modifying step droppings at j th stage by generating Rj from Bin(n − x − r − dj , pj ). 1

X Specify the value of n,k, T=(T1 , T2 , . . . , Tk )and p X Specify the value of model parameters α and θ

2 Set i = 0, dsum = 0, rsum = 0

3 i =i +1

4 If i = k + 1, exit the algorithm

5 Generate xi as a binomial random variable with (n − dsum − rsum)

and [F (Ti ) − F (Ti−1 )]/[1 − F (Ti−1 )] 6 Set dsum = dsum + di

7 If i < k, Generate ri from B(n − dsum − rsum, pi )

Else ri = (n − dsum − rsum) 8 Set rsum = rsum + ri

9 Go to step iii.

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ML, EM and bayes estimates of parameters along with ML estimate of reliability and hazard on the bases of simulated data for α = 2.5, θ = 2 1

Dropping Scheme 03 1 2 5 3 4

αML

θML

α ˆ EM

θÊM

α ˆB

θˆB

SˆML (t = 1)

3.072(1.1609) 3.0936(1.3329) 4.5742(4.2195) 4.2535(3.1504) 4.2493(3.1037) 3.5187(1.8215)

2.2027(0.5126) 2.1844(0.5697) 2.3974(0.8522) 2.3918(0.876) 2.3863(0.8205) 2.2976(0.7256)

– 3.0936(1.3329) 4.5742(4.2195) 4.2535(3.1504) 4.2493(3.1037) 3.5187(1.8215)

– 2.1844(0.5697) 2.3974(0.8522) 2.3918(0.876) 2.3863(0.8205) 2.2976(0.7256)

2.9853(0.7594) 3.0866(0.9275) 4.592(1.4053) 4.191(2.6922) 3.9366(2.6659) 3.3828(3.7513)

2.0697(0.2792) 2.0389(0.3441) 2.1024(0.4667) 2.2711(0.5589) 2.2595(0.5982) 2.1936(0.674)

0.2954 0.2975 0.2869 0.2893 0.2886 0.2929

1.9722 1.9609 2.1766 2.1692 2.1621 2.0704

30

0 1 2 3 4 5

2.8437(0.8030) 2.8430(0.8694) 3.069(1.3157) 3.0194(1.2536) 3.0598(1.2116) 2.9323(1.0941)

2.1146(0.4054) 2.1005(0.4128) 2.2074(0.6575) 2.1693(0.6293) 2.1617(0.5704) 2.1377(0.5208)

– 2.8430(0.8694) 3.0690(1.3157) 3.0194(1.2536) 3.0598(1.2116) 2.9323(1.0940)

– 2.1005(0.4128) 2.2074(0.6575) 2.1693(0.6293) 2.1617(0.5704) 2.1377(0.5208)

2.5995(0.5193) 2.6555(0.6105) 2.8678(1.0634) 2.8014(1.0399) 2.8984(0.9592) 2.7674(0.8894)

1.9845(0.2524) 1.9907(0.2257) 2.0814(0.483) 2.0551(0.4667) 2.0304(0.4672) 2.0013(0.3746)

0.3021 0.3032 0.2979 0.2990 0.3019 0.2998

1.8874 1.8778 1.9889 1.9536 1.9370 1.9185

40

0 1 2 3 4 5

2.7522(0.7113) 2.7533(0.8041) 2.9138(1.0462) 2.9759(1.0445) 2.8454(0.9456) 2.8313(0.8899)

2.0954(0.3489) 2.0776(0.3893) 2.1162(0.7111) 2.1556(0.7052) 2.1801(0.5425) 2.1106(0.4869)

– 2.7533(0.8041) 2.9138(1.0462) 2.9759(1.0448) 2.8454(0.9455) 2.8313(0.8899)

– 2.0776(0.3893) 2.1556(0.7112) 2.1555(0.7049) 2.1806(0.5426) 2.1101(0.4869)

2.591(0.5347) 2.5881(0.6730) 2.7162(0.9159) 2.8396(0.8938) 2.6756(0.8132) 2.6577(0.6952)

1.9956(0.2440) 1.9806(0.2539) 2.076(0.6126) 2.0388(0.3861) 2.0332(0.3189) 1.9689(0.5889)

0.3000 0.3033 0.2980 0.2971 0.3016 0.2997

1.8913 1.8565 1.9392 1.9585 1.8914 1.8948

50

0 1 5 2 3 4

2.7681(0.7113) 2.7532(0.8041) 2.8145(0.9059) 2.8326(0.8993) 2.8009(0.8623) 2.7499(0.8065)

2.0954(0.3488) 2.0776(0.3893) 2.1098(0.6149) 2.1195(0.4954) 2.1017(0.4742) 2.0883(0.4323)

– 2.7532(0.8041) 2.8145(0.9059) 2.8326(0.8993) 2.8009(0.8623) 2.7499(0.8065)

– 2.0776(0.3893) 2.1098(0.6149) 2.1195(0.4954) 2.1017(0.4742) 2.0883(0.4323)

2.5336(0.6450) 2.4924(0.6998) 2.5954(0.7606) 2.5351(0.8321) 2.5879(0.7338) 2.5242(0.7548)

2.0335(0.1833) 2.0157(0.2412) 2.0466(0.4659) 2.0126(0.4803) 2.0022(0.411) 1.9891(0.3039)

0.3016 0.3033 0.3019 0.2995 0.3028 0.3018

1.8707 1.8559 1.8942 1.9041 1.8837 1.8709

n

20

1

HˆML (t = 1)

2

Here, true value of reliability at time 1 is S(1) = 0.3048 True value of hazard rate at time 1 is H(1) = 1.7851 3 Arun Kaushik (BHU) PTI Interval December 20, 2015 23 / 33 0 means complete case, when no dropping and data points collected continuously 2

ML, EM and bayes estimates of parameters along with ML estimate of reliability and hazard on the bases of simulated data for n = 30 α

θ

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5

0.5 0.5 0.5 1.5 1.5 1.5 2.5 2.5 2.5 0.5 0.5 0.5 1.5 1.5 1.5 2.5 2.5 2.5 0.5 0.5 0.5 1.5 1.5 1.5 2.5 2.5 2.5

Dropping Scheme 0 1 4 0 1 4 0 1 4 0 1 4 0 1 4 0 1 4 0 1 4 0 1 4 0 1 4

αˆ ML

θˆML

0.5449(0.0152) 0.5455(0.0155) 0.5483(0.0164) 0.5346(0.0146) 0.5416(0.0146) 0.5469(0.0151) 0.5334(0.0145) 0.5367(0.0150) 0.5448(0.0152) 1.6645(0.2268) 1.6711(0.2277) 1.6811(0.2311) 1.6888(0.2350) 1.6926(0.2358) 1.6953(0.2363) 1.6877(0.2310) 1.6909(0.2318) 1.6991(0.2321) 2.8912(0.9119) 2.8916(0.9124) 2.8990(0.9133) 2.8226(0.8472) 2.8289(0.8475) 2.8355(0.8478) 2.8473(0.8631) 2.8498(0.8641) 2.8523(0.8645)

0.5647(0.0282) 0.5717(0.0287) 0.5784(0.0293) 1.6681(0.2519) 1.6698(0.2528) 1.6701(0.2528) 2.7835(0.7001) 2.7828(0.7004) 2.7859(0.7012) 0.5363(0.0134) 0.5433(0.0136) 0.5430(0.0139) 1.6039(0.1192) 1.6067(0.1196) 1.6097(0.1203) 2.6877(0.3341) 2.6915(0.3346) 2.6983(0.3352) 0.5320(0.0107) 0.5369(0.0123) 0.5365(0.0131) 1.5908(0.0959) 1.5912(0.0965) 1.5980(0.0965) 2.6456(0.2645) 2.6449(0.2655) 2.6446(0.2663)

Arun Kaushik (BHU)

αˆ EM

θÊM

0.5455(0.0158) 0.5718(0.0287) 0.5482(0.0164) 0.5785(0.0294) 0.5414(0.0146) 1.6697(0.2529) 0.5471(0.0152) 1.6701(0.2528) 0.5365(0.015) 2.7831(0.7002) 0.5449(0.0151) 2.7858(0.7015) 1.6708(0.2371) 0.5432(0.0137) 1.6809(0.2273) 0.5433(0.014) 1.6928(0.2359) 1.6065(0.1195) 1.6954(0.2364) 1.61(0.1201) 1.6908(0.2317) 2.6918(0.3346) 1.6993(0.232) 2.6981(0.3355) 2.8916(0.9125) 0.5367(0.0113) 2.8988(0.9134) 0.5368(0.0124) 2.8288(0.8478) 1.5915(0.0961) 2.8356(0.8478) 1.5981(0.0966) 2.8497(0.864) 2.6448(0.2658) 2.8524(0.8643) 2.6443(0.2666)

PTI Interval

SˆML (t = 1)

HˆML (t = 1)

αˆ B

θˆB

0.3691(0.0045) 0.3685(0.0046) 0.3650(0.0048) 0.1157(0.0024) 0.1091(0.0026) 0.1104(0.0027) 0.0422(0.0007) 0.0359(0.0008) 0.0356(0.0009) 0.7552(0.0043) 0.7699(0.0044) 0.7714(0.0045) 0.3140(0.0048) 0.3179(0.0050) 0.3192(0.0052) 0.1182(0.0021) 0.1168(0.0024) 0.1132(0.0026) 0.9078(0.0017) 0.9253(0.0018) 0.9232(0.0021) 0.4636(0.0051) 0.4795(0.0053) 0.4743(0.0054) 0.1909(0.0038) 0.1931(0.0039) 0.1902(0.0043)

0.6977(0.0217) 0.7090(0.0219) 0.7195(0.0223) 1.7587(0.2476) 1.7632(0.2478) 1.7579(0.2479) 2.8343(0.7332) 2.8309(0.7333) 2.8273(0.7336) 0.3868(0.0075) 0.3903(0.0075) 0.3833(0.0078) 1.4903(0.1136) 1.4886(0.1138) 1.4923(0.1140) 2.6306(0.3633) 2.6292(0.3636) 2.6366(0.3638) 0.1984(0.0038) 0.1975(0.0041) 0.1955(0.0044) 1.293(0.0737) 1.2745(0.0740) 1.2854(0.0740) 2.4825(0.3134) 2.4767(0.3141) 2.4665(0.3144)

0.5163(0.0150) 0.5201(0.0153) 0.5198(0.0154) 0.5100(0.0129) 0.5187(0.0137) 0.5231(0.0151) 0.5130(0.0131) 0.5091(0.0143) 0.5220(0.0147) 1.5798(0.2241) 1.5839(0.2256) 1.5884(0.2282) 1.5911(0.2339) 1.5899(0.2354) 1.5885(0.2271) 1.5848(0.2245) 1.5886(0.2243) 1.5947(0.2319) 2.6953(0.9069) 2.6881(0.9105) 2.6941(0.9101) 2.6549(0.8414) 2.6644(0.8453) 2.6610(0.8458) 2.6651(0.8566) 2.6650(0.8583) 2.6697(0.8582)

0.5179(0.0267) 0.5163(0.0269) 0.5210(0.0278) 1.5549(0.2468) 1.5519(0.2485) 1.5468(0.2499) 2.5849(0.6971) 2.5920(0.6970) 2.5897(0.6932) 0.5108(0.0080) 0.5120(0.0095) 0.5048(0.0122) 1.5269(0.1119) 1.5269(0.1140) 1.5288(0.1089) 2.5540(0.3241) 2.5579(0.3238) 2.5588(0.3251) 0.5013(0.0103) 0.5043(0.0116) 0.5106(0.0128) 1.5273(0.0908) 1.5256(0.0951) 1.5247(0.0958) 2.5405(0.2532) 2.5433(0.2557) 2.5429(0.2560)

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Average Bayes estimates(MSE in brackets) and 95% HPD intervals based on simulated data by scheme A-1 with taking different choice of prior

g1 (α) G (4, 2) 4 G(4,2) G(4,2) G(1,0.5) G(1,0.5) G(1,0.5) G(0.4,0.2)

4

g2 (θ) G (4, 2) G(1,0.5) G(0.4,0.2) G(4,2) G(1,0.5) G(0.4,0.2) G(0.4,0.2)

α ˆB 1.9030(0.8063) 1.9152(0.8342) 1.8542(0.9134) 1.8866(0.9075) 1.8866(0.9275) 1.8671(0.9324) 1.8539(0.9453)

HPD Interval (3.6263, 2.1042) (3.7263, 2.0055) (3.7963, 1.8692) (3.7701, 1.9506) (3.9263, 1.8069) (3.9292, 1.7864) (4.5138, 1.7001)

θˆB 2.0112(0.2760) 2.0276(0.3158) 2.0412(0.3564) 2.0198(0.3023) 2.0389(0.3441) 2.0500(0.3674) 2.0567(0.3623)

HPD Interval (1.3451, 2.6745) (1.3071, 2.7326) (1.0128, 2.9745) (1.2016, 2.8618) (1.1129, 2.9045) (0.9976, 3.0198) (0.9900, 3.1199)

G (a, b) denotes the gamma prior with shape parameter a and scale parameter b Arun Kaushik (BHU)

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1

Background

2

Estimating Model

3


4


5

Application

6


7

Conclusions

Arun Kaushik

Conclusion I In this work we have considered both classical and Bayesian analysis for the progressive type-i interval censored data, when the lifetime of the items follows exponentiated exponential distribution. The ML estimates do not have explicit forms. Modified Newton-Raphson and EM algorithm has been used to compute the MLEs and it works quite well. We have proposed to use MCMC technique to compute the Bayes estimates when the shape and scale parameters have independent gamma priors. Further, we present a simulation study from real data set and compute the ML and Bayes estimate of parameters; also, compare these estimates with estimates under complete sample. On the bases of our study we propose the estimation procedure under progressive type-I interval censoring is performed similar to estimation under complete sample. Arun Kaushik (BHU)

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Conclusion II

The performance of proposed Bayes estimators α ˆ B and θˆB perform well in (in sense of smaller MSE) comparison to other competitive estimators. It is clearly noticed that performance of considered estimators under PTI interval censoring without dropping is similar to that of complete case, and performance under PTI interval censoring with droppings, is well but slightly poor. Therefore, one can conduct PTI interval, where continuously monitoring also possible, with slightly less efficiency but in very smaller cost for with suitable choice of inspection times.

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Conclusion III

It will be interesting to develop statistical inference of the unknown parameters in presence of covariates. Further, we have considered dropping probabilities at each stages are fixed, but in real life phenomena these may be random and randomness govern by beta, uniform or other supported distribution. Also, it is noticed here that the inspection times having important role in PTI interval censoring, therefore it is necessary, to develop a method for the choice of inspection times. More work is needed in these directions.

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References I

R Aggarwala. Progressive interval censoring: Some mathematical results with applications to inference. Communications in Statistics - Theory and Methods, 30(8-9):1921–1935, jul 2001. S K Ashour and W M Afify. Statistical analysis of exponentiated weibull family under type i progressive interval censoring with random removals. Journal of Applied Sciences Research, 3(12):1851–1863, 2007. N Balakrishnan and E Cramer. The Art of Progressive Censoring. Springer New York, 2014.

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References II

J O Berger and D Sun. Bayesian analysis for the poly-weibull distribution. Journal of the American Statistical Association, 88:1412–1418, 1993. P P Carbone, L E Kellerhouse, and E A Gehan. Plasmacytic myeloma: A study of the relationship of survival to various clinical manifestations and anomalous protein type in 112 patients. The American Journal of Medicine, 42(6):937 – 948, 1967. D G Chen and Y L Lio. Parameter estimations for generalized exponential distribution under progressive type-i interval censoring. Computational Statistics and Data Analysis, 54(6):1581–1591, 2010.

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References III A P Dempster, N M Laird, and D B Rubin. Maximum likelihood from incomplete data via the em algorithm. Journal of the Royal statistical Society, 39(1):1–38, 1977. A Gelmen, J Carlin, H Stern, and D Rubin. Bayesian Data Analysis. Text in Statistical Science, Chapman & Hall, 1995. J F Lawless. Statistical Models and Methods for Lifetime Data. Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, USA, 2002. I Natzoufras. Bayesian Modeling Using WinBugs. Series B, Wiley, 2009. Arun Kaushik (BHU)

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References IV S K Singh, U Singh, M Kumar, and GP Singh. Estimation of parameters of exponentiated pareto distribution for progressive type-ii censored data with binomial random removals scheme. Electronic Journal of Applied Statistical Analysis, 6(2), 2013. S K Tse, C Yang, and H K Yuen. Statistical analysis of weibull distributed lifetime data under type ii progressive censoring with binomial removals. Journal of Applied Statistics, 27(8):1033–1043, November 2000. H K Yuen and S K Tse. Parameters estimation for weibull distributed lifetimes under progressive censoring with random removeals. Journal of Statistical Computation and Simulation, 55(1-2):57–71, September 1996. Arun Kaushik (BHU)

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