Estimation for the generalized Gompertz distribution of hybrid progressive censored samples

Abstract

In this paper, Bayesian and non Bayesian methods are adapted to determined interval and point estimations under Type-I hybrid Progressively censored for generalized Gompertz distribution. The special case from MCMC adjust Metropolis-Hastings within Gibbs sampling in the case Bayesian procedure is used to solve the double integrations. Numerical examples including generation data sets are shown to make a conclusion on methods estimations used here.

Keywords

Type-I hybrid Progressive Bayesian and Non-Bayesian estimations Markov chain Mont-Carlo generalized Gompertz distribution

1 Introduction

The Gompertz distribution has very important attention. Many application this distribution including modeling human mortality, growth model, tumor growth and fitting actuarial tables are considered as applications the Gompertz distribution [1 –4]. El-Gohary et al. (2013) have presented a generalized Gompertz distribution (GGD) [2]. This new distribution generalizes some distributions including Gompertz, exponential and generalized exponential distributions. Cordeiro et al. (2016) have studied general mathematical properties and applications for the generalized Gompertz distribution (exponentiated Gompertz distribution) [3 –5].

The probability, cumulative functions and hazard rate function, for GGD with two shape parameters θ, δ and one scale parameter β are given by; respectively:

$\begin{matrix} f (x; δ, β, θ) & = & β θ [e^{- \frac{β}{δ} (e^{δ x} - 1) + δ x}] \\ {[1 - e^{- \frac{β}{δ} (e^{δ x} - 1)}]}^{θ - 1}, \end{matrix}$ (1) $F (x; δ, β, θ) = {[1 - e^{- \frac{β}{δ} (e^{δ x} - 1)}]}^{θ},$ (2) and $h (x) = \frac{β θ [e^{- \frac{β}{δ} (e^{δ x} - 1) + δ x}] {[1 - e^{- \frac{β}{δ} (e^{δ x} - 1)}]}^{θ - 1}}{{1 - [1 - e^{- \frac{β}{δ} (e^{δ x} - 1)}]}^{θ}},$ (3)

HRF of GGD satisfies the following:

The hazard rate function is increasing function if δ > 0 or constant function if δ = 0.

The hazard function is increasing function when θ > 1; and

The hazard rate function will be either decreasing if δ = 0 or bath-tub if δ > 0 for θ < 1 .

where δ > 0, θ > 0, β > 0, x > 0 and t > 0.

The organization paper are, we introduce the model description provide the necessary assumptions in Section 2. The MLEs are obtained and their exact distributions are provided in Section 3. The construction confidence intervals is provided in Section 4. The Bayes procedure is presented in Section 5. Bayesian estimation using MCMC approach is obtained in Section 6. In Section 7, some numerical results are found. Finally, discussions, remarks, and conclusions will be given in Section 8.

2 The model description

In recent few years, an important role was played by censoring schemes. A complete information on failure times may be unable to obtain for all experimental items in many cases. The second reason for using censoring schemes is reducing cost and time. The flexibility allowing removing items during the experiment may be not found in many kinds censoring schemes except in the progressive censoring. It has an advantage property that the experimenter can remove at specific steps one or more items. Many new researches in Type-I and Type-II hybrid progressive censoring schemes have appeared recently. The Type-II progressively hybrid is taking a lot time to obtain the maximum number predetermined number failures m and time T to terminate the experiment. So that the Type-I hybrid progressive censoring scheme (TIHPCS) is considered suitable and then suggested here.

TIHPCS is suggested here to make an inference including interval and point estimations for unknown parameters GGD. In a Type-I censoring scheme, the time elapsed and the number failures the experiment are determined in fact before the beginning the experiment. The describing TIHPCS is in follow: Assuming that n identical items are observed in a particular experiment. Let X₁, . . . . , X_m be and the lifetimes the m items, m < n. Also, let R₁, . . . , R_m are integers greater than zero and predetermined beforehand, where $n = m + \sum_{i = 1}^{m} R_{i}$ . R₁ units are randomly removed when first failure occur. From the remaining n - 1 units, R₂ units the rest n - R₁ - 1 are randomly removed when the second failure occur. This process will continue until reach at min {m, T} i.e. If the m^th failure happen before a prescribed time T, the experiment will be finished at X_(m:m:n) time point. In the other hand, the test quit at time T satisfying X_(j:m:n) < T < X_(j+1:m:n) (see Fig 1). The life test is terminated at a random time T₁ = min (X_(m:m:n), T) and the observed sample take one the following two cases (see Kayal et al. (2017): $\begin{matrix} CaseI {X_{(1 : m : n)}, X_{(2 : m : n)}, \dots, X_{(m : m : n)}}, \\ if X_{(m : m : n)} < T \\ CaseII {X_{(1 : m : n)}, X_{(2 : m : n)}, \dots, X_{(j : m : n)}}, if \\ X_{(j : m : n)} < T < X_{(j + 1 : m : n)} . \end{matrix}$ (4)

Fig. 1

Experiment terminate at m^th failure i.e. x_m < T.

Fig. 2

Experiment terminate at time T, i.e. T < x_m.

Fig. 3

The graph the likelihood and the scale parameter β.

Fig. 4

The graph the likelihood and the shape parameter θ.

Many papers have appeared in progressively hybrid and the other censoring types as you see in ref’s [6 –26]. For extensive survey, see book "Art Progressive Censoring" by Balakrishnan, and Cramer, (2014) [14].

3 The maximum likelihood method

This section deals with obtaining interval and point estimation for the unknown parameters θ, δ and β GGD by MLE technique. The likelihood function can written as: $Case I : L \propto \prod_{i = 1}^{m} f (x_{i}) {[1 - F (x_{i})]}^{R_{i}},$ (5)

$Case II : L \propto \prod_{i = 1}^{J} f (x_{i}) {[1 - F (x_{i})]}^{R_{i}} {[1 - F (T)]}^{R_{j}^{*}},$ (6)

where $R_{j}^{*} = n - j - \sum_{i = 1}^{j} R_{i}$ .

Substituting from Equations (1 - 2) in Eqs. (5 - 6); yields: $\begin{matrix} Case I : & L \propto \prod_{i = 1}^{m} β θ [e^{- \frac{β}{δ} (e^{δ x_{i}} - 1) + δ x_{i}}] \\ \times {[1 - e^{- \frac{β}{δ} (e^{δ x_{i}} - 1)}]}^{θ - 1} \\ \times {({1 - [1 - e^{- \frac{β}{δ} (e^{δ x_{i}} - 1)}]}^{θ})}^{R_{i}}, \end{matrix}$ (7) $\begin{matrix} Case II : & L \propto \prod_{i = 1}^{j} β θ [e^{- \frac{β}{δ} (e^{δ x_{i}} - 1) + δ x_{i}}] \\ \times {[1 - e^{- \frac{β}{δ} (e^{δ x_{i}} - 1)}]}^{θ - 1} \\ \times {({1 - [1 - e^{- \frac{β}{δ} (e^{δ x_{i}} - 1)}]}^{θ})}^{R_{i}} \\ \times {({1 - [1 - e^{- \frac{β}{δ} (e^{δ T} - 1)}]}^{θ})}^{R_{j}^{*}} \end{matrix}$ (8)

The likelihood in the two cases can be written as: $\begin{matrix} L \propto & \prod_{i = 1}^{d} β θ [e^{- \frac{β}{δ} (e^{δ x_{i}} - 1) + δ x_{i}}] \\ \times {[1 - e^{- \frac{β}{δ} (e^{δ x_{i}} - 1)}]}^{θ - 1} \\ \times {({1 - [1 - e^{- \frac{β}{δ} (e^{δ x_{i}} - 1)}]}^{θ})}^{R_{i}} \\ \times w (θ, β, δ; T), \end{matrix}$ (9) where $d = {\begin{matrix} m; m < T \\ j; m > T \end{matrix}$ and $\begin{matrix} w (θ, β, δ; T) \\ = {\begin{matrix} 1; m < T \\ e^{R_{j}^{*} log ({1 - [1 - e^{- \frac{β}{δ} (e^{δ T} - 1)}]}^{θ})}; m > T \end{matrix} \end{matrix}$

From Equation (9), the associated log-likelihood function is:

$\begin{matrix} log & L \propto d log θ + d log β + δ \sum_{i = 1}^{d} x_{i} \\ - \frac{β}{δ} \sum_{i = 1}^{d} e^{(δ x_{i} - 1)} + (θ - 1) \sum_{i = 1}^{d} log [1 - e^{- \frac{β}{δ} (e^{δ x_{i}} - 1)}] \\ + \sum_{i = 1}^{d} R_{i} log ({1 - [1 - e^{- \frac{β}{δ} (e^{δ x_{i}} - 1)}]}^{θ}) \\ + log w (θ, β, δ; T), \end{matrix}$ (10)

differentiating Eq. (10) by θ, β, and δ, and solving the non-linear Equations will give the estimators θ, δ and β.

4 Asymptotic confidence intervals

The main aim this section is find confidence interval estimation for unknown parameters. For this purpose, the Fisher information matrix I = [I_i,j] i, j = 1, 2 is given by (see Greene (2000) [18], Agresti (2002) [5]): $I = - [\begin{matrix} \frac{\partial^{2} log L}{\partial θ^{2}} & \frac{\partial^{2} log L}{\partial θ \partial β} & \frac{\partial^{2} log L}{\partial θ \partial δ} \\ \frac{\partial^{2} log L}{\partial β \partial θ} & \frac{\partial^{2} log L}{\partial β^{2}} & \frac{\partial^{2} log L}{\partial β \partial δ} \\ \frac{\partial^{2} log L}{\partial δ \partial θ} & \frac{\partial^{2} log L}{\partial δ \partial β} & \frac{\partial^{2} log L}{\partial δ^{2}} \end{matrix}] = [\begin{matrix} I_{11} & I_{12} & I_{13} \\ I_{21} & I_{22} & I_{23} \\ I_{31} & I_{32} & I_{33} \end{matrix}] .$ (11)

The variance-covariance matrix approximated as $\begin{matrix} [V_{ij}] = [I]^{- 1} & = - {[\begin{matrix} \frac{\partial^{2} log L}{\partial θ^{2}} & \frac{\partial^{2} log L}{\partial θ \partial β} & \frac{\partial^{2} log L}{\partial θ \partial δ} \\ \frac{\partial^{2} log L}{\partial β \partial θ} & \frac{\partial^{2} log L}{\partial β^{2}} & \frac{\partial^{2} log L}{\partial β \partial δ} \\ \frac{\partial^{2} log L}{\partial δ \partial θ} & \frac{\partial^{2} log L}{\partial δ \partial β} & \frac{\partial^{2} log L}{\partial δ^{2}} \end{matrix}]}^{- 1} \\ = {[\begin{matrix} V_{11} & V_{12} & V_{13} \\ V_{21} & V_{22} & V_{23} \\ V_{31} & V_{32} & V_{33} \end{matrix}]}_{(\hat{θ}, \hat{β}, \hat{δ})} . \end{matrix}$ (12)

The confidence interval with level significance (η) for θ, β, δ; respectively are:

$\begin{matrix} \hat{θ} \pm Z_{(η / 2)} \sqrt{{\hat{V}}_{11}}, \hat{β} \pm Z_{(η / 2)} \sqrt{{\hat{V}}_{22}} \\ and \hat{δ} \pm Z_{(η / 2)} \sqrt{{\hat{V}}_{33}}, \end{matrix}$ (13) where ${\hat{V}}_{11}$ , ${\hat{V}}_{22}$ and ${\hat{V}}_{33}$ are the diagonal elements the $[V_{ij}] (\hat{θ}, \hat{β}, \hat{δ})$ and z_η/2 is the value obtained from the standard normal distribution with level significance η.

5 Bayesian procedure

The target this section is using Bayesian procedure for estimation purpose for unknown parameters GGD.

Suppose that X₁, X₂, . . . , X_d denotes a Type-I progressive hybrid censored sample drawn from a Burr X (θ, β, δ) distribution. It is assumed that θ and β are independent variables and have Gamma(a, b) and Gamma(p, q) distributions; respectively as a prior distributions. $π (θ) \propto θ^{a - 1} e^{- b θ}, θ, > 0, a > 0, b > 0,$ (14) $π (β) \propto β^{p - 1} e^{- q β}, β > 0, p > 0, q > 0 .$ (15) and $π (δ) \propto δ^{u - 1} e^{- v δ}, δ > 0, u > 0, v > 0 .$ (16)

The joint prior distribution θ, β and δ is: $g (θ, β, δ) = L (θ, β, δ | x) π (θ) π (β) π (δ) .$ (17)

From Eq. (17) the posterior density function θ, β and δ denoted by π^∗ (θ, β, δ|x), can written in the form: $\begin{matrix} π^{*} (θ, β, δ | x) \\ = \frac{L (θ, β, δ | x) π (θ) π (β) π (δ)}{\int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} L (θ, β, δ | x) π (θ) π (β) π (δ) d θ d β d δ} . \end{matrix}$ (18)

Under squared error loss function, the Bayes estimator is the posterior mean and giving by:

$\begin{matrix} \hat{g} (θ, β, δ | x) & = E (g (θ, β, δ | x)) \\ = \frac{\int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} g (θ, β, δ) L (θ, β, δ | x) π (θ) π (β) π (δ)}{\int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} L (θ, β, δ | x) π (θ) π (β) π (δ) d θ d β d δ} . \end{matrix}$ (19)

Its very hard to solve Eq. (19). We expect that the MCMC method will be the suitable method to solve the triple integration. Unfortunately, the MCMC was not good and the calculations still hard to evaluate the Eq.(19). So, we assumed that the parameter δ is known and we deal with two parameters only in the next sections. The idea MCMC is using the posterior distributions to generate samples and then compute the Bayes estimator $\hat{g} (θ, β | x; δ)$ .

6 The MCMC approach

MCMC approach widely used in the case Bayesian procedure especially in the case difficulty evaluating integrations. This method has presented in (1953) by Metropolis et al. [22] and extended later in (1970) by Hastings [19]. As expected, the MCMC was the suitable method to estimate $\hat{θ}$ and $\hat{β}$ by Bayesian approach. The joint-posterior density function Eq. (19) can be written in the form: $\begin{matrix} g (θ, β, δ | x) \propto θ^{d + a - 1} β^{d + p - 1} δ^{d + u - 1} e^{- (b θ + q β + v δ)} \\ \times \prod_{i = 1}^{d} [e^{- \frac{β}{δ} (e^{δ x_{i}} - 1) + δ x_{i}}] {[1 - e^{- \frac{β}{δ} (e^{δ x_{i}} - 1)}]}^{θ - 1} \\ \times {({1 - [1 - e^{- \frac{β}{δ} (e^{δ x_{i}} - 1)}]}^{θ})}^{R_{i}} \times w (θ, β, δ; T) . \end{matrix}$ (20) where $w (θ, β, δ; T) = {\begin{matrix} 1; m < T \\ e^{R_{j}^{*} log ({1 - [1 - e^{- \frac{β}{δ} (e^{δ T} - 1)}]}^{θ})}; m > T \end{matrix}$

From Eq. (20), the prior density function θ given δ, β and β given δ, θ are, respectively: $\begin{matrix} g_{1} (θ | β, δ; x) & \propto θ^{d + a - 1} e^{- (b θ)} \prod_{i = 1}^{d} {[e^{- \frac{β}{δ} (e^{δ x_{i}} - 1)}]}^{θ - 1} \\ \times {({1 - [1 - e^{- \frac{β}{δ} (e^{δ x_{i}} - 1)}]}^{θ})}^{R_{i}} \\ \times w (θ, β, δ; T) . \end{matrix}$ (21) and $\begin{matrix} g_{2} (β | θ, δ; x) \propto β^{d + p - 1} e^{- (q β)} \prod_{i = 1}^{d} [e^{- \frac{β}{δ} (e^{δ x_{i}} - 1)}] \\ \times & {[1 - e^{- \frac{β}{δ} (e^{δ x_{i}} - 1)}]}^{θ - 1} {({1 - [1 - e^{- \frac{β}{δ} (e^{δ x_{i}} - 1)}]}^{θ})}^{R_{i}} \\ \times w (θ, β, δ; T) . \end{matrix}$ (22)

The Eqs (21 - 22) can not be analytically reduced to well-known distributions. The normal distribution was suggested as a proposal distribution. The MCMC method with normal which taken as a proposal distribution will be used to give credible intervals for θ and β. Fortunately, the results were very wonderful. The algorithm Metropolis-Hastings method can be written as:

Start with β⁽⁰⁾, θ⁽⁰⁾

Set τ = 1 .

Generate θ^τ from g₁ with the normal N (θ^τ-1, V₁₁). Where V obtained in Eq. (11).

Generate β^τ from g₂ with the normal N (β^τ-1, V₂₂) .

Compute θ^τ and β^τ .

Set τ = τ + 1 .

Repeat 3-6 steps N times.

Rearrange the values θ_i and β_i, i = M + 1, M + 2, ·· · , N .

The Bayes estimates θ and β can be obtained as:

$E (θ | data; δ) = \frac{1}{N - M} \sum_{i = M + 1}^{N} θ_{i}$ and

$E (β | data; δ) = \frac{1}{N - M} \sum_{i = M + 1}^{N} β_{i}$ ,

where M is burn-in.

100 (1 - η)% credible intervals θ and β can be computed as: $(θ_{{\frac{η}{2} (N - M)}}, θ_{{\frac{1 - η}{2} (N - M)}})$ and

$(β_{{\frac{η}{2} (N - M)}}, β_{{\frac{1 - η}{2} (N - M)}}) .$

7 Simulation study

In the simulation studies, all computations are performed using Mathematica 8 program with 4 GB RAM and processor Core i7. Monte Carlo simulations are employed to compare the different methods of estimation discussed in the preceding sections. samples have generated from Type-I hybrid progressive samples from GGD with different censoring schemes R contains N = 11000 values with discard the first M = 1000 values in the case MCMC as burn-in. Mathematica ver. 8.0 program is used to generate samples and then used for the estimation process. In Tables (2 - 7) below, interval and point estimation are calculated for θ and β by maximum likelihood (ML) and Bayes procedure with MCMC technique including non-informative prior denoted by (MCMC0) and informative prior denoted by (MCMC1) are calculated numerically. Point estimation (P.E.), mean squared error (MSE), lower limit interval estimation (LL), upper limit interval estimation (UL) and interval length (Len) using maximum likelihood and Bayesian including informative and non-informative priors methods are compared via MSE. All results are obtained at different values T = 1, 5 ; θ = 2 ; β = 3 ; t = 0.3 ; a = 12 ; b = 0.25 ; p = 10 ; q = 0.2 ; δ = 1.5 ; N = 11000 ; M = 1000 ; η = 0.05 and different censoring schemes (CS) (see Table 1).

Table 1
Censoring schemes with different values for n and m, where 0^r means that 0 repeated r times

Censoring scheme n m R

CS ₁ 10 5 R₁ = {1, 0, 1, 1, 2}

CS ₂ 20 5 R₂ = {1, 4, 3, 4, 3}

CS ₃ 20 10 R₃ = {1, 0, 3, 1, 1, 0, 1, 0, 2, 1}

CS ₄ 30 10 R₄ = {2, 1, 4, 1, 3, 4, 1, 2, 1, 1}

CS ₅ 30 15 R₅ = {1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 2, 0, 0, 1, 1}

CS ₆ 40 20 R₆ = {1, 1, 2, 3, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 1}

CS ₇ 40 30 R₇ = {0⁸, 2, 0⁶, 3, 1, 0³, 1, 0³, 1, 0⁴, 2}

CS ₈ 50 20 R₈ = {0⁸, 2, 0⁶, 8, 3, 0³, 5, 0⁸, 2}

CS ₉ 50 25 R₉ = {2, 1, 1, 1, 0, 1, 1, 0, 2, 0, 1, 2, 0, 1, 0, 2, 3, 0, 1, 0, 1, 0, 2, 1, 2}

CS ₁₀ 50 30 R₁₀ = {1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 2}

Censoring scheme	n	m	R
CS ₁	10	5	R₁ = {1, 0, 1, 1, 2}
CS ₂	20	5	R₂ = {1, 4, 3, 4, 3}
CS ₃	20	10	R₃ = {1, 0, 3, 1, 1, 0, 1, 0, 2, 1}
CS ₄	30	10	R₄ = {2, 1, 4, 1, 3, 4, 1, 2, 1, 1}
CS ₅	30	15	R₅ = {1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 2, 0, 0, 1, 1}
CS ₆	40	20	R₆ = {1, 1, 2, 3, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 1}
CS ₇	40	30	R₇ = {0⁸, 2, 0⁶, 3, 1, 0³, 1, 0³, 1, 0⁴, 2}
CS ₈	50	20	R₈ = {0⁸, 2, 0⁶, 8, 3, 0³, 5, 0⁸, 2}
CS ₉	50	25	R₉ = {2, 1, 1, 1, 0, 1, 1, 0, 2, 0, 1, 2, 0, 1, 0, 2, 3, 0, 1, 0, 1, 0, 2, 1, 2}
CS ₁₀	50	30	R₁₀ = {1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 2}

Table 2

Point estimation (P.E.), mean squared error (MSE), lower limit interval estimation (LL), upper limit interval estimation (UL) and interval length (Len) using maximum likelihood and Bayesian including informative and non-informative for beta and theta of GGD

	n = 10, m = 5, CS = R₁, T = 1
	ML		MCMC1		MCMC0
	β	θ	β	θ	β	θ
P . E .	4.0006	2.6341	3.1647	2.0262	3.4759	2.3789
MSE	1.0013	0.402	0.0271	0.0007	0.2265	0.1436
LL	0.1794	0.05166	1.9485	1.1987	0.5666	0.5341
UL	7.8219	5.7848	4.6128	3.1564	7.41	6.1544
Len	7.6426	6.3014	2.6643	1.9577	6.8434	5.6203
	n = 20, m = 5, CS = R₂, T = 1
P . E .	1.8171	1.3511	2.7618	1.8496	1.7054	1.3067
MSE	1.3992	0.421	0.0567	0.0226	1.676	0.4807
LL	0.05945	0.1171	1.6425	1.1561	0.1314	0.4167
UL	4.2287	2.5852	4.1995	2.7643	4.368	2.8055
Len	4.8232	2.4681	2.557	1.6082	4.2366	2.3888
	n = 20, m = 10, CS = R₃, T = 1
P . E .	3.998	1.8765	3.4646	1.774	3.8808	1.8856
MSE	0.996	0.0153	0.2159	0.0511	0.7758	0.0131
LL	0.9494	0.3731	2.2	1.1047	1.2899	0.7359
UL	7.0465	3.3799	5.0384	2.5933	7.6231	4.0501
Len	6.0971	3.0068	2.8383	1.4887	6.3332	3.3142
	n = 30, m = 10, CS = R₄, T = 1
P . E .	2.0871	1.4186	2.7067	1.7673	1.985	1.7673
MSE	0.8334	0.338	0.086	0.0541	1.0302	0.3839
LL	0.2821	0.4537	1.6772	1.1684	0.5742	0.6295
UL	3.8921	2.3836	3.9756	2.563	3.8749	2.4976
Len	3.61	1.9299	2.2984	1.3946	3.3006	1.8681

Table 3

	n = 30, m = 15, CS = R₅, T = 1
	ML		MCMC1		MCMC0
	β	θ	β	θ	β	θ
P . E .	9.0958	9.6128	3.9402	2.4653	8.4509	8.9265
MSE	0.5632	0.1243	0.884	0.2165	0.8789	0.1576
LL	5.0524	0.4999	2.6517	1.59	4.8611	3.1087
UL	13.1391	18.7257	5.4012	3.63	12.333	19.8695
Len	8.0867	18.2258	2.7494	2.04	7.472	16.7608
	n = 40, m = 20, CS = R₆, T = 1
P . E .	2.8759	1.827	2.9485	1.8905	2.7506	1.7762
MSE	0.0154	0.0299	0.0027	0.012	0.0622	0.0501
LL	1.3717	0.8338	1.9891	1.2772	1.378	0.9587
UL	4.3801	2.8201	4.0429	2.6496	4.4175	2.9475
Len	3.0085	1.9863	2.0539	1.3724	3.0396	1.9887
	n = 40, m = 30, CS = R₇, T = 1
P . E .	2.9731	1.9878	2.9967	2.0196	2.9252	1.9706
MSE	0.0007	0.0001	0 .	0.0004	0.0056	0.0009
LL	1.7255	0.9721	2.1611	1.3506	1.7165	1.1067
UL	4.2208	3.0036	3.9894	2.8849	4.244	3.1169
Len	2.4953	2.0315	1.8283	1.5343	2.5276	2.0103
	n = 50, m = 20, CS = R₈, T = 1
P . E .	2.0445	1.5355	2.4517	1.7996	1.9716	1.5027
MSE	0.9131	0.2158	0.3006	0.0402	1.0576	0.2473
LL	0.8825	0.7696	1.6349	1.2447	0.9623	0.8769
UL	3.2064	2.3013	3.372	2.4962	3.2287	2.3686
Len	2.324	1.5317	1.7371	1.2515	2.2664	1.4916

Table 4

		n = 50, m = 25, CS = R₉, T = 1
	ML		MCMC1		MCMC0
	β	θ	β	θ	β	θ
P . E .	2.2971	1.561	2.6206	1.7824	2.2676	1.5564
MSE	0.4941	0.1928	0.144	0.0473	0.5364	0.1968
LL	1.1498	0.8139	1.7639	1.224	1.1609	0.8906
UL	3.4443	2.308	3.5934	2.4421	3.4804	2.4022
Len	2.2944	1.4941	1.8295	1.2181	2.3195	1.5116
		n = 60, m = 30, CS = R₁₀, T = 1
	ML		MCMC1		MCMC0
	β	θ	β	θ	β	θ
P . E .	3.7994	2.2869	3.437	2.0577	3.7414	2.2744
MSE	0.639	0.0823	0.191	0.0033	0.5496	0.0753
LL	2.2707	1.1593	2.4434	1.4139	2.3508	1.3524
UL	5.328	3.4144	4.5946	2.8774	5.3182	3.5884
Len	3.0573	2.2551	2.1513	1.4635	2.9674	2.236

Table 5

	n = 10, m = 5, CS = R₁, T = 5
P . E .	2.1859	1.7685	2.7285	2.0492	2.0375	1.7332
MSE	0.6627	0.0536	0.0737	0.0024	0.9264	0.0712
LL	0.02252	0.02177	1.6266	1.1947	0.2606	0.443
UL	4.5971	3.7547	4.0078	3.1192	4.391	4.2008
Len	4.8222	3.9723	2.3812	1.9245	4.1304	3.7578
	n = 20, m = 5, CS = R₂, T = 5
P . E .	4.2065	3.6803	2.9279	2.3094	3.9478	3.8547
MSE	1.4557	2.8235	0.0052	0.0958	0.8983	3.4399
LL	0.1241	0.0712	1.7504	1.435	0.8374	0.9938
UL	8.2889	8.0727	4.3687	3.4831	8.4991	10.5997
Len	8.1648	8.7847	2.6183	2.0481	7.6617	9.6059
	n = 20, m = 10, CS = R₃, T = 5
P . E .	5.5973	3.3168	3.5756	2.0371	5.447	3.383
MSE	6.7459	1.734	0.3313	0.0014	5.9879	1.9128
LL	2.0369	0.4016	2.2947	1.2541	2.3001	1.2826
UL	9.1577	6.232	5.1584	3.0807	9.177	7.1839
Len	7.1207	5.8304	2.8637	1.8265	6.8768	5.9013

Table 6

	n = 30, m = 10, CS = R₄, T = 5
	ML		MCMC1		MCMC0
	β	θ	β	θ	β	θ
P . E .	2.2086	1.6277	2.6745	1.9105	2.1639	1.6261
MSE	0.6263	0.1386	0.106	0.008	0.6991	0.1398
LL	0.4134	0.4998	1.6725	1.2204	0.6567	0.7082
UL	4.0038	2.7556	3.8604	2.7941	4.198	3.073
Len	3.5904	2.2558	2.1879	1.5736	3.5413	2.3648
	n = 30, m = 15, CS = R₅, T = 5
P . E .	1.8027	1.2128	2.438	1.5796	1.7447	1.1834
MSE	1.4336	0.6197	0.3159	0.1768	1.5758	0.6669
LL	0.5528	0.5129	1.5874	1.0302	0.69	0.6253
UL	3.0525	1.9127	3.4138	2.2879	3.0611	1.9846
Len	2.4996	1.3998	1.8264	1.2577	2.3711	1.3593
	n = 40, m = 20, CS = R₆, T = 5
P . E .	2.9286	1.8657	2.9782	1.9228	2.8151	1.8282
MSE	0.0051	0.018	0.0005	0.006	0.0342	0.0295
LL	1.2966	0.7837	2.0417	1.3036	1.314	0.9441
UL	4.5605	2.9476	4.0995	2.7177	4.4914	3.089
Len	3.2639	2.1639	2.0578	1.4141	3.1774	2.1449
	n = 40, m = 30, CS = R₇, T = 5
P . E .	2.4612	1.4589	2.7701	1.6773	2.3947	1.4302
MSE	0.2903	0.2928	0.0529	0.1041	0.3664	0.3247
LL	1.3426	0.7603	1.9627	1.143	1.3977	0.8474
UL	3.5798	2.1574	3.7051	2.34	3.527	2.2177
Len	2.2372	1.3971	1.7424	1.197	2.1293	1.3703

Table 7

	n = 50, m = 20, CS = R₈, T = 5
	ML		MCMC1		MCMC0
	β	θ	β	θ	β	θ
P . E .	4.7098	2.6262	3.6939	2.0787	4.4867	2.5247
MSE	2.9235	0.3922	0.4815	0.0062	2.2103	0.2753
LL	2.4658	1.1648	2.5307	1.3984	2.481	1.3982
UL	6.9538	4.0877	5.0694	2.9239	6.7623	4.1123
Len	4.488	2.9229	2.5387	1.5256	4.2812	2.7141
	n = 50, m = 25, CS = R₉, T = 5
P . E .	3.8121	3.0077	3.2361	2.4105	3.6823	2.9469
MSE	0.6594	1.0155	0.0557	0.1685	0.4655	0.8965
LL	2.2074	1.3242	2.3209	1.6383	2.2528	1.6323
UL	5.4167	4.6912	4.2437	3.3968	5.2924	4.8434
Len	3.2093	3.367	1.9228	1.7585	3.0396	3.2111
	n = 50, m = 30, CS = R₁₀, T = 5
P . E .	3.6499	2.9059	3.3929	2.6028	3.1229	2.6623
MSE	0.4223	0.8207	0.1544	0.3634	0.0151	0.4386
LL	2.2635	1.3967	3.1017	2.3622	2.7245	2.0617
UL	5.0362	4.4151	3.8024	2.9193	3.6272	3.2089
Len	2.7728	3.0184	0.7007	0.5571	0.9026	1.1472

8 Results and discussions

In this paper, the maximum likelihood and Bayesian methods are used to make interval and point estimation for the GGD in the case TIHPCS using Mathematica 8 program with 4 GB RAM and processor Core i7. However, the calculations in the case the three parameters GGD under TIHPCS has taken a too long time to evaluate. In our opinion, this due to the two exponentials in the probability and cumulative functions. So, We assumed the shape parameters delta is known and the scale and the shape parameters (β, θ) are unknown. Also, in the case Bayesian, the joint posterior had a complicated form to find the double integrations. So, Markov Chain Monte Carlo (MCMC) approximation is used to solve the hard integrations in the case informative and non-informative priors. Comparisons between the methods inference are presented. To a comparative study purpose, a simulation study was achieved to compare the performance the proposed methods for different numbers n, m, T and different censoring schemes CS. In spite the likelihood function l (θ, β|δ ; X) is unique, exist and has a uni-mode in case θ and β see Figs. (3, 4).

The mean squared error for the likelihood was unstable or had highly oscillation which changes from 0.007 to 37.1584. According to the results obtained, we see that the main reason that the roots is very sensitive to the generated sample and the form ML obtaind in Eqs. (6, 7) and its differentiations by (β, θ) and this lead to the MSE is very sensitive to the generated sample. On the contrary, in the case Bayesian procedure with MCMC approximation the MSE and Len gave an excellent results. From the results Tables [2 - 7], we observe the following.

Remarks

In case informative prior Bayesian method (MCMC1), MSE and Len is less than non-informative prior (MCMC0).

By increasing the failure times m and for fixed the sample size n, the MSEs are decreased.

For fixed values n, m the MSEs for θ less than the MSEs for β in most cases.

The interval length (Len) in the case MCMC1 is the smallest among the other methods.

The results in the case (MCMC0) are close to ML results as expected.

9 Conclusion

From numerical results section, Its clear to observe that in the case three parameters Gompertz (generalized Gompertz) distribution based on Type-I hybrid progressive censoring, the results depending on MSE in the case point estimation and Len in the case interval estimation have shown to us a strong evidence to recommend that the Bayesian method with informative prior is the best method among all the methods used here and then in the second order the method Bayesian with non-informative prior. But the maximum likelihood will be disregarded. So, we recommend that the experiments lifetimes based on Type-I hybrid progressive when the considered distribution is GGD should use Bayesian procedure for the estimation process.

Footnotes

Acknowledgment

This project was funded by the Deanship Scientific Research (DSR), King Abdulaziz University, Jeddah under grant no. (G-667-247-39). The authors, therefore, acknowledge with thanks DSR technical and financial support.

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