Inferences of inverted gompertz parameters from an adaptive type-ii progressive hybrid censoring

A new two-parameter inverted Gompertz distribution with an upside-down bathtub-shaped failure rate is investigated in the presence of incomplete (censored) data. Reliability experimenters favor employing censoring strategies to collect data in order to strike a compromise between the time needed to complete the test, the required sample size, and the cost. Recently, an adaptive Type-II hybrid progressively censoring strategy has been proposed to improve the effectiveness of statistical inference. Therefore, using this strategy, this study explores the classical and Bayesian estimates of the inverted Gompertz distribution. The distribution parameters, reliability, and hazard functions are estimated using maximum likelihood and Bayesian estimation methods. On the presumption that the gamma priors are independent, symmetric and asymmetric loss functions are used to create the Bayesian estimation. The Markov chain Monte Carlo approach is used to collect samples from the entire conditional distributions and then acquire the Bayes estimates since the joint posterior distribution has a difficult shape. The highest posterior density and asymptotic confidence intervals are also obtained. Through simulated research, the efficacy of the various recommended strategies is contrasted. The best progressive censoring schemes are also shown, and various optimality criteria are investigated. Two real data sets, representing the lifetimes of mechanical components and the Boeing 720 jet airplane, are also looked at to show how the recommended point and interval estimators may be used. When the experimenter’s main concern is the number of failures, the results of the simulation research and data analysis showed that the proposed scheme is adaptive and highly useful in concluding the experiment.


Introduction
Gompertz distribution occupies an important member in modelling human mortality, behavioral and actuarial sciences.Historically, several works have contributed to the statistical methodology and characterization of this lifetime model.Recently, Eliwa et al [1] introduced the two-parameter inverted Gompertz (IG) distribution with an upside-down (inverted) bathtub hazard rate shape.However, a lifetime random variable X is said to have the IG(λ, μ) with shape parameter λ > 0 and scale parameter μ > 0, if its probability density function (PDF), f ( • ), cumulative distribution function (CDF), F( • ), and hazard rate function (HRF), h( • ) are given by respectively, and also its reliability function (RF) provided by R( • ) = 1 − F( • ).Setting μ → 0 in (1), the inverseexponential (IE) distribution is obtained as a special case.Recently, Elshahhat et al [2] proposed a new threeparameter distribution called the extended-IG distribution as an extension of the IE, IG, and generalized-IE distributions.
The most widely-used multistage censoring is known as progressive Type-II censoring (PT2C) in which n independent units are entered into a test, m is a prefixed number of failed items, and progressive censoring ( ) S S S , , m 1 = ¼ is also prefixed.At each time of the i th failure observed (say X i:m:n ) for i = 1,K,m − 1, S i surviving units are randomly selected and removed from the test.Lastly, when the last failure X m:m:n occurred, all remaining units will be removed;for additional details, see Balakrishnan and Cramer [3].Similar to the PT2C procedure, Childs et al [4] proposed two plans of progressive hybrid censoring in which the test based on Type-I is stopped at ( ) T min T X , m m n : : = * such T is a predetermined time.The main issue with Type-I progressive hybrid censoring is that the target sample size is chosen at random and may turn out to be too small.Consequently, the accuracy of statistical computations becomes inefficient.To handle this problem, adaptive Type-II progressive hybrid censoring (AT2PHC) was proposed by Ng et al [5].Today, it has become one of the most flexible censoring plans.Under this mechanism, the duration test is allowed to run over the time T and some values of S i for i = 1,K,m − 1 could possibly be revised during the test.If X m:m:m < T, the test stops at X m:m:m with the same PT2C strategy.Otherwise, where D denotes the number of observed failures up to T, we stop the progressive censoring by placing S i = 0 for i = D + 1, D + 2,K,m − 1, and all the remaining units S n m S m i D i 1 = -å = are removed at the last failure X m:m:n observed.This modification ensures that the test will stop when we have the appropriate number of failures m, and that the overall test duration will not deviate too much from the optimal time T. Suppose that is an observed AT2PHC sample from a continuous population with PDF f (x) and CDF F(x), then the joint likelihood function can be expressed as : : 1 : : : : where C is a constant and q is a parametric vector.Different researches based on the proposed censoring have been conducted by several authors in the literature; for example, see Nassar and Abo-Kasem [6], Mohie El-Din et al [7], Ateya and Mohammed [8], Liu and Gui [9], Elshahhat and Nassar [10], Panahi and Asadi [11], Helu and Samawi [12], El-Sherpieny et al [13], Alotaibi et al [14], Mohammed et al [15], and references cited therein.
To the best of our knowledge, despite the fact that IG distribution has gotten a lot of attention from various authors, the challenge of estimating the IG parameters and/or its RF or HRF in the presence of data gathered via AT2PHC sampling has yet to be studied.We are driven to do this kind of work, however, for two reasons: (i) The IG model's HRF exhibits an upside-down bathtub-shaped failure rate, which is preferable in many practical domains; (ii) The IG model's adaptability in modelling various data sets of various geometries; (iii) The importance of AT2PHC in improving the effectiveness of statistical inference by avoiding obtaining observable small sample sizes serves as a motivation for this work.Therefore, this article offers an analysis of AT2PHC lifetime data when each test unit follows the IG distribution.As a consequence, the study's goals are fivefold: • Obtain the maximum likelihood estimators (MLEs) as well as their approximate confidence intervals (ACIs) of the unknown model parameters λ and μ, in addition to the reliability indices R(t) and h(t).
• Obtain the Bayes estimators of λ, μ, R(t), and h(t) in addition to their highest posterior density (HPD) intervals from independent gamma priors assumption against two loss functions, namely the squared error (SE) and general entropy (GE) loss functions.Monte Carlo Markov-chain (MCMC) techniques will be utilized to approximate the Bayes' point and interval estimators.
• It is impossible to determine which method theoretically produces the best estimations.Thus, based on various accuracy criteria, namely: root mean squared-error, mean relative absolute bias, average confidence length, and coverage percentage, a thorough simulation study is conducted to examine the behavior of the various estimations and enable comparison.
• Explore several optimality metrics-based frameworks to choose the best progressive sampling plan from the proposed distribution.
• Analysis of two different real-world applications is presented to evaluate how the offered approaches operate in practice and discuss how to choose the best censoring design.
The remainder of the paper is organized as follows: The maximum likelihood and Bayes inferences of the unknown parameters and reliability characteristics are presented in sections 2 and 3, respectively.Section 4 reports the simulation outcomes.Two real-world applications are offered in section 5. Section 6 discusses various methods for selecting the best censoring strategy, and section 7 concludes the study.

Likelihood estimation
Let x 1:m:n < L < x D:m:n < T < x D+1:m:n < L < x m:m:n with (S 1 ,K,S D , 0,K,0, S m ) be AT2PHC order statistics of size m taken from the IG distribution with PDF and CDF given, respectively, by (1) and (2).In this case, one can rewrite the likelihood function (4) based on (1) and (2), after ignoring the constant term, as follows: where x i = x i:m:n for simplicity, and Practically, it is more convenient to work with the log-likelihood function rather than the likelihood function itself.Therefore, by taking the natural logarithm of the likelihood function in (5) Let l and m denote the MLEs of the unknown parameters λ and μ, respectively.These estimators can be acquired by maximizing the objective function ℓwith respect to λ and μ.An alternative approach to obtaining the needed estimators is to solve the following two normal equations simultaneously: x e e  for i = 1, 2,K,m.It is clear from (7) and (8), that the MLEs l and m don't have closed forms Numerical methods may be implemented to solve these equations to produce l and m.We can get the MLEs of R(t) and h(t) by using the invariance property of the MLEs l and m as follows: , and After getting the point estimates of λ, μ, R(t), and h(t), it is also of interest to construct the associated confidence intervals.This objective can be achieved by employing the asymptotic properties of the MLEs.It is known, based on the theory of large samples, that the asymptotic distribution of ( ˆˆ) , T l m is normal distribution with a mean ( ˆˆ) , l m and a variance-covariance matrix I −1 (λ, μ) which is obtained by taking the inverse of the Fisher information (FI) matrix.It is not easy to obtain the elements of the FI matrix due to its intractable elements.Therefore, we can use the observed FI matrix to obtain ( ˆˆ) I , 1 l m as an estimate of I −1 (λ, μ), where where the main diagonal elements are the estimated variances of l and m, respectively, and the elements of the observed FI matrix can be easily obtained from (6) as follows: ; 2 2 ;

2
; ; ; Thus, two-sided 100(1 − α)% ACIs for λ and μ are given, respectively, by where z 2 a denotes the upper 2 a percentage point of the standard normal distribution, where ŝll and ŝmm are the main diagonal elements of (2), respectively.
To create the ACIs of the reliability indices R(t) and h(t), we propose to consider the delta method in order to approximate the variance of their estimators ˆ( ) R t and ˆ( ) h t , respectively; for more details, see Greene [16].Under regularity conditions, the distribution of the MLE ˆ( ) 1 e e 1 e e 1 , where v e 1 Hence, the two-sided ACIs of R(t) and h(t) for a distinct time t at the confidence level 100(1 − α) are obtained, respectively, as where a i > 0 and b i > 0 for i = 1, 2, are assumed to be known.Combining (5) and (9), the joint posterior PDF of λ and μ, can be written as ´-- where the normalized constant  is given by ´-- Utilizing the SE loss, which is the most popular symmetric loss function, the Bayes estimator (say ˜(•) J ) of any function of λ and μ, say ϑ(λ, μ), is given by the posterior expectation.The SE loss (say η S ) and its Bayes estimator (say ˜( ) , S J l m ) are defined as respectively, for more details, see Martz and Waller [17].
On the other hand, one of the useful asymmetric losses is called the GE loss function and is given by It is noted, from (12), that the smallest error occurs at and when putting γ = − 1, the Bayes estimator via GE loss coincides with the Bayes estimator via SE loss.Further, when γ > 0, a positive error has a more serious effect than a negative error, whereas for γ < 0 a negative error has a more serious effect than a positive error.Using (12), the Bayes estimator ˜( ) , J l m of ϑ(λ, μ) is given by for more details see Dey et al [18].First, the conditional posterior PDFs of λ and μ must be obtained as ; Due to the nonlinear formulas of λ and μ as in (13) and (14), respectively, there is no closed form solution for the Bayes estimators of λ, μ, R(t), or h(t).As a result, using the SE (or GE) loss, we propose to use the MCMC approach to calculate Bayes estimates and construct the associated HPD intervals.Therefore, the Metropolis-Hastings (M-H) algorithm with a normal proposal distribution is suggested to obtain the Bayes estimates along with their HPD interval estimates of λ, μ, R(t), and h(t).One may refer to Gelman et al [19] for additional details.Now, to collect the MCMC sample from (13) and (14), do the following procedure as Step 1. Set the start values of (λ, μ), say ( ˆˆ) Step 2. Set j = 1.
Step 10.Compute the Bayes estimates of λ, μ, R(t) or h(t) (say θ for brevity) under SE and GE loss functions respectively as Step 11.Create the HPD interval of θ by sorting its MCMC samples in ascending order as ( . Thus, following Chen and Shao [20], the 100(1 − α)% HPD interval estimator for θ is given by where the highest integer less than or equal to x is referred by [x].

Monte-Carlo simulations
To evaluate the effectiveness of the derived point/interval estimators of λ, μ, R(t), and h(t), developed in the proceeding sections, we implemented extensive Monte-Carlo simulations.

Simulation scenarios
Using several choices of T(Threshold time), n(Total experimental items), m(Effective sample length), and S(Progressive pattern), we propose the following procedure to simulate an AT2PHC sample: Step 1. Set the actual values of IG(λ, μ).

b. Set
Step 3. Determine the value of D at T.
Step 4. Determine the case type of AT2PHC sample as: a.If X m < T, stop the experiment at X m .
b.If X m > T, set S i = 0, for i = D + 1,K,m − 1, and stop the experiment at X m .
Next, to assess the acquired estimates of the IG parameters λ, μ, R(t), and h(t), we repeated the AT2PHC steps 1,000 times from IG(0.5, 0.8).Moreover, to assess the acquired estimates of R(t) and h(t), when t = 0.5 their true values are taken as 0.915 47 and 0.914 68, respectively.Since the AT2PHC test is terminated when the number of failed subjects reaches (or surpasses) a particular value m, for given n( = 40, 80), different failure percentage (FP%) levels are utilized to get m as m n (=50, 80)%.Also, by setting T( = 1.5, 3.5), three progressive mechanisms S are used called: * .It should be noted here that there is no limit on the number of iterations or the proposed censoring settings.Once the AT2PHC samples are obtained, by R 4.2.2 programming software, the MLEs as well as the 95% ACIs of λ, μ, R(t), and h(t) are calculated via the maxLik' package, see Henningsen and Toomet [21].On the other hand, the Bayes estimates against SE and GE (for γ( = − 2, + 2)) loss functions as well as the 95% HPD interval estimates of λ, μ, R(t), and h(t) are evaluated via the coda' package, see Plummer et al [22].Via running the MCMC technique 12,000 times, then eliminating the first 2,000 iterations from each Markovian chain, the Bayes' inferences are developed based on two prior sets called Prior-1: For clarification, for ξ = 1, 2, 3, 4, the average estimates (Av.Es) of λ, μ, R(t) or h(t) (say θ) are given by where ( )  i q is the estimate of θ at i th sample, θ 1 = λ, θ 2 = μ, θ 3 = R(t), and θ 4 = h(t).
Evaluation of the point estimates of θ is made based on their root mean squared-errors (RMSEs) and mean relative absolute biases (MRABs) as On the other hand, the comparison between interval estimates of θ is made based on their respective average confidence lengths (ACLs) and coverage percentages (CPs) as where 1( • ) is the indicator function, is two bounds of (1 − α)% ACI/HPD interval values of θ ξ .

Simulation outputs
Heat-map is one of the best data visualizations in R software.Therefore, in this study, the simulated RMSE, MRAB, ACL, and CP values of λ, μ, R(t), and h(t) are plotted with a heat-map and displayed in figures 1-4, respectively.The numerical findings of λ, μ, R(t), and h(t) are provided in appendix [table A1-A8].Clearly, some notations of the proposed methods, based on Prior-1 (say P1) for example, have been considered, such as: 'SE-P1' refers to the Bayes estimates relative to the SE loss; 'GE1-P1' refers to the Bayes estimates relative to the GE loss (for γ = − 2); 'GE2-P1' refers to the Bayes estimates relative to the GE loss (for γ = + 2) and 'HPD-P1' refers to the HPD interval.
From figures 1-4, in terms of the lowest values of RMSE, MRAB, and ACL as well as the highest value of CP, we draw the following observations: • All acquired estimates of λ, μ, R(t), or h(t) have good behavior, it is the best of our general observation.
• As n(or m) increases, all examined estimates perform well.A similar comment is obtained when S i m i 1 å = narrowed down also.
• Bayes estimates (or HPD intervals) of λ, μ, R(t), or h(t), due to gamma prior information, behave satisfactorily compared to the likelihood estimates.
• Bayes' point (or HPD interval) estimates with respect to Prior-2 have more accurate results than others.This is due to the fact that the associated variance of Prior-2 is smaller than others.• Asymmetric Bayes estimates (from GE loss) of λ, μ, R(t), and h(t) are overestimates for (γ < 0) while for (γ > 0) are underestimates.
• As T increases, in most tests, it is noted that -The RMSEs and MRABs of all unknown parameters of λ, μ, R(t), and h(t) increase.
-The ACLs of λ and R(t) decrease while those of μ and h(t) increase.
-The CP values of λ and R(t) increase while those of μ and h(t) decrease.
• Comparing the proposed progressive mechanisms, it can be seen that -The acquired point and interval estimates of λ perform better using Scheme-2 (i.e., middle censoring) than others.
-The acquired point and interval estimates of μ, R(t) and h(t) perform better using Scheme-3 (i.e., right censoring) than others.
• To summarize, the Bayes approach via the Metropolis-Hastings procedure is recommended to estimate all unknown IG parameters of life in the presence of data collected from Type-II adaptive progressive hybrid censoring.

Real-Life applications
To clarify the usefulness of the suggested inferential approaches and to illustrate the relevance of the study findings to actual situations, this section provides two different applications by analyzing real-life data sets drawn from the engineering field.

Mechanical component data
This application analyzes the failure times of twenty mechanical components reported by Murthy et al [24] and re-discussed by Alotaibi et al [25].In table 1, each point in the original mechanical components data is multiplied by ten for convenience.To check whether the fit of the IG distribution to the mechanical components data is sufficient or not, the Kolmogorov-Smirnov (KS) distance with its P-value is used.To establish this goal, from table 1, the MLEs with their standard errors (St.Es) of λ and μ must be calculated as 0.0579(0.0425)and 3.6776 (0.7043), respectively.Thus, the KS(P-value) becomes 0.1336(0.868).Obviously, the IG model fits the mechanical component data well.To determine the fit result with graphical tools, three well-known plots are considered: figure 5  existed and unique; and the estimated RF supports the same fitting findings.It is preferable to set the estimates ˆ0.0579 l @ and ˆ3.6776 m @ as initial values for running the next calculations.To illustrate the estimates of λ, μ, R(t) and h(t) investigated in this study, taking m = 10 with different choices of S and T, three artificial AT2PHC samples are generated from the mechanical components data and reported in table 2. From table 2, the point estimates (including maximum likelihood and Bayes MCMC with their St.Es) and interval estimates (including ACI and HPD interval estimates with their lengths) of λ, μ, R(t) and h(t) (at t = 1) are calculated and presented in tables 3-4.Using improper gamma priors, by running the MCMC sampler 50,000 times and ignoring the first 10,000 times as burn-in, the Bayes estimates are obtained based on SE and GE (for γ(= -5, -0.05, +5)) loss functions.Because we don't have any extra historical information about the IG parameters from the given data set, tables 3-4 showed that the point (or interval) estimates of λ, μ, R(t) and h(t) developed by the likelihood (or Bayes) approach are very close to each other as expected.Further, some popular statistical characteristics of λ, μ, R(t), and h(t) namely: mean, mode, first-quartile Q 1 , median Q 2 , third-quartile Q 3 , standard deviation (St.D) and skewness are computed based on 40,000 MCMC draws and provided in table 5. To examine the convergence of the simulated Markovian chains, the corresponding density and trace plots of λ, μ, R(t), and h(t) are shown in figure 6.It shows that the simulated MCMC estimates via the M-H algorithm converge appropriately.It also demonstrates that the estimates of μ and h(t) have a fairly-symmetrical distribution, while those of λ and R(t) have positive and negative-skewed, respectively.For specification; sample mean and two-sided HPD interval are represented by solid (-) and dashed (---) lines, respectively.

Boeing 720 jet data
Boeing 720 is an American narrow-body airliner produced by Boeing Commercial Airplanes.In this application, to examine the proposed methodologies in real phenomena, a real data set consisting of interval times (in hours) of consecutive air-conditioning-system failures of the Boeing 720 jet airplane '7910' is analyzed.This data set was first reported by Proschan [26] and later re-analyzed by Gupta and Waleed [27].In table 6, the complete Boeing 720 jet data is listed in ascending order.Before progressing further, to see whether the proposed model fits the  Boeing 720 jet data, the KS statistic (with its P-value) and MLEs (with their St.Es) of λ and μ are obtained.However, from table 6, the MLEs (St.Es) of λ and μ are 35.064(14.084)and 6.9957(12.642),respectively, and the KS (P-value) is 0.1452(0.910).These results concluded that the IG distribution fits the Boeing 720 jet data quite well.Further, figure 7 displays (a) estimated/empirical scaled TTT transform; (b) contour of the estimated loglikelihood function; and (c) estimated/empirical reliability function.As we anticipate, figure 7 indicates that the decreasing hazard rate shape is suitable to fit the IG model; the MLEs l and m existed and are unique also; and the fitted RF supports our fitted results.We also suggest considering these estimates ˆ35.064 l @ and ˆ6.9957 m @ as initial guesses to carry out any upcoming analytics.From Boeing 720 jet airplanes, three different AT2PHC samples (when m = 7 and several combinations of S and T) are generated and provided in table 7.Then, we evaluated the acquired point and interval estimates obtained by likelihood and Bayesian approaches of λ, μ, R(t), and h(t) (at t = 50) and are presented in tables 8-9, respectively.Repeating the MCMC sampler 50,000 times and eliminating the first 10,000 iterations as burn-in, the Bayes estimates against both SE and GE (for γ( = − 5, − 0.05, + 5)) loss functions are calculated based on improper gamma priors.It is observed, from tables 8-9, that the point and interval inferences derived from the MCMC technique behave better than those obtained from the frequentist approach in terms of minimum standard-error and interval length values.Useful statistics of the simulated Markovian chains of λ, μ, R(t), and h (t) are obtained and reported in table 10.Moreover, from figure 8, it is noticed that the MCMC procedure converges sufficiently and that the estimates of all unknown parameters are almost symmetrically distributed.

Optimum PCST2 mechanisms
In the context of reliability methodology, the examiner may want to choose the 'optimal' censoring strategy from a group of all possible censoring schemes in order to offer the most information about his unknown parameter(s).Firstly, Balakrishnan and Aggarwala [28] studied this issue based on various scenarios.However, several optimality criteria have been suggested, and numerous findings regarding the best censoring have been examined.Based on the availability of units, experimental facilities, and cost considerations, such that the values of T, n, and m must be determined in advance, the optimal censoring ( ) S S S S , , , m 1 2

=
¼ can be proposed, for additional details see Ng et al [29].Thus, to get the optimum PCST2 mechanism, various criteria are presented in table 11.Several works in the literature have considered the problem of comparing two (or more) different progressive censoring designs, for example see Pradhan and Kundu [30], Elshahhat and Rastogi [31], Ashour et al [32], Elshahhat and Abu El Azm [33], among others.
It is clear, from table 11, that the criterion  1 aims to maximize the main diagonal informations of the observed Fisher matrix; the criteria  i , 2, 3 i = aimed to minimize the trace and determinant of the variancecovariance matrix; and the criterion  4 aims to minimize the variance of the logarithmic MLE of the ò th quantile, denoted by ( ( ˆ))   var log ¡ , where 0 < ò < 1.So, from (2), the logarithmic for ˆ ¡ of the IG model is

Concluding remarks
In this work, we have obtained the maximum likelihood and Bayes estimators of the unknown parameters, reliability, and hazard rate functions of the inverted Gompertz distribution based on an adaptive progressive Type-II hybrid censoring scheme.The Bayes estimators based on squared error and general entropy loss functions are acquired using the Markov chain Monte Carlo method.The asymptotic normality of the maximum likelihood estimators is used to derive approximative confidence intervals for the unknown parameters, reliability, and hazard rate functions.The largest posterior density credible intervals are also obtained.Four optimality criteria are studied together with the best progressive censoring schemes.A simulation study is performed to compare the performance of different point and interval estimators while accounting for different sample sizes and censoring techniques.The simulation's outcomes demonstrated that the Bayesian technique provides more accurate estimates than the maximum likelihood strategy.We looked at two actual data sets for Mechanical components and Boeing 720 jet data in order to show how the recommended estimators operate in practical settings.According to the research, the inverted Gompertz distribution is a suitable model for these data, and when adaptive progressively Type-II hybrid censored inverted Gompertz data are present, the Bayesian estimation approach is suggested for estimating the unknown parameters.For further investigation, it is possible to use different estimating techniques, such as the maximum product of spacing estimation approach, which may be a viable replacement for the maximum likelihood method, to examine the estimation of the suggested model's reliability features.It is also possible to incorporate the competing risk model or accelerated life tests into the methodologies established in this study.

Figure 1 .
Figure 1.Heat-maps for the simulation results of λ.

Figure 2 .
Figure 2. Heat-maps for the simulation results of μ.
(a) estimated/empirical scaled total time on test (TTT) transform; figure 5(b) contour of the estimated log-likelihood function; and figure 5(c) estimated/empirical reliability function.It is clear, from figure 5, that an inverse-bathtub hazard rate shape is more appropriate to fit the IG model; the MLEs l and m are

Figure 3 .
Figure 3. Heat-maps for the simulation results of R(t).

Figure 4 .
Figure 4. Heat-maps for the simulation results of h(t).

Table 1 .
Failure times of mechanical components.

Table 2 .
Three AT2PHC samples from mechanical components data.

Table 4 .
Interval estimates of λ, μ, R(t) and h(t) from mechanical components data.

Table 5 .
Summary for MCMC iterations of λ, μ, R(t) and h(t) from mechanical components data.

Table 7 .
Three AT2PHC samples from Boeing 720 jet data.

Table 6 .
Failures of air conditioning system on Boeing 720 jet airplanes.whilethatused in Sample S 3 is the best under  2 .From Boeing 720 jet data, table13shows that the PCST2 mechanism used in Sample S 2 is the best under  i i =

Table 10 .
Summary for MCMC iterations of λ, μ, R(t) and h(t) from Boeing 720 jet data.

Table 12 .
Optimum PCST2 mechanisms from mechanical components data.