Quasi-Bayesian estimation of time to test transform under asymmetric loss functions

Introduction

In the 1970s the total time on test plot, and its theoretical counterpart, the scaled TTT transform, was presented by R. E. Barlow et al. as a tool for model identification based on data representing lives of non-repairable equipment. Later, Klefsjo (1982) [4] presented some relationship between the TTT transform and other ageing properties (with their duals) of random variable. Chacko et al. (2010) [2] discussed use of TTT transform in identifying failure rate model of semi-Markov reliability system. Total Time on Test transform plots are ALSO useful for analyzing non-negative data. The plots help in choosing a mathematical model for the data and provide information about failure rate. Also incomplete data can be analyzed and there is a theoretical basis for such an analysis. The TTT-Transform has also been found quite useful in theoretical applications such as looking for test statistics for particular purposes and to study their power. Kochar et al. (2002) [5] defined TTT transform order and Shaked and Shanthikumar (2007) [8] studied it explicitly. Nair et al. (2008) [6] provided applications of TTT of order n in reliability analysis. The scaled total time on test (TTT) transform of F is defined as

The main common types of Pareto distribution are known as Pareto Type I, II, III, IV, and Feller Pareto distributions. One of the popular hierarchies of Pareto distribution is Pareto Type II which has been named as Lomax distribution. Lomax distribution is an advantageous lifetime distribution in reliability analysis. The applicability of the Lomax distribution is not restricted only to reliability field, but it has broad application in the field of economics, actuarial statistics, queuing problems, biological sciences, etc. Lomax distribution has been applied in a variety of fields such as engineering and reliability and life testing. Lomax distribution has been used as an alternative to the exponential, gamma and Weibull distributions for heavy tailed data by Bryson (1974) [1] . Golaup et al. (2005) [3] introduced the size distribution of computer files on servers using Lomax distribution. Nasiri and Hosseini (2012) [7] also studied Lomax distribution regarding the MLE and various Bayesian estimation based on record values. The probability density function of Lomax distribution is given by

Where 𝜃 and 𝜆 are shape and scale parameters respectively. For the above model, the TTT simplifies to

Quasi-bayesian estimation

The quasi-likelihood function was introduced by Wedderburn (1974), to be used for estimating the unknown parameters in generalized linear models. If the underlying distribution comes from a natural exponential family the maximum quasi-likelihood estimate maximizes the likelihood function and so it has full asymptotic efficiency; under more general distributions there is some loss of efficiency.

Wedderburn defined the quasi-likelihood function as

Where 𝜇 = 𝐸(𝑥), 𝑉(𝜇) = 𝑉𝑎𝑟(𝑥) and 𝑜(𝑥) is some function of x only. The variance assumption is generalized to 𝑉𝑎𝑟(𝑥) = 𝜙𝑉𝑎𝑟(𝜇) where the variance function V(.) is assumed to be known and the parameter 𝜙 may be unknown. For a sample 𝑥 = (x 1 , x 2 ,… x n ) of size n from (1.2), the quasi-likelihood function simplifies to

The natural exponent of Q(𝑥, 𝜃, 𝜆) is the likelihood function and is given as

In this section, we derive the quasi Bayesian estimates of TTT under The extension of Jeffrey's prior and conjugate prior by using different loss function.

Quasi-Bayesian estimation of TTT under The Extension of Jeffrey's prior by using different loss function i) 𝝀 is known

Using the likelihood function as given in (2.3) and the extended Jeffrey's prior, the posterior density of θ is derived as follows:

Let ∅(𝑡) be a parameter itself denoted by ∅ for simplicity. Replacing θ in (2.4) in terms of ∅ by that (1.3), we get the posterior of the TTT as

The symbol C with suffixes stands for the normalizing constants.

• The Quasi Bayes estimator of TTT under Entropy Loss Function is given by

• The Quasi Bayes estimator of TTT under Precautionary Loss function is given by

ii) 𝝀 is unknown Using the likelihood function as given in (2.3) and the extended Jeffrey's prior, the posterior density of θ is derived as follows,

Replacing θ in (2.4) in terms of ∅ by that (1.2), we get the posterior of the TTT as

• The Quasi Bayes estimator of TTT under Entropy Loss Function is given by

• The Quasi Bayes estimator of TTT under Precautionary Loss function is given by

Quasi-Bayesian estimation of TTT under Conjugate prior by using different loss function i) 𝝀 is known

Here we suggest the conjugate prior distribution for the parameters and is given by • The Quasi Bayes estimator of TTT under Entropy Loss Function is given by

• The Quasi Bayes estimator of TTT under Precautionary Loss function is given by

ii) 𝝀 is unknown

In the inference problem considered here we assume that the parameter 𝜆 is unknown and in this case the joint prior distribution for the parameters as

References

1. Bryson MC. Heavy-tailed distributions: properties and tests, Technometrics 1974;16(1):61-68.
2. Chacko VM, Praveena PC, Manoharan M. Ageing properties of a semi-Markov system and Total Time on test transforms, In. Trans. Mathematical Science and Computers 2010;3:257-275.
3. Golaup A, Holland O, Aghvami AH. Concept and optimization of an effective packet scheduling algorithm for multimedia traffic over HSDPA, In Personal, Indoor and Mobile 100 Radio Communications, 2005. PIMRC 2005. IEEE 16th International Symposium on 2005;3:1693-1697.
4. Klefsjo B. On Ageing Properties and Total Time on Test Transforms, Scand. J Statist 1982;9:37-41.
5. Kochar SC, Li X, Shaked M. The total time test transform and the excess wealth stochastic orders of distribution, Adv. Appl. Prob 2002;34:826-845.
6. Nair NU, Sankaran PG, Vinesh Kumar B. Total time on test transforms of order n and their implications in reliability analysis, J Appl. Prob 2008;45:1126-1139.
7. Nasiri P, Hosseini S. Statistical Inferences for Lomax distribution based on record values (Bayesian and Classical), Journal of Modern Applied Statistical Methods 2012;11:179-189.
8. Shaked M, Shanthikumar JG. Stochastic Orders, Springer, New York 2007.