A MIXTURE OF EXPONENTIAL AND IFR GAMMA DISTRIBUTIONS HAVING AN UPSIDEDOWN BATHTUB-SHAPED FAILURE RATE

2012 ◽  
Vol 26 (4) ◽  
pp. 573-580 ◽  
Author(s):  
Henry W. Block ◽  
Naftali A. Langberg ◽  
Thomas H. Savits
Keyword(s):  
Exponential Distribution ◽  
Failure Rate ◽  
Gamma Distribution ◽  
Increasing Failure Rate ◽  
Bathtub Shape ◽  
Gamma Distributions ◽  
The Right

We consider a mixture of one exponential distribution and one gamma distribution with increasing failure rate. For the right choice of parameters, it is shown that its failure rate has an upsidedown bathtub shape failure rate. We also consider a mixture of a family of exponentials and a family of gamma distributions and obtain a similar result.

scholarly journals Continuous Mixtures of Exponentials and IFR Gammas Having Bathtub-Shaped Failure Rates

2010 ◽  
Vol 47 (04) ◽  
pp. 899-907 ◽  
Author(s):  
Henry W. Block ◽  
Naftali A. Langberg ◽  
Thomas H. Savits ◽  
Jie Wang
Keyword(s):  
Exponential Distribution ◽  
Failure Rate ◽  
Gamma Distribution ◽  
Failure Rates ◽  
Increasing Failure Rate ◽  
Continuous Mixture ◽  
Mixtures Of Exponentials ◽  
The Right

It can be seen that a mixture of an exponential distribution and a gamma distribution with increasing failure rate for the right choice of parameters can yield a distribution with a bathtub-shaped failure rate. In this paper we consider a continuous mixture of exponentials and a continuous mixture of gammas with increasing failure rates and show that the resulting mixture has a bathtub-shaped failure rate.

scholarly journals Continuous Mixtures of Exponentials and IFR Gammas Having Bathtub-Shaped Failure Rates

2010 ◽  
Vol 47 (4) ◽  
pp. 899-907 ◽  
Author(s):  
Henry W. Block ◽  
Naftali A. Langberg ◽  
Thomas H. Savits ◽  
Jie Wang
Keyword(s):  
Exponential Distribution ◽  
Failure Rate ◽  
Gamma Distribution ◽  
Failure Rates ◽  
Increasing Failure Rate ◽  
Continuous Mixture ◽  
Mixtures Of Exponentials ◽  
The Right

It can be seen that a mixture of an exponential distribution and a gamma distribution with increasing failure rate for the right choice of parameters can yield a distribution with a bathtub-shaped failure rate. In this paper we consider a continuous mixture of exponentials and a continuous mixture of gammas with increasing failure rates and show that the resulting mixture has a bathtub-shaped failure rate.

A multivariate IFR class

1985 ◽  
Vol 22 (01) ◽  
pp. 197-204 ◽  
Author(s):  
Thomas H. Savits
Keyword(s):  
Exponential Distribution ◽  
Failure Rate ◽  
Random Vector ◽  
Increasing Failure Rate ◽  
Rate Distribution ◽  
Weak Limits ◽  
Multivariate Exponential Distribution ◽  
Multivariate Exponential

A non-negative random vector T is said to have a multivariate increasing failure rate distribution (MIFR) if and only if E[h(x, T)] is log concave in x for all functions h(x, t) which are log concave in (x, t) and are non-decreasing and continuous in t for each fixed x. This class of distributions is closed under deletion, conjunction, convolution and weak limits. It contains the multivariate exponential distribution of Marshall and Olkin and those distributions having a log concave density. Also, it follows that if T is MIFR and ψ is non-decreasing, non-negative and concave then ψ (T) is IFR.

A multivariate IFR class

1985 ◽  
Vol 22 (1) ◽  
pp. 197-204 ◽  
Author(s):  
Thomas H. Savits
Keyword(s):  
Exponential Distribution ◽  
Failure Rate ◽  
Random Vector ◽  
Increasing Failure Rate ◽  
Rate Distribution ◽  
Weak Limits ◽  
Multivariate Exponential Distribution ◽  
Multivariate Exponential

A non-negative random vector T is said to have a multivariate increasing failure rate distribution (MIFR) if and only if E[h(x, T)] is log concave in x for all functions h(x, t) which are log concave in (x, t) and are non-decreasing and continuous in t for each fixed x. This class of distributions is closed under deletion, conjunction, convolution and weak limits. It contains the multivariate exponential distribution of Marshall and Olkin and those distributions having a log concave density. Also, it follows that if T is MIFR and ψ is non-decreasing, non-negative and concave then ψ (T) is IFR.

OPTIMAL BURN-IN PROCEDURES IN A GENERALIZED ENVIRONMENT

2005 ◽  
Vol 12 (03) ◽  
pp. 189-201 ◽  
Author(s):  
JI HWAN CHA ◽  
JIE MI
Keyword(s):  
Lower Bound ◽  
Failure Rate ◽  
Rate Function ◽  
General Assumption ◽  
Lifetime Distribution ◽  
Failure Rate Function ◽  
Increasing Failure Rate ◽  
Bathtub Shape ◽  
Early Failures ◽  
Special Case

Burn-in procedure is a manufacturing technique that is intended to eliminate early failures. In the literature, assuming that the failure rate function of the products has a bathtub shape the properties on optimal burn-in have been investigated. In this paper burn-in problem is studied under a more general assumption on the shape of the failure rate function of the products which includes the traditional bathtub shaped failure rate function as a special case. An upper bound for the optimal burn-in time is presented under the assumption of eventually increasing failure rate function. Furthermore, it is also shown that a nontrivial lower bound for the optimal burn-in time can be derived if the underlying lifetime distribution has a large initial failure rate.

scholarly journals Inference for the Exponential Distribution under Generalized Progressively Hybrid Censored Data from Partially Accelerated Life Tests with a Time Transformation Function

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1510
Author(s):  
Alaa H. Abdel-Hamid ◽  
Atef F. Hashem
Keyword(s):  
Exponential Distribution ◽  
Failure Rate ◽  
Censored Data ◽  
Sufficient Conditions ◽  
Transformation Function ◽  
Point Estimation ◽  
Estimation Methods ◽  
Time Transformation ◽  
Monte Carlo Simulation Study ◽  
Accelerated Life

In this article, the tampered failure rate model is used in partially accelerated life testing. A non-decreasing time function, often called a ‘‘time transformation function", is proposed to tamper the failure rate under design conditions. Different types of the proposed function, which have sufficient conditions in order to be accelerating functions, are investigated. A baseline failure rate of the exponential distribution is considered. Some point estimation methods, as well as approximate confidence intervals, for the parameters involved are discussed based on generalized progressively hybrid censored data. The determination of the optimal stress change time is discussed under two different criteria of optimality. A real dataset is employed to explain the theoretical outcomes discussed in this article. Finally, a Monte Carlo simulation study is carried out to examine the performance of the estimation methods and the optimality criteria.

scholarly journals Clinical Evaluation of Bracket Bonding Using Two Different Polymerization Sources

2008 ◽  
Vol 78 (5) ◽  
pp. 922-925 ◽  
Author(s):  
Nikolaos S. Koupis ◽  
Theodore Eliades ◽  
Athanasios E. Athanasiou
Keyword(s):  
Failure Rate ◽  
Orthodontic Treatment ◽  
Light Emitting Diode ◽  
Rank Test ◽  
Dental Arch ◽  
Halogen Lamp ◽  
Failure Rates ◽  
Bond Failure ◽  
Significant Difference ◽  
The Right

Abstract Objective: To comparatively assess clinical failure rate of brackets cured with two different photopolymerization sources after nine months of orthodontic treatment. Materials and Methods: The sample of this study comprised 30 patients who received comprehensive orthodontic treatment by means of fixed appliances. Using the same adhesive, 600 stainless steel brackets were directly bonded and light cured for 10 seconds with the light-emitting diode (LED) lamp or for 20 seconds with the conventional halogen lamp. A split-mouth design randomly alternated from patient to patient was applied. Failure rates were recorded for nine months and analyzed with Pearson χ2 test, and log-rank test at α = .05 level of significance. Results: The overall failure rate recorded with the halogen unit (3.33%) was not significantly different from the failure rate for the LED lamp (5.00%). Significantly more failures were found in boys compared with girls, in the mandibular dental arch compared with the maxillary arch, and in posterior segments compared with anterior segments. However, no significant difference was found between the right and left segments. Conclusion: Both light-curing units showed sufficiently low bond failure rates. LED curing units are an advantageous alternative to conventional halogen sources in orthodontics because they enable a reduced chair-time bonding procedure without significantly affecting bond failure rate.

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