1. DEPARTMENT OF STATISTICS LOYOLA COLLEGE CHENNAI – DISSERTATION SUBMITTED BY TUNNY SEBASTIAN (07 MST 09) UNDER THE GUIDANCE OF DR. B. CHANDRASEKAR, M.Sc., M.Phil., Ph.D.

2. BIVARIATE AND TRIVARIATE EXPONENTIAL DISTRIBUTIONS IN RELIABILITY THEORY

3. CHAPTERS 1. Marshall – Olkin Bivariate Exponential Distribution
0. Introduction and Summary
1. Marshall – Olkin Bivariate Exponential Distribution
2. Inference for Marshall - Olkin Bivariate Exponential Distribution
3. Absolutely Continuous Trivariate Exponential Distribution - Two Parameter Case  
4. Absolutely Continuous Trivariate Exponential Distribution - Three Parameter Case

4. CHAPTER 1 MARSHALL – OLKIN BIVARIATE EXPONENTIAL DISTRIBUTION

5. Marshall and Olkin (1967) proposed a Bivariate Exponential Distribution by supposing that failure is caused by three types of Poisson shocks on a system containing two components and obtained some properties of the distribution.
Consider the two – component system with components 1 and 2.
Let
X: Failure time for component 1
Y: Failure time for component 2
λ1, λ2 and λ12 are the intensity parameters of the three poisson processes.

6. The joint survival function is
The joint pdf due to Bemis et al (1972) is
Clearly,

7. Remark
The pdf can be viewed as a mixture of three pdfs with the mixing proportions and , where .
That is
and

8. Theorem 1.2.1 Theorem 1.2.2 If follow MOBVE then and If follow MOBVE
Then follows exponential with parameter

9. Theorem 1.2.3
Let be a random sample from
MOBVE . If
Then
where

10. (i) The marginal distribution of is exponential with the parameter
Theorem 1.2.4
(i) The marginal distribution of is exponential with the parameter
(ii) The marginal distribution of is exponential with the parameter
Mean and Variance

11. Theorem 1.2.5 The moment generating function of is pdf approach
Definition approach

12. E(XY) using Theorem 1.2.6 MGF approach through the coefficients
MGF approach using the derivatives
pdf approach
The covariance between and is
Theorem 1.2.6
The correlation coefficient between X and Y is

13. Conditional Distributions and Regression
Theorem 1.3.1
The conditional pdf of given is

14. The conditional pdf of given is

15. Corollary
Remark
The regression equations are not linear.

16. Reliability Measures Standby System The system failure time is
Since X and Y are exponential,

17. Series System The failure time of the system is Since
and the reliability function is

18. Parallel System The failure time of the system is
Reliability Function is
and

19. CHAPTER 2 INFERENCE FOR MARSHALL - OLKIN BIVARIATE EXPONENTIAL DISTRIBUTION

20. Maximum Likelihood Estimators
Suppose is a random sample from MOBVE distribution having pdf
Let

21. The likelihood function is

22. The likelihood equations are
and
Solving the above equations we get the Maximum Likelihood Estimators and

23. Case(i) : , for one i (a) Case (ii) : for any two i .
The estimates are given by
(b)
Case (ii) : for any two i .
The separate estimate does not exist for the parameters.

24. Moment type Estimation
Estimators

25. Minimal Sufficient Statistic
By factorization criterion
is a sufficient statistic.
Fisher's Information Matrix

26. Inverse of the matrix is

27. Following Lehman and Casella(1998), we obtained CR lower bound for is
Cramer – Rao lower bound
Following Lehman and Casella(1998), we obtained CR lower bound for is
for is
for is

28. CHAPTER 3 ABSOLUTELY CONTINUOUS TRIVARIATE EXPONENTIAL DISTRIBUTION – TWO PARAMETER CASE

29. This chapter deals with the two parameter absolutely
continuous trivariate exponential (ACTVE) distribution due to Weier and Basu (1980). As pointed out by Sen Gupta (1995), ACTVE distribution is a special case of Marshall - Olkin trivariate exponential distribution.
The pdf of the two - parameter ACTVE is
Note that independence corresponds to

30. Theorem 3.2.1 The bivariate marginal distribution of has the pdf
is distributed as the weighted average of two equal marginal ACBVE distributions with parameters and and corresponding weights
and respectively.

31. Theorem 3.2.2
The Marginal distribution of has the pdf

32. Remark Note that Where is the pdf of , is the pdf of , and

33. Theorem 3.3.1
Following Lehmann and Casella(1998), we derive the MGF of as

34. Theorem 3.3.2 Let be a random sample of size Then
is complete sufficient with MGF

35. Remark
The MGF of at is
The MGF of at is

36. Reliability Measures
Consider a three component system having the joint distribution, which is a two - parameter ACTVE.
Standby System
System failure time is
From Remark 3.3.1

37. Reliability Function
Where
and

38. Mean Time Before Failure is
Parallel System
System failure time is
By Remark 3.3.1,

39. Reliability Function
Mean Time Before Failure is

40. Series System
The system failure time is
The reliability of the system is
Hence

41. Remark
Given a random sample from a two-parameter ACTVE, then the statistic
is complete sufficient. Thus the UMVUEs of the MTBFs are
(Standby System)
(ii) (Parallel System).

42. CHAPTER 4 ABSOLUTELY CONTINUOUS TRIVARIATE EXPONENTIAL DISTRIBUTION – THREE PARAMETER CASE

43. The pdf of the three - parameter ACTVE is
Note that corresponds to the independence.

44. Theorem 4.2.1
The bivariate marginal distribution of has the pdf
where

45. is the pdf of equal marginal ACBVE
and is the pdf given by

46. Theorem 4.2.2
The marginal distribution of has the pdf

47. Theorem 4.3.1 is the pdf of and is the pdf of
Following Lehmann and Casella(1998), the MGF of is

48. Theorem 4.3.2 Let be a random sample of size . Then the statistic
is a complete sufficient statistic and its moment generating function is

49. Reliability Measures
Consider a three component system having the joint distribution, which is a three - parameter ACTVE.
Standby System
The system failure time is

50. The reliability of the system is
where
Thus

51. Parallel System
The system failure time is
The reliability function is
where

52. and
Thus

53. Series System
The system failure time is
The reliability of the system is
Hence

54. Remark Given a random sample from a three - parameter ACTVE, then the
statistic
is complete sufficient. Thus the UMVUEs
of the MTBFs are
(i) (Standby System)
(ii) (Parallel System).

55. Remark
The two - parameter ACTVE is a particular case of the three - parameter ACTVE. The results of Chapter 4 reduce to those in Chapter 3 when and is replaced by

56. REFERENCES

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67. THANK YOU