DEPARTMENT OF STATISTICS LOYOLA COLLEGE CHENNAI – 600 034 DISSERTATION SUBMITTED BY TUNNY SEBASTIAN (07 MST 09) UNDER THE GUIDANCE OF DR. B. CHANDRASEKAR, M.Sc., M.Phil., Ph.D.

BIVARIATE AND TRIVARIATE EXPONENTIAL DISTRIBUTIONS IN RELIABILITY THEORY

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

CHAPTER 1 MARSHALL – OLKIN BIVARIATE EXPONENTIAL DISTRIBUTION

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.

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

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

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

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

(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

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

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

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

The conditional pdf of given is

Corollary Remark The regression equations are not linear.

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

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

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

CHAPTER 2 INFERENCE FOR MARSHALL - OLKIN BIVARIATE EXPONENTIAL DISTRIBUTION

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

The likelihood function is

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

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.

Moment type Estimation Estimators

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

Inverse of the matrix is

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

CHAPTER 3 ABSOLUTELY CONTINUOUS TRIVARIATE EXPONENTIAL DISTRIBUTION – TWO PARAMETER CASE

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

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.

Theorem 3.2.2 The Marginal distribution of has the pdf

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

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

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

Remark The MGF of at is The MGF of at is

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

Reliability Function Where and

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

Reliability Function Mean Time Before Failure is

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

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).

CHAPTER 4 ABSOLUTELY CONTINUOUS TRIVARIATE EXPONENTIAL DISTRIBUTION – THREE PARAMETER CASE

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

Theorem 4.2.1 The bivariate marginal distribution of has the pdf where

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

Theorem 4.2.2 The marginal distribution of has the pdf

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

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

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

The reliability of the system is where Thus

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

and Thus

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

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).

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

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