## Doctoral Theses

### On Nonparametric Families of Life Distributions: Some Issues and Applications.

2-28-1994

2-28-1995

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Basu, Sujit Kumar (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

In reliability theury, one in primarily concerned with the study of the lifetime of a unit. The unit may be a mechanical device or a component of such a device; it may also be a living organism. By lifetime, we usually mean the duration for which the unit under study continues to perform cartain specific functions before it passes on to what is popularly known as the 'failed state'. More generally, however, lifetime can be, and often is, interpreted as the time to occurrence of a certain event such as the elapsed time before a broken down machine starts functioning again or the time period for which a person uses a particular brand of a certain consumer product etc.It is the practice of reliabilists to consider lifetime as a non-negative random variable. We shall generically denote it by X and assume that X is continuous. For z20, F(2) := P[X S a) Is called the distribution function (d.f.) of X and F(z) := P(X > =) denotes the corresponding survival function.The d.f. Fis a ife d.f. If it satisfies F(0-) = 0. Note that the survival function, which is also referred to as the reli-. ability function, defines the probability that the unit will function at least for e units of time. For absolutely continuous X with probability density function (p.d.f.) f(z), the failure rate function of the unit is defined by rala) := for ax 0 satisfying F) > 0.It is easy to see that rf(x) has a nice physical interpretation; heuristically, re(a)da can be looked upon as the probability that the unit alive at age will fail in the interval (a, z + da) where dz is taken to be small. The above function is basic in 1 reliability theory and is variously known as the hazard rate function, intensity function and also force of mortality.Another function of fundamental importance in the study of life lengths is the Mean Residual Life (MRL) function (see Guess and Proschan (1985)), which is defined as eF(z) := E(X - 2|X > 2) where ep(z) is given by (1/F=)) Fe)dt, for z2 0 satisfying F(2) > 0 0, when m) = 0. er(z) As indicated above, this function gives the expected residual life of the unit given that it has survived upto age x.The importance of the functions introduced above lies in their use in the study of the ageing pattern of units. In fact, various ageing criteria have been defined in the literature based on the behaviour of these functions. A unit is said to age positively (negatively) if its residual life tends to decrease (increase), in some probabilistic sense, with increase in age. The following definitions, which can be found in Bryson and Siddiqui (1969), Barlow and Proschan (1975), Rolski (1975), KlefsjÃ¶ (1982a, 1982b) and Hollander and Proschan (1984), illustrate how the failure rate and MRL functions are used to introduce various notions of ageing.

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843443

ISILib-TH247

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

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