الفهرس 
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Abstract
The vital needs of modem science and technology insistantly put forward the task of developing a systematic approach to the study of any phenomena and processes occurring in the world. It is quite natural therefore, that such an approach is also required in studying the problem of reliability of technical products and system. The theory of reliability is a branch of statistical technology that deals with the general regularities to observe in the design, testing, manufacture, acceptance and use of products to obtain the maximum efficiency of the products. Reliability is a measure of the degree of successful performance of a functional under the required conditions of operation. The quantification of reliability, therefore, resolves itself into how success may be measured or, conversely, what measure can be attached to failure. Reliability plays a significant role in modern life where reliability is utilized in many fields, in production, planning, and design. In addition, the application of reliability can be seen in engineering, social, biological, sciences, maintenance, and biornetrics. It is perhaps useful at this stage to see how this idea of the measurement of success or failure concerning reliability has developed over the past few decades. In the expansion of the aircraft industry after the First World War, the fact that an aircraft engine might fail was partly instrumental in the development of multi-engines aircraft. Comparisons were made between one and two and between two and four engine aircraft from the point of successful flights. The compression tended to be purely qualitative at this stage and title attempt was made to express the reliability in terms, say, of the proportion of flight which were deemed successful for the different engine configurations. This led, in the 1 93OYs, to the concept of expressing reliability or unreliability in the form of an average number of failures or as a mean failure-rate for aircraft. from this beginnings, various inference were made as to what should be the reliability criteria for aircraft and the necessary safety levels began to be expressed in terms of ~naximuln permissible failure-rate. During the Second World War (1 935-1944) Gennans began to use lnatheinatical reliability pattern by introducing a research in reliability of n-units (devices) connected in parallel which started to be introduced by Lusser. In 1940’s requirements were given for aircraft in such terins, as the accident rate should not exceed, on average, one per 100,000 hours of flying time. During the 1960 decade, it was deduced that an aircraft has been involved in a fatal landing approximately once in every million landings that has taken place, at the start of the automatic landing system development. The degree of reliability required for the system was specified in terins of the fatal landing risk not being greater than one per 1 o7 landing. from the 195OYs, the nuclear studies have grown up in an atmosphere where reliability has been of paramount importance from safety point of view. More recently, other aspects of nuclear power reactors have been expressed in numerical reliability forms. The nuclear industry is not the only one where hazards may arise due to the unreliable operation of plant or cquipmcnt, so, considcration Imvc bccn givcn to thc reliability of turbines, circulators, boiler feed pumps and standby electrical supply system. The electrical supply industry has used reliability evaluation techniques to determine costs of supply configurations and availability of supplies to a particular consumer, for instance, might be expressed in terms of an average 30 minutes loss of s~~pply per year of’ usage. This brief review of the way in which the interest in reliability has been developed leads scientists to take this advantage for new branches of reliability such that classes of life distributions. Statisticians and reliability analysts have shown growing interest in modeling survival data using classification of life distribution based on different aspect of aging concepts that describe how a population of units or system improved or deteriorates with age. There have been significant developments, in the last two decades, of ways and means of statistical theory to solve or model different problems in reliability engineering. The theory of reliability had its effective origin by works of Bmlow et a1 (1 963) and Bryson and Siddiqui (1 969). Eversince, statisticians and reliability analysts have introduced various classes of life distributions and their dual to describe several types of deterioration or determination (improvement) that accompanied aging. The main classes of life distributions which have been introduced in the literature are based on Increasing Failure Rate (IFR), Increasing Failure Rate Average (IFRA), Decreasing Mean Residual Life (DMRL), Decreasing Mean Residual Life Average (DMRLA), New Better than Used (NBU), New Better than Used in Expectation (NBUE), New Better than Used Failure Rate (NBUFR), New Better than Average Failure Rate (NBAFR) and Harmonic New Better than Used in Expectation (KNBUE) concepts of aging. For detailed discussions on properties and some possible applications we refer to Barlow and Proclzan (1 969), Bryson and Sidrliqui ( 1 969), Rolslci (1 975), Loh (1 984)) Deslzpand and Koclzar and Singh (1 986) and Abouammnh and Alzmed (1 988). Bryson and SicZclique (1 969) have given the relationships between various classes of life distributions with aging properties exiting at that e . time. r Many other statisticians and reliability analysts, for examples, Alzmed (1 990), Con and Wnng (1 991), AI-Znic1(1994), Hencli et nl(1996), and Abounnzmnh et a1 (1996) have introduced the following additional concepts of aging classes such as Generalized Harmonic New Better than Used in Expectation (GHNBUE), New Better than Used in Convex (NBUC), Used Better than Age (UBA), New Better than Used in Average at Specific Interval (NBUASI) and New Renewal Better than Used (NRBU) respectively. I The problem of comparing two probabilities of failures always arises when it is necessary to compare the reliability of systems of the same type si~nultaneously operating in different conditions or the same system used d~iring various periods of operation. It is also frequently required to colnpare the reliability of groups of systems of various types having operated the same number of hours. We shall describe area of statistical tests which make it possible to distinguish effectively between the null hypothesis [F is exponential distribution] versus the alternative hypothesis [F has a classes of life distribution but not exponential]. During recent years some tests, for examples, TTT-test, Un-test, &-test, J (F,, a, I<)-test and Q,-test, have been suggested for testing the null hypothesis versus the alternative hypothesis. Such tests were proposed e.g. by Prochan and Pyke (1 967) when V = IFR, by Proclznn and Cnnzpo (1 975) when, V = IFRA, by HollnnrZnr and Proclznn (1 975) when, V = DMRL, and by Hendi and Andy (1 994) when, V = NBUA, by About~mnzolz and Al-Sndi ( 1 996) when, V = NRBU, and by Abounnznzoh and newby (1989) when V = NBURFR and V = NBAR.FR. This thesis contains five Chapters, the first Chapter is a historical background on the reliability theory and the classes of life distributions with detail. In Chapter 2 we introduce some classes of life distributions with some details. This Chapter is divided to three types, existing classes, renewal classes and the third is a combination between the existing and renewal classes of life distribution. In Chapter 3 we su~ninarize the relationships between every type of classes of life distributions and give a diagram for all relationships between all classes of life distribution. In Chapter 4 we introduce some classes of life distributions with some tests procedure su~ninarized and followed by the definition of censored data on its four ltinds with details and two ltinds of tests [Bnrlow - Prosclznn test & I!aplnn - Meier estimator of the s~lrvivor function] which based on censored data followed by two numerical examples by the end of each test is introduced. Chapter 5 deals with testing problems a new test statistic concerning GHNBUE is formed and computed based on two types of tests [Empirical test and TTT-test]. A simulation via a made up Pascal program is done to calculate the test statistics critical values and its power at a set of values of n is done. Chapter 5 is contributed to this area of research. Finally we give an appendix which contain the used program and a list of references which we consider as a bibliography of the topic.