الفهرس 
INTRODUCTIONOnly 14 pages are availabe for public view
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Abstract
Sometimes the classical distributions do not give a good fit to some real data. For example, the normal and logistic distributions are not suitable to describe asymmetric data. To overcome this problem some methods for generating new distributions by adding one or more parameters to known traditional distributions have recently been studied in the statistical literature. One of these methods is the compounding of distributions based on failures of a system. This technique could be used to generate new distributions which are more flexible in modeling such data mentioned above. When the parameters in the new distribution increase, shapes of the hazard rate function (HRF) increase.
In reliability, engineering, actuarial literature, demographic, econometric, biomedical and physical studies, the units might fail owing to one of several risk factors. These risk factors affect the units in order to fail them. If the risks are latent, then a problem arises about which factor was responsible for the component failure and hence the lifetime associated with a particular risk cannot be observed. It can be observed only, in such situations, the maximum (minimum) lifetime value among all risks. This is called in the statistical literature ”parallel (series) system”.
The two models of series and parallel systems can be combined to produce new systems such as series-parallel and parallel-series systems. We use the concept of compounding distributions to generate new distributions based on failures of series-parallel and parallel-series systems in which the number of units in each subsystem is random and the number of these subsystems is also random.
Censoring is very common in life tests. Reducing the total test time and associated cost are the major reasons for censoring. It could be applied when the distribution of exact lifetimes are known for only a portion of the units and the remainder lifetimes are known only to exceed certain values under a
life test. Type-I and type-II are two commonly used censoring schemes (CSs).
A mixture of type-I and type-II CSs is called hybrid CS. These types of censoring cannot allow the experimenter to remove units from a life test at various stages during the experiment. This problem can be overcome by applying progressive CSs. The progressive type-II censoring and type-I progressive hybrid censoring are considered generalization forms of censoring. They include the conventional censoring as a special case. Compared with the conventional censoring, however, the progressive censoring provides higher flexibility to the experimenter in the design stage by allowing the removal of test units at non-terminal time points and thus, it proves to be highly efficient and effective in utilizing the available resources. Another advantage of progressive censoring is that the degeneration information of the test units is obtained from those removed units.
The accelerated life tests (ALTs) are preferred to be used in manufacturing industries to obtain enough failure data, in a short period of time, to make inference regarding its relationship with external stress variables. In ALTs, the test items are subjected to higher than usual levels of stress (such as pressure, temperature, humidity, etc.) to induce early failures. Data collected at such accelerated conditions are then extrapolated through a physically appropriate statistical model to estimate the lifetime distribution at normal use condition.
Prediction is considered an important problem in the statistical inference. It has many applications in the fields of quality control, reliability, medical sciences, business, engineering, meteorology and other areas as well. It is the problem of inferring the values of unknown observables (future observations), or functions of such observables, from current available (informative) observations. One-sample and two-sample schemes are two commonly used schemes of prediction. A predictor could be a point or an interval predictor.
The thesis consists of six chapters. Chapter 1 is the current introductory chapter. In Chapter 2, the Poisson-Lomax distribution (PLD), with decreasing and upside-down bathtub shapes of hazard rate, is considered as a lifetime distribution. Its genesis may appear in the complementary risks (CR) model (parallel system). Based on progressive type-II censoring, the maximum likelihood (ML), unweighted least squares (UWLS), weighted least squares (WLS) and Bayes (under linear-exponential (LINEX) and general entropy (GE) loss functions) estimation methods are considered to estimate the involved parameters. The performance of these methods is compared through an extensive numerical simulation, based on mean of mean squared errors (MMSEs) and mean of relative absolute biases (MRABs).
In Chapter 3, a new class of distributions is obtained by compounding zero truncated Poisson and lifetime distributions based on failures of a series-parallel system. A specialization is paved to a new distribution, called doubly Poisson-exponential distribution (DPED), which may represent the lifetime of a series-parallel system. The new distribution can be obtained by compounding two zero truncated Poisson distributions with an exponential distribution (ED). Among its motivations is that its HRF can take different shapes such as decreasing, increasing and upside-down bathtub depending on the values of its parameters. Some properties of the new distribution are discussed. Based on progressive type-II censoring, some estimation methods [ML, moments, least squares (LS), WLS and Bayes (under LINEX and GE loss functions) estimations] are used to estimate the involved parameters. The Bayes estimates are obtained using Markov chain Monte Carlo (MCMC) algorithm. The performance of these methods is compared through an extensive numerical simulation, based on MMSEs and MRABs. A real data set is used to compare the new distribution with other five distributions.
In Chapter 4, a new class of distributions is obtained by compounding zero truncated Poisson and lifetime distributions based on failures of a parallel-series system. A specialization is paved to a new distribution, called exponential-doubly Poisson distribution (EDPD), which may represent the lifetime of a parallel-series system. The new distribution can be obtained by compounding two zero truncated Poisson distributions with an ED. Among its motivations is that its HRF can take different shapes such as decreasing, increasing and upside-down bathtub depending on the values of its parameters. Some properties of the new distribution are discussed. Based on progressive type-II censoring, some estimation methods for the involved parameters are considered. The methods are ML, moments, LS, WLS and Bayes (under LINEX and GE loss functions) estimations. Bayes estimates for the parameters are obtained using MCMC algorithm. The performance of these methods is compared through an extensive numerical simulation, based on MMSEs and MRABs. Two real data sets are used to compare the new distribution with other five distributions.
In Chapter 5, a new class of distributions is obtained by compounding geometric, zero truncated Poisson and lifetime distributions based on failures of a parallel-series system. A specialization is paved to a new distribution, called geometric-Poisson-Rayleigh distribution (GPRD), which may represent the lifetime of a parallel-series system. The new distribution can be obtained by compounding a Rayleigh distribution (RD) with geometric and zero truncated Poisson distributions. Among its motivations is that its HRF can take different shapes such as increasing, upside-down bathtub and increasing-decreasing-increasing. Some properties of the new distribution are discussed. A real data set is used to compare the new distribution with other six distributions. The progressive-stress ALT is applied to products with a non-linear increasing function of time. Based on type-I progressive hybrid censoring with binomial removals, the ML and Bayes (under LINEX and GE loss functions) estimation methods are considered to estimate the involved parameters. Some point predictors such as the ML predictor (MLP), conditional median predictor (CMP), best unbiased predictor (BUP) and Bayes point predictor under LINEX loss function (BPL) and GE loss function (BPG) for future order statistics are obtained. Prediction intervals (PIs) for future order statistics are also discussed. The Bayes estimates are obtained using MCMC algorithm. Finally, a simulation study is performed and numerical computations are carried out to compare the performance of the different methods of estimation and prediction.
In Chapter 6, a new class of distributions is obtained by compounding discrete distributions with a mixture of continuous distributions based on failures of a parallel-series system. The failure times of all units included in the system are distributed according to a mixture of h components each of which represents a different cause of failure. A specialization is paved to a new distribution, called two exponentials-Poisson-geometric distribution (TEPGD), which is obtained by compounding a mixture of two exponential distributions (MTED) with geometric and zero truncated Poisson distributions. Some special cases of the new distribution are discussed. The HRF of the new distribution has different shapes such as decreasing, increasing, upside-down bathtub, bathtub-shaped and increasing-decreasing-increasing, which make it suitable to fit several real data. Three real data sets are considered to compare the new distribution with other six distributions.