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Lifetime Data and Concepts
Published in Prabhanjan Narayanachar Tattar, H. J. Vaman, Survival Analysis, 2022
Prabhanjan Narayanachar Tattar, H. J. Vaman
The strength of the exponential distribution is in its simplicity, and that itself becomes a bane in other setups. Since the distribution is governed by a single parameter, it will not be able to explain the data when there is more variability across the observations. Generalization of the exponential distribution is thus important and unequivocally required. Gamma and Weibull distributions are important alternative contenders for the exponential distribution and Figure 1.2 shows a plot of the survival function for these two distributions with different choices of parameters. Details of the parameters and exact form follow in the next chapter.
Survival Modeling II: Time-to-Event Regression Models
Published in Gary L. Rosner, Purushottam W. Laud, Wesley O. Johnson, Bayesian Thinking in Biostatistics, 2021
Gary L. Rosner, Purushottam W. Laud, Wesley O. Johnson
Piecewise Exponential Model. We now describe a fairly straightforward statistical model for the baseline hazard function. The exponential distribution is a simple one-parameter model that sometimes is useful when modeling times until an event. The usefulness of this sampling model derives from its relationship to a Poisson process. That is, if one considers that the number of events that occur in an interval follows a Poisson distribution with parameter λ, then the time in between the events will follow an exponential distribution with parameter λ. We have seen that the gamma distribution is conjugate for the exponential distribution, so a gamma prior on λ will lead to a gamma posterior.
MODELS TO AID IN PLANNING CANCER SCREENING PROGRAMS
Published in Richard G. Cornell, Statistical Methods for Cancer Studies, 2020
Michael Shwartz, Alonzo L. Plough
van Oortmarssen (1979) has analyzed a model of cervical cancer with disease states dysplasia, carcinoma in situ and four stages of invasive disease. Two types of disease are assumed, a slow progressing disease which occurs in women under age 35 and a fast progressing disease which occurs in women over age 35. Spontaneous regression is not allowed. Duration in each disease state is described by an exponential distribution. Dysplasia, “a state whose detection does not alter the possibilities of developing cancer” and thus more “a state of (very) high risk than a
A new class of skew distributions with climate data analysis
Published in Journal of Applied Statistics, 2021
Hassan S. Bakouch, Meitner Cadena, Christophe Chesneau
Some of these distributions generalize well-known distributions. Two of their simple cases are discussed below with interpretations. For x>0, (the sf related to the exponential distribution with parameter 1), we have For x>0, we have 3) under some calibration on the parameters. Let 3) with
Design of variables sampling plans based on lifetime-performance index in presence of hybrid censoring scheme
Published in Journal of Applied Statistics, 2019
Ritwik Bhattacharya, Muhammad Aslam
Let us consider X representing the lifetime random variable. In practice, it is not always clock-time or chronological. For example, a number of pages output for a computer printer machine can be considered as its lifetime. However, lifetime clearly represents a larger-the-better type quality characteristic, that is, longer lifetime represents a better quality product. Therefore, a quantity called lower specification limit, say L, is generally associated with the lifetime. Lifetime exceeds L is highly expected. A dimensionless process capability index 18]) X following as exponential distribution with cumulative distribution function is given by 3]), and secondly, the maximum likelihood estimator of the model parameter exists with its explicit form. Moreover, the exact sampling distribution of the estimator is derived in the literature.
A moment-based empirical likelihood ratio test for exponentiality using the probability integral transformation
Published in Journal of Applied Statistics, 2019
Besides the normal distribution, the exponential distribution is one of the most applied continuous distribution in statistical sciences. In practice, it is always vital to test for departures from exponentiality before further inferences are done. The effort of developing goodness-of-fit (GoF) tests to detect these departures is therefore of paramount importance. Among the several techniques for developing these GoF tests for exponentiality, the characterization of the distribution plays an important role and can be used as a basic ingredient for exponentiality testing. Of these tests, we have those which the characterization is based on moments of order statistics (see [32]), memoryless property (e.g. [4,23]), identically distributed random variables (e.g. [51]), uniformly distributed random variables and normalized spacings (e.g. [65]) and many others. In practice, the alternatives to the exponential distribution are often not known hence tests for exponentiality which focus on detecting all distributional departures from exponentiality are of paramount importance. Such tests are called omnibus tests.