Describing Within-Person Change Over Time
Lesa Hoffman in Longitudinal Analysis, 2015
The name of the exponential model is derived from the sign of its rate parameter, and not the direction of its change. That is, a negative exponential model could describe a positive rate of change that is slowing down as it approaches an upper asymptote or a negative rate of change that is slowing down as it approaches a lower asymptote (as we will see in our RT data). In contrast, a positive exponential model would include a positive rate parameter to describe a function that is moving away from the asymptote instead (i.e., the function is speeding up). Positive exponential functions are more commonly see in biological applications (e.g., exponential growth of bacteria over time), although they may have other applications as well.
London’s population and hospitals: 1801–1971
Leslie Mayhew in Urban Hospital Location, 2018
The most convenient way to characterise the population of a city is with the aid of a population density function. Apart from concurring with the theory presented in earlier chapters (particularly Ch. 2), a descriptive model based on a suitable density function is consistent with the intended approach. The critical question is the choice of function and the adequacy with which it portrays essential features of the population of the city. In a wide and growing collection of studies, there is considerable agreement that the distribution of population densities in cities follows a negative exponential distribution. Clark (1951), Muth (1961), Mills (1970) and Bussiere (1972) are representative of early work in this area and in recent years this list has considerably expanded. Most of the possible alternatives to the negative exponential are in fact close mathematical relatives, but the empirical evidence for selecting one rather than another tends in fact to be fairly weak. Zielinski (1980) gives an abridged chronology of examples as indications of the range of variation that has been considered.
S
Filomena Pereira-Maxwell in Medical Statistics, 2018
A depiction of a non-linear relationship between two continuous variables, characterized by an initial slow or no growth phase, followed by growth acceleration in the form of a steeper segment, and a final deceleration or stagnant phase as an asymptotic upper value is approached. This is shown in Figure S.3 (MITCHELL & JAKUBOWSKI, 1999), which models the increasing complication rate with age, in a group of 1172 patients who underwent surgical treatment of unruptured intracranial aneurysms. As the negative exponential curve, this curve is characteristic of constrained population growth, although it differs in that it has an initial phase of slow growth. Sigmoid or S-shaped curves are fitted through non-linear regression. The specific curve obtained depends on the magnitude and direction of the parameters of a given model, and the values of the x-variable. Reverse sigmoid curves - in which there is a slow decline, followed by rapid decline, followed by a return to a slow decline - may also be obtained. HAMILTON (1992; 2012) describes logistic and Gompertz curves as examples of symmetrical and asymmetrical sigmoid curves. Sigmoid curves are also characteristic of cumulative distribution functions (for the Normal, t and logistic probability distributions), and may be fitted to model cumulative response. See also exponential curve, U-shaped curve, J-shaped curve, spline regression.
A new alternative quantile regression model for the bounded response with educational measurements applications of OECD countries
Published in Journal of Applied Statistics, 2023
Mustafa Ç. Korkmaz, Christophe Chesneau, Zehra Sedef Korkmaz
Recently, the work on the unit distribution has increased with great interest in many different fields. This is mainly motivated by the practitioner's dissatisfaction with the classic unit distributions. For instance, the beta distribution can be inadequate in order to both model and predict based on the real data phenomena. The beta distribution does not take into account the events of the end zone or more flexibility in specifying the variance. For this aspect, we may refer to [6]. In light of this, the existing unit distributions have generally been elaborated by transforming well-known probability distributions. The main interest of using the transformation based on these distributions is that they do not add new parameters to them on the unit interval. To transform a positive random variable (rv) into new unit rvs, the most used transformation is centered on the negative exponential function. For instance, the Kumaraswamy [31], log-Lindley [17], unit-Weibull [37], unit Gompertz [35], log-xgamma [4], unit inverse Gaussian [16], unit generalized half normal [27], log-weighted exponential [2] and log-extended exponential geometric [22] distributions have been obtained via this method. One may see [3,18,20,26,34] for other unit models that were obtained with other transformation methods. These proposed unit distributions can present more flexible density shapes on the
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.
Testing symmetry based on empirical likelihood
Published in Journal of Applied Statistics, 2018
Jun Zhang, Jing Zhang, Xuehu Zhu, Tao Lu
The exact formula of X follows an exponential distribution α: et al. [30] obtained that α, however, the Weibull density is asymmetric in the strict mathematical sense for any given α, owing to the very thin and negligible tail. The visual impression of the density curve at
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