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In this paper, a novel test statistic is introduced to evaluate the goodness of fit of lifetime distributions when dealing with Type II censored data. The test statistic is derived from a local linear regression-based estimator of the Kullback–Leibler information. Extensive analysis is conducted to examine the properties of this test statistic, highlighting its nonnegative nature, as for KL information. The proposed test statistic is applied to various distributions, namely exponential, Weibull, log-normal, and Pareto. Critical values and Type I error rates for the tests are determined, demonstrating their exceptional accuracy and reliability. To further assess their performance, the proposed tests are subjected to Monte Carlo simulations, comparing their power values against alternative tests currently in use. Finally, the proposed method is applied to three real data sets from the engineering reliability aspect to prove their practical versatility.

The recent stimulating proposal of a "Gauge Theory of Finance" by Ilinsky et al. is connected here with traditional approaches. First, the derivation of the log-normal distribution is shown to be equivalent both in information and mathematical content to the simpler and well-known derivation, dating back from Bachelier and Samuelson. Similarly, the re-derivation of Black–Scholes equation is shown equivalent to the standard one because the limit of no uncertainty is equivalent to the standard risk-free replication argument. Both re-derivations of the log-normality and Black–Scholes result do not provide a test of the theory because it is not uniquely specified in the limits where these results apply. Third, the choice of the exponential form a la Boltzmann, of the weight of a given market configuration, is a key postulate that requires justification. In addition, the "Gauge Theory of Finance" seems to lead to "virtual" arbitrage opportunities for a pure Markov random walk market when there should be none. These remarks are offered in the hope to improve the formulation of the "Gauge Theory of Finance" into a coherent and useful framework.

We introduce a class of one-dimensional linear kinetic equations of Boltzmann and Fokker–Planck type, describing the dynamics of individuals of a multi-agent society questing for high status in the social hierarchy. At the Boltzmann level, the microscopic variation of the status of agents around a universal desired target, is built up introducing as main criterion for the change of status a suitable value function in the spirit of the prospect theory of Kahneman and Twersky. In the asymptotics of grazing interactions, the solution density of the Boltzmann-type kinetic equation is shown to converge towards the solution of a Fokker–Planck type equation with variable coefficients of diffusion and drift, characterized by the mathematical properties of the value function. The steady states of the statistical distribution of the social status predicted by the Fokker–Planck equations belong to the class of Amoroso distributions with Pareto tails, which correspond to the emergence of a social elite. The details of the microscopic kinetic interaction allow to clarify the meaning of the various parameters characterizing the resulting equilibrium. Numerical results then show that the steady state of the underlying kinetic equation is close to Amoroso distribution even in an intermediate regime in which interactions are not grazing.

In two-channel microarray experiments, the image analysis extracts red and green fluorescence intensities. The ratio of the two fluorescence intensities represents the relative abundance of the corresponding DNA sequence. The subsequent analysis is performed by taking a log-transformation of this ratio. Therefore, the statistical analyses depend on accuracy of the ratios calculated from the image analysis. However, not many studies have been proposed for developing more reliable ratio statistics. In this paper, we consider a new type of log-transformed ratio statistic. We compare the new ratio statistic with the conventional ratio statistic commonly used in two-channel microarray experiments. First, under the specific log-normal distributional assumption, we compare analytically the new statistics with the conventional ratio statistic. Second, we compare those ratio statistics using a two-channel microarray data obtained by hybridizing a mixture of mouse RNA and yeast in vitro transcript (IVT). Both comparisons show that the proposed ratio statistic performs better than the conventional one.

In this paper, a new heavy-tailed distribution is used to model data with a strong right tail, as often occuring in practical situations. The proposed distribution is derived from the log-normal distribution, by using odd log-logistic distribution. Statistical properties of this distribution, including hazard function, moments, quantile function, and asymptotics, are derived. The unknown parameters are estimated by the maximum likelihood estimation procedure. For different parameter settings and sample sizes, a simulation study is performed and the performance of the new distribution is compared to beta log-normal. The new lifetime model can be very useful and its superiority is illustrated by means of two real data sets.

In this chapter, we first review the basic theory of normal and log-normal distribution and their relationship, then bivariate and multivariate normal density function are analyzed in detail. Next, we discuss American options in terms of random dividend payment. We then use bivariate normal density function to analyze American options with random dividend payment. Computer programs are used to show how American co-options can be evaluated. Finally, pricing option bounds are analyzed in some detail.

It is well known that both normal and log-normal distributions are important to understand Black & Scholes-type European and American options. Therefore, we first review the basic theory of normal and log-normal distributions and their relationship, then bivariate and multivariate normal density functions are analyzed in detail. Next, we discuss American options in terms of random dividend payment. We then use bivariate normal density function to analyze American options with random dividend payment. Excel programs are used to show how American co-options can be evaluated. Finally, pricing option bounds are analyzed in some detail.