Moment Generating Function Research Articles

ODD GOMPERTZ-G FAMILY OF DISTRIBUTION, ITS PROPERTIES AND APPLICATIONS

In this research paper, we introduced a novel generator derived from the continuous Gompertz distribution, known as the odd Gompertz-G distribution family. We conducted an in-depth analysis of the statistical characteristics of this family, including moments, moment-generating functions, quantile functions, survival functions, hazard functions, entropies, and order statistics. Within this family, we also derived a specific distribution called the odd Gompertz-Exponential distribution. To evaluate the reliability of the distribution's parameters, we employed Monte Carlo simulations. Furthermore, we assessed the applicability of this newly proposed distribution family by examining its performance on real-world data and the results demonstrate that the new model (OG-E) outperformed its comparators under consideration.

Analyzing the interest rate risk of equity-indexed annuities via scenario matrices

The financial return of equity-indexed annuities depends on an underlying fund or investment portfolio complemented by an investment guarantee. We discuss a so-called cliquet-style or ratchet-type guarantee granting a minimum annual return. Its path-dependent payoff complicates valuation and risk management, especially if interest rates are modelled stochastically. We develop a novel scenario-matrix (SM) method. In the example of a Vasicek-Black-Scholes model, we derive closed-form expressions for the value and moment-generating function of the final payoff in terms of the scenario matrix. This allows efficient evaluation of values and various risk measures, avoiding Monte-Carlo simulation or numerical Fourier inversion. In numerical tests, this procedure proves to converge quickly and outperforms the existing approaches in the literature in terms of computation time and accuracy.

Modified Threshold-based Spectrum Sensing Approach for VANETs

The Primary User (PU) signal detection in Cognitive Radio (CR) is crucial and is achieved through spectrum sensing techniques. The Energy Detection method is a commonly used technique, and selecting a proper threshold is essential to enhance the efficiency of the CR system. This research paper demonstrates the maximum achievable throughput and validates a Modified Threshold (MT) approach. The authors consider a scenario with multiple antennas at the receiver, where these antennas are correlated and subjected to mobility effects, and they employ the Energy Detection (ED) for spectrum sensing. The study analyzes the system's performance over a Nakagami-m fading channel, considering available correlations among the antenna elements. To compute important statistical values, the Moment Generating Function (MGF) method is employed. The research employs specialized mathematical functions, such as the Lauricella and confluent hypergeometric functions, to derive closed-form expressions for the Probability of Detection when employing the diversity technique. The results indicate a significant enhancement in the performance of the proposed algorithm when utilizing the modified threshold parameter across a wide range of Signal to Noise Ratio (SNR) values. Additionally, increasing the number of branches in the antenna system further improves detection performance. Interestingly, under high fading parameter conditions (m=4), the detection probability is found to be superior with exponential correlation among the L antenna elements compared to other available correlated branches.

Exact solution to a first-passage problem for an Ornstein–Uhlenbeck process with jumps and its integral

Let dX(t)=Y(t)dt, where Y(t) is an Ornstein–Uhlenbeck process with Poissonian jumps, and let T(x,y) be the first time that X(t)+Y(t)=0, given that X(0)=x and Y(0)=y. The moment-generating function of T(x,y) is obtained in the case when the jumps are exponentially distributed by solving the integro-differential equation it satisfies, subject to the appropriate boundary conditions. The case when the jumps are uniformly distributed is also considered.

A New Approach of Presenting Lindley Expontiated Gumbel Distribution with Application to Environmental Data

The Lindley exponentiated Gumbel (LEGu) Distribution, a new family of probability distribution developed for modelling environmental data is introduced in this research paper. The investigation defines the new distribution's basic statistical properties, such as its shape, density function, hazard rate function, moment generation function, and maximum likelihood estimates of its model parameters. The study shows that, in comparison to other baseline distributions and comparators, increasing the number of parameters in the distribution improves its robustness and adaptability, making it more flexible in accommodating different types of data. The model is now more adaptable and can handle a wider range of data sets, making it a useful and effective tool for environmental modelling. The creation of the Lindley exponentiated Gumbel Distribution and the statistical qualities that go along with it bring up new possibilities for the analysis of environmental data, making a significant contribution to the larger initiatives in climate research and decision-making. The results of this study have the potential to advance our comprehension of environmental mechanisms and raise the precision of climate assessments and projections.

Bulk Behaviour of Ground States for Relativistic Schrödinger Operators with Compactly Supported Potentials

We propose a probabilistic representation of the ground states of massive and massless Schrödinger operators with a potential well in which the behaviour inside the well is described in terms of the moment-generating function of the first exit time from the well and the outside behaviour in terms of the Laplace transform of the first entrance time into the well. This allows an analysis of their behaviour at short to mid-range from the origin. In a first part, we derive precise estimates on these two functionals for stable and relativistic stable processes. Next, by combining scaling properties and heat kernel estimates, we derive explicit local rates of the ground states of the given family of non-local Schrödinger operators both inside and outside the well. We also show how this approach extends to fully supported decaying potentials. By an analysis close-by to the edge of the potential well, we furthermore show that the ground state changes regularity, which depends qualitatively on the fractional power of the non-local operator.

Recurrence Relations for Moment Generating Function Based on Progressive First Failure Censoring from Generalized Pareto Distribution and Characterization

In this article, we establish recurrence relations (RR) for single and product moment generating function (MGF) based on based on progressive first failure censoring (PFFC) for generalized Pareto distribution (GPD). Characterization for GPD using RR of single and product MGF of PFFC are also obtained. Further, the results are specialized to the progressively type-II right censored (PTIIRCOS).

A Hanbury Brown and Twiss renascence: measurement of photon correlations yields spatio-temporal coherence

Investigations of photon correlations similar to the Hanbury Brown and Twiss experiment are since the advent of quantum optics in the center of both fundamental and application-oriented research. We derive a factorial moment-generating function of the probability distribution of electromagnetic radiation emitted by a source and absorbed by a photodetector from which we calculate an analytical expression for the second-order correlation coefficient g^{(2)} which reflects spatial coherence properties of the emitted light. The measurement of g^{(2)} thus allows to retrieve the spatial coherence parameter of the light source in terms of the combination of source area, source-detector distance, wavelength and detector area. The validity of the concept is proven by investigating g^{(2)} of true thermal light from a Xenon arc lamp and from the sun for various detector areas, in excellent agreement with the theory. Finally, we suggest a novel scheme for determining light source diameters by exploiting the spatially dependent statistics and confirm its validity by exemplary calculations. These results based on radiation thermodynamics and photon statistics give fresh insight into quantum optical properties of classical light sources, photon correlations and photodetection with promising applications perspectives, more than 68 years after the original Hanbury Brown and Twiss experiment.

Probability thermodynamics and probability quantum field

We introduce probability thermodynamics and probability quantum fields. By probability we mean that there is an unknown operator, physical or nonphysical, whose eigenvalues obey a certain statistical distribution. Eigenvalue spectra define spectral functions. Various thermodynamic quantities in thermodynamics and effective actions in quantum field theory are all spectral functions. In the scheme, eigenvalues obey a probability distribution, so a probability distribution determines a family of spectral functions in thermodynamics and quantum field theory. This leads to probability thermodynamics and probability quantum fields determined by a probability distribution. In constructing spectral functions, we encounter a problem. The conventional definition of spectral functions applies only to lower bounded spectra. In our scheme, however, there are two types of spectra: lower bounded spectra, corresponding to the probability distribution with nonnegative random variables, and the lower unbounded spectra, corresponding to probability distributions with negative random variables. To take the lower unbounded spectra into account, we generalize the definition of spectral functions by analytical continuation. In some cases, we encounter divergences. We remove the divergence by a renormalization procedure. In virtue of spectral theory in physics, we generalize some concepts in probability theory. For example, the moment-generating function in probability theory does not always exist. We redefine the moment-generating function as the generalized heat kernel introduced in this paper, which makes the concept definable when the definition in probability theory fails. We construct examples corresponding to some probability distributions. Thermodynamic quantities, vacuum amplitudes, one-loop effective actions, and vacuum energies for various probability distributions are presented.

ALAMOUTI SPACE-TIME CODING FOR VEHICULAR COMMUNICATIONS IN THE PRESENCE OF CHANNEL ESTIMATION ERRORS

In this paper, the error performance of the Alamouti space-time coding (STC) scheme is investigated for vehicle-to-vehicle (V2V) communication systems over imperfect cascaded fading channels. In vehicular communication systems, perfect knowledge of the channel state information is not available to the users at all times due to the rapid movement of communicating vehicles and fast change of the rich scattering environment which makes the fading effects in wireless channels more severe. Therefore, in the analysis, we consider the erroneous estimation of the channel gain, which is more realistic for practical scenarios. For this purpose, we first derive the moment-generation function (MGF) of the channel fading coefficient with estimation error in the case of the cascaded Nakagami-m fading conditions. Then, using the MGF, we obtain the closed-form symbol-error-rate (SER) expressions of Alamouti STC with two transmitting and L receiving antennas for the M-PSK and M-QAM modulation schemes. Then, the exact ergodic capacity expression is derived for the proposed system. Furthermore, the analytical results are verified through Monte-Carlo simulations. Numerical results show that the SER performance of V2V communication systems can be improved significantly by using Alamouti STC even in case of harsh fading conditions and full channel-state information is not available due to estimation errors.

An assessment of model risk in pricing wind derivatives

AbstractWind derivatives are financial instruments designed to mitigate losses caused by adverse wind conditions. With the rapid growth of wind power capacity due to efforts to reduce carbon emissions, the demand for wind derivatives to manage uncertainty in wind power production is expected to increase. However, existing wind derivative literature often assumes normally distributed wind speed, despite the presence of skewness and leptokurtosis in historical wind speed data. This paper investigates how the misspecification of wind speed models affects wind derivative prices and proposes the use of the generalized hyperbolic distribution to account for non-normality. The study develops risk-neutral approaches for pricing wind derivatives using the conditional Esscher transform, which can accommodate stochastic processes with any distribution, provided the moment-generating function exists. The analysis demonstrates that model risk varies depending on the choice of the underlying index and the derivative’s payoff structure. Therefore, caution should be exercised when choosing wind speed models. Essentially, model risk cannot be ignored in pricing wind speed derivatives.

A New Odd Beta Prime-Burr X Distribution with Applications to Petroleum Rock Sample Data and COVID-19 Mortality Rate

In this article, we pioneer a new Burr X distribution using the odd beta prime generalized (OBP-G) family of distributions called the OBP-Burr X (OBPBX) distribution. The density function of this model is symmetric, left-skewed, right-skewed, and reversed-J, while the hazard function is monotonically increasing, decreasing, bathtub, and N-shaped, making it suitable for modeling skewed data and failure rates. Various statistical properties of the new model are obtained, such as moments, moment-generating function, entropies, quantile function, and limit behavior. The maximum-likelihood-estimation procedure is utilized to determine the parameters of the model. A Monte Carlo simulation study is implemented to ascertain the efficiency of maximum-likelihood estimators. The findings demonstrate the empirical application and flexibility of the OBPBX distribution, as showcased through its analysis of petroleum rock samples and COVID-19 mortality data, along with its superior performance compared to well-known extended versions of the Burr X distribution. We anticipate that the new distribution will attract a wider readership and provide a vital tool for modeling various phenomena in different domains.

Odd Chen Exponential Distribution: Properties and Applications

In this study, a new statistical distribution with three parameters called the Odd-Chen Exponential has been proposed. The statistical properties of the proposed distribution, such as the quantile, moments, incomplete moments, moment-generating function, and mean residual life, were developed. The density shows different shapes, making it more flexible for analyzing different forms of data. The hazard function also exhibits different shapes, including the well-known bathtub shape, which means that the distribution is flexible with real-life data. To estimate the distribution parameters, ordinary least squares estimators, Cramér-von Mises estimators, and maximum likelihood estimators were derived. The results were compared using a Monte Carlo simulation. Two-time datasets; one from the mining field and the other from survival analysis, were used to check the applicability of the proposed distribution. The results revealed that the OCE distribution performed better than the Odd Chen Weibull, Odd Chen Rayleigh, Rayleigh, Cauchy, Generalised Inverse Weibull and the Modified Extended Chen distributions.

Double Reconfigurable Intelligent Surface to Improve Error Performance and Coverage Probability of Wireless Communication System

This paper investigates the symbol error rate (SER) and coverage probability of single reconfigurable intelligent surface (S-RIS) and double reconfigurable intelligent surface (D-RIS) systems operating over Weibull fading channels. In the S-RIS system, the RIS is positioned between the base station and user link, while in the D-RIS system, RIS-1 and RIS-2 are placed near the base station and user respectively. By leveraging the central limit theorem, we derive the statistical parameters of the received signal-to-noise ratio (SNR) and subsequently closed-form expressions for SER and coverage probability of both systems using the moment-generating function. Also, we determine the optimal location of the RISs in D-RIS system from the simulations of SER. Comparative analysis reveals that the D-RIS system exhibits superior error performance and provides better coverage than the S-RIS system, despite having an equal number of reflecting elements. In addition, we provide an energy consumption gain analysis of the D-RIS system.

Joint Distribution of Ages of Information in Networks

We study a general setting of status updating systems in which a set of source nodes provide status updates about some physical process(es) to a set of monitors. The freshness of information available at each monitor is quantified in terms of the Age of Information (AoI), and the vector of AoI processes at the monitors (or equivalently the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">age vector</i> ) models the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">continuous</i> state of the system. While the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">marginal</i> distributional properties of each AoI process have been studied for a variety of settings using the stochastic hybrid system (SHS) approach, we lack a counterpart of this approach to systematically study their <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">joint</i> distributional properties. Developing such a framework is the main contribution of this paper. In particular, we model the discrete state of the system as a finite-state continuous-time Markov chain, and describe the coupled evolution of the continuous and discrete states of the system by a piecewise linear SHS with linear reset maps. Using the notion of tensors, we first derive first-order linear differential equations for the temporal evolution of both the joint moments and the joint moment generating function (MGF) for an arbitrary set of age processes. We then characterize the conditions under which the derived differential equations are asymptotically stable. The generality of our framework is demonstrated by recovering several existing results as special cases. Finally, we apply our framework to derive closed-form expressions of the stationary joint MGF in a multi-source updating system under non-preemptive and source-agnostic/ source-aware preemptive in service queueing disciplines.

A statistical mechanical model of supercooled water based on minimal clusters of correlated molecules.

In this paper, we apply a theoretical model for fluid state thermodynamics to investigate simulated water in supercooled conditions. This model, which we recently proposed and applied to sub- and super-critical fluid water [Zanetti-Polzi et al., J. Chem. Phys. 156(4), 44506 (2022)], is based on a combination of the moment-generating functions of the enthalpy and volume fluctuations as provided by two gamma distributions and provides the free energy of the system as well as other relevant thermodynamic quantities. The application we make here provides a thermodynamic description of supercooled water fully consistent with that expected by crossing the liquid-liquid Widom line, indicating the presence of two distinct liquid states. In particular, the present model accurately reproduces the Widom line temperatures estimated with other two-state models and well describes the heat capacity anomalies. Differently from previous models, according to our description, a cluster of molecules that extends beyond the first hydration shell is necessary to discriminate between the statistical fluctuation regimes typical of the two liquid states.

Correlations in randomly stacked solids.

Packing of spheres is a problem with a long history dating back to Kepler's conjecture in 1611. The highest density is realized in face-centered-cubic (FCC) and hexagonal-close-packed (HCP) arrangements. These are only limiting examples of an infinite family of maximal-density structures called Barlow stackings. They are constructed by stacking triangular layers, with each layer shifted with respect to the one below. At the other extreme, Torquato-Stillinger stackings are believed to yield the lowest possible density while preserving mechanical stability. They form an infinite family of structures composed of stacked honeycomb layers. In this article, we characterize layer-correlations in both families when the stacking is random. To do so, we take advantage of the Hägg code-a mapping between a Barlow stacking and a one-dimensional Ising magnet. The layer correlation is related to a moment-generating function of the Ising model. We first determine the layer correlation for random Barlow stacking, finding exponential decay. We next introduce a bias favoring one of two stacking chiralities-equivalent to a magnetic field in the Ising model. Although this bias favors FCC ordering, there is no long-ranged order as correlations still decay exponentially. Finally, we consider Torquato-Stillinger stackings, which map to a combination of an Ising magnet and a three-state Potts model. With random stacking, the correlations decay exponentially with a form that is similar to the Barlow problem. We discuss relevance to ordering in clusters of stacked solids and for layer-deposition-based synthesis methods.

A new reduced contagious zero-inflated model: An application to count data

In this paper, a reduced one-parameter contagious distribution was developed from the joint distribution of three-parameter gamma and Poisson distributions, on which Lakshmi’s three-parameter gamma distribution is based to model a count data. The distribution properties and some common descriptive measures relating to this contagious distribution are derived. The behavior of the probability mass function with changes in parameters was also studied. The parameter estimation by the maximum likelihood and moment-generating function methods is discussed. A simulation study was carried out with the proposed model to check for consistency and bias. The new model show consistency as the sample size increases. The model was applied to a real-life dataset and was seen to be more flexible in capturing excess zero, under, and over-dispersion in count data and proved to be a useful alternative to some existing zero-inflated models.

Gamma-shadowed two-ray with diffuse power composite fading model

In this paper, a novel gamma-shadowed two-ray with diffuse power (GS-TWDP) composite fading model is proposed by assuming that the shadowing-caused mean power variations of the received signal, affected by TWDP multipath fading, are gamma distributed. For the proposed model, the essential statistics such as probability density function (PDF), cumulative distribution function (CDF), moment generating function (MGF) and higher order moments are obtained and used to derive the approximate general order rectangular quadrature amplitude (RQAM) average symbol error probability (ASEP). That expression enabled us to investigate the effect of shadowing on overall error probability, in case when signal is propagating in channels with TWDP multipath fading. The proposed model is verified by comparing analytically obtained results with those measured at 28 GHz and reported in literature, which makes the GS-TWDP model a promising candidate for modeling propagation at millimeter wave (mmWave) frequencies. Analytical results are validated by Monte-Carlo simulation.

Modified generalized Weibull distribution: theory and applications

This article presents and investigates a modified version of the Weibull distribution that incorporates four parameters and can effectively represent a hazard rate function with a shape resembling a bathtub. Its significance in the fields of lifetime and reliability stems from its ability to model both increasing and decreasing failure rates. The proposed distribution encompasses several well-known models such as the Weibull, extreme value, exponentiated Weibull, generalized Rayleigh, and modified Weibull distributions. The paper derives key mathematical statistics of the proposed distribution, including the quantile function, moments, moment-generating function, and order statistics density. Various mathematical properties of the proposed model are established, and the unknown parameters of the distribution are estimated using different estimation techniques. Furthermore, the effectiveness of these estimators is assessed through numerical simulation studies. Finally, the paper applies the new model and compares it with various existing distributions by analyzing two real-life time data sets.