Generalized Lambda Distribution Research Articles

Uncertainty quantification with high-dimensional correlated process variations for an in-situ thermal expansion coefficient test structure

The impacts of process variations on the π-shaped thermal expansion coefficient (TEC) in-situ test structure are analyzed. Correlations among high-dimensional design parameters have to be taken into account for this type of structure, while the widely studied polynomial chaos expansion (PCE) technique is incapable of solving such problems. This paper presents a novel method for high-dimensional uncertainty quantification problems with non-Gaussian correlated variants. As a pre-processing method, the Spearman correlation coefficient and generalized lambda distribution are merged to eliminate the correlation. Then a compressed-sensing based Legendre-sparse solver is applied benefiting from the uniformly distributed parameters derived from the pre-processing and thus the cost of rebuilding correlated basis functions is eliminated. This method has demonstrated superior performance over existing methods in the analysis of TEC in-situ test structures. This analysis shows that the measurement accuracy of the test structure demonstrates good performance and is insensitive to process errors, which is consistent with experimental data. The presented technique has the potential to be applied in the design of various microelectromechanical system (MEMS) devices.

Identifying the Probability Distribution of Exchange Rates in Sri Lanka

The exchange rate is one of the main components that represents the status of a country’s economic condition. This study mainly focuses on identifying an appropriate distribution that can capture the asymmetric behaviour of important currency exchange rates related to Sri Lanka. The true behaviour of the exchange rates of USD, EURO, JPY, AUD, GBP, CHF, SGD and CAD in terms of LKR was determined using flexible distributions of generalised lambda distribution (GLD), Normal Inverse Gaussian and Skew-Normal. The parameter estimation was carried out by maximum likelihood estimation (MLE), while for GLD, few more parameter estimation techniques were applied. To understand how well the fitted distributions identify the real behaviour of data, we conducted the goodness-of-fit (GOF) tests. The results from Cramer–Von Mises (CvM) and Anderson–Darling (AD) GOF tests pointed out that the exchange rates data follow a GLD with the parameter estimation of maximum product of spacings (MPS). Further, plots of histograms with theoretical densities, cumulative distribution functions (CDFs) with empirical CDF and Quantile–Quantile plots illustrated the same idea. This study presented the findings based on different suitable parameter estimation techniques and this study is the first investigation on the distribution identification of exchange rates in Sri Lanka. JEL Codes: C12,C13,G10

Reliability Evaluation Method of Pseudo-failure Life Distribution for Industrial Robot Servo System Based on The Generalized Lambda Distribution

A reliability evaluation method of pseudo-failure life distribution for Industrial Robot Servo System (IRSS) based on The Generalized Lambda Distribution (GLD) was proposed in the paper. The IRSS pseudo-failure life GLD distribution was established. Quantile estimation method was applied to estimate the GLD parameters. The proposed method had no need of presetting specific distribution morphology, and thus improve the accuracy of IRSS pseudo-failure reliability modeling. In the application experiment, the IRSS pseudo-failure life distribution reliability evaluation method based on GLD was used to obtain the IRSS failure life, the reliability evaluation result was consistent with the design failure life, which indicated the validity of the proposed method.

On the goodness-of-fits of the generalized lambda distribution on high-frequency stock index returns

On the goodness-of-fits of the generalized lambda distribution on high-frequency stock index returns

Modified information criterion for testing changes in generalized lambda distribution model based on confidence distribution

Modified information criterion for testing changes in generalized lambda distribution model based on confidence distribution

Evaluating risk in precious metal prices with generalised lambda, generalised pareto and generalised extreme value distributions

In this study we investigate the performance of the generalised lambda distribution (GLD), the generalised Pareto distribution (GPD) and the generalised extreme value distribution (GEVD) in modelling daily platinum, gold and silver price log-returns. Our primary goal is to compare GLD against GPD, and GEVD, in the estimation of Value-at-Risk (VaR) and expected shortfall (ES) as per the international Basel regulatory framework. Our analyses show that GPD and GLD generally outperform GEVD for VaR and ES estimation for negative precious metal returns. For gold, the GPD stands out as the most suitable model. For platinum, GPD and GLD are equally adequate, especially at the 1% VaR level. For silver, GLD is the most suitable at 1% VaR level, whereas GPD is the best model at 0.1%. This study has shown that GLD is a suitable model for extreme risk in precious metal prices and can be used for the estimation of VaR and ES values. Keywords: Expected shortfall, Generalised extreme value, Generalised lambda, Generalised Pareto, Precious metal returns, Value-at-Risk

Probabilistic power flow computation using nested point estimate method

The probabilistic power flow (PPF) computation involves quantifying and propagating uncertainty over hundreds of variables. Zhao's point estimate method (PEM) has been widely used for PPF computation, and the computational burden increases linearly with respect to the number of PPF inputs. In this paper, Zhao's PEM is reformulated as the Kronecker product of Gauss-Hermite quadrature and an identity matrix. Following this formulation, a new PEM termed Nested PEM is developed based on Hadamard matrix, which can reduce the computational burden nearly by half relative to Zhao's PEM. In order to consider the dependence among PPF inputs, the extended generalized lambda distribution (EGLD) is employed to model marginal distributions of correlated PPF inputs, and the t-copula is introduced to map PPF problem to the independent standard normal space. Finally, case studies are conducted on a modified IEEE 118-bus system to illustrate the proposed PEM.

Value at Risk estimation using GAS models with heavy tailed distributions for cryptocurrencies

Risk management and prediction of market losses of cryptocurrencies are of notable value to risk managers, portfolio managers, financial market researchers and academics. One of the most common measures of an asset’s risk is Value-at-Risk (VaR). This paper evaluates and compares the performance of generalized autoregressive score (GAS) combined with heavy-tailed distributions, in estimating the VaR of two well-known cryptocurrencies’ returns, namely Bitcoin returns and Ethereum returns. In this paper, we proposed a VaR model for Bitcoin and Ethereum returns, namely the GAS model combined with the generalized lambda distribution (GLD), referred to as the GAS-GLD model. The relative performance of the GAS-GLD models was compared to the models proposed by Troster et al. (2018), in other words, GAS models combined with asymmetric Laplace distribution (ALD), the asymmetric Student’s t-distribution (AST) and the skew Student’s t-distribution (SSTD). The Kupiec likelihood ratio test was used to assess the adequacy of the proposed models. The principal findings suggest that the GAS models with heavy-tailed innovation distributions are, in fact, appropriate for modelling cryptocurrency returns, with the GAS-GLD being the most adequate for the Bitcoin returns at various VaR levels, and both GAS-SSTD, GAS-ALD and GAS-GLD models being the most appropriate for the Ethereum returns at the VaR levels used in this study.

An application of Multiple Imputation under the Two Generalized Parametric Families

Multiple imputation under the multivariate normality assumption has often been regarded as a viable model-based approach in dealing with incomplete continuous data. Considering the fact that real data rarely conform with normality, there has been a growing attention to generalized classes of distributions that cover a broader range of skewness and elongation behavior compared to the normal distribution. In this regard, two recent works have shown that creating imputations under Fleishman’s power polynomials and the generalized lambda distribution may be a promising tool. In this article, essential distributional characteristics of these families are illustrated along with a description of how they can be used to create multiply imputed data sets. Furthermore, an application is presented using a data example from psychiatric research. Multiple imputation under these families that span most of the feasible area in the symmetry-peakedness plane appears to have substantial potential of capturing real missing-data trends that can be encountered in clinical practice.

Gaussian quadrature method for GLD parameter estimation

Generalized lambda distribution (GLD) is a very flexible distribution that has many applications in estimating the distribution of real data. Its flexibility is due to its λs parameters that each of them somehow determine the form of distribution. Since the shape of distribution is very sensitive to the changes of each λs , the smallest error in estimating each parameter results in an error in other parameters, and then resulting in a total error in estimating the data distribution. One of the most accurate and widely used methods for estimating GLD parameters is the momentum method, which requires integral calculations. This paper uses Gaussian quadrature to integrate GLD equations. The results show that in terms of both percentile index and momentum comparison, the new method leads to more accurate calculation of the GLD parameters.

Reliability-Aware Statistical BSIM Compact Model Parameter Generation Methodology

This article presents an accurate, reliability-aware statistical Berkeley short-channel IGFET Model 4 (BSIM4) compact model parameter generation methodology using the generalized lambda distribution (GLD) method. Using this methodology, compact model parameter sets can be generated “on the fly” beyond the limitation of the number of compact models extracted from the TCAD simulation. The generated unlimited parameter sets enable circuit designer for the statistical circuit simulation. An analytical model has been developed to interpolate TCAD simulation data points, which enables statistical compact model parameter sets to be generated at any aging level. The capability to generate such intermediate aging model parameters at trap densities that were not physically simulated has important application in statistical circuit simulation, opening up the possibility to include accurately reliability assessment in circuit design. An aging model that can transfer trap density to stress/aging time is an integral part of the presented methodology. The accuracy of the compact model parameter generation methodology is validated by comparing the new generated compact model parameter sets at an interpolated trap density, against physical “atomistic” 3-D TCAD simulation. The compact model parameter generation methodology enables the accurate investigation of the influence of statistical variability and bias temperature instability (BTI)-induced aging at circuit level.

An efficient estimator of the parameters of the generalized lambda distribution

Estimation of the four generalized lambda distribution parameters is not straightforward, and available estimators that perform best have large computation times. In this paper, we introduce a simple two-step estimator of the parameters that is comparatively very quick to compute and performs well when compared with other methods. This computational efficiency makes the use of bootstrapping to obtain interval estimators for the parameters possible. Simulations are used to assess the performance of the new estimators and applications to several data sets are included.

Modeling the joint distribution of firm size and firm age based on grouped data.

The firm size distribution is highly skewed to the right and often follows a power law. In practice, it is common that firm size and firm age data are aggregated and released as grouped data to avoid disclosure of confidential information. We investigate multiple parametric methods for firm size and firm age modeling based on grouped data, and propose to estimate the joint distribution of firm size and firm age using the Plackett copula. The goodness-of-fit of the estimated marginal distributions are benchmarked with respect to the fit to the whole data and to the upper tails, respectively. The utilization of the proposed methods are demonstrated via an application to the 1977-2014 US firm data. Results show that the generalized lambda distribution has overall better performance in modeling both firm size and firm age data. The exponentiated Weibull distribution also works well in modeling the firm size data. As a by-product, the estimated parameter of the Plackett copula provides a measure of the association between firm size and firm age.

Two Stages Fitting Techniques using Generalized Lambda Distribution: Application on Malaysian Financial Return

The underline distribution assumption used in the analysis of share market returns is crucial in risk management. An important aspect related to stock return modelling is to obtain accurate prediction. This paper presents an innovative fitting method called two stages (TS) method for modelling daily stock returns. The proposed approach by first establishing trend in the series, and then separately performing L-moment estimation on the generalized lambda distribution (GLD) parameter. The performance of the TS-GLD models had been evaluated using Monte Carlo simulation and Malaysian Kuala Lumpur Composite Index (KLCI) returns from year 2001 to 2015. Based on k-sample Anderson darling goodness of fit test, the two stages GLD model in location parameter (GLD.1) performed well in all studied cases. The GLD.1 model benefits risk management by providing effective distribution fitting.

The distribution of extreme share return in different Malaysian economic circumstances

This study evaluated the performance of probability distribution in various financial periods by investigating the effect of economic cycle on extreme stock return activity. Malaysian stock price KLCI data from 1994–2008 were split into three economy periods correspond to the growth, financial crisis, and the recovery. Four prevalent distributions specifically generalized lambda distribution (GLD), generalized extreme value (GEV), generalized logistic (GLO), and generalized pareto (GPA) were employed to model weekly and monthly maximum and minimum Kuala Lumpur Composite Index (KLCI) share returns. The L-moment approach was used to estimate the parameter while k-sample Anderson darling (k-ad) test was applied to measure the goodness of fit estimation. In conclusion, GLD is the most appropriate distribution representing a weekly maximum minimum return for overall three economic scenarios in Malaysia.

REPLICATION-BASED EMULATION OF THE RESPONSE DISTRIBUTION OF STOCHASTIC SIMULATORS USING GENERALIZED LAMBDA DISTRIBUTIONS

Due to limited computational power, performing uncertainty quantification analyses with complex computational models can be a challenging task. This is exacerbated in the context of stochastic simulators, the response of which to a given set of input parameters, rather than being a deterministic value, is a random variable with unknown probability density function (PDF). Of interest in this paper is the construction of a surrogate that can accurately predict this response PDF for any input parameters. We suggest using a flexible distribution family -- the generalized lambda distribution -- to approximate the response PDF. The associated distribution parameters are cast as functions of input parameters and represented by sparse polynomial chaos expansions. To build such a surrogate model, we propose an approach based on a local inference of the response PDF at each point of the experimental design based on replicated model evaluations. Two versions of this framework are proposed and compared on analytical examples and case studies.

Modelling return behaviour of global real estate investment trusts equities

PurposeThis paper aims to contribute to knowledge by investigating the return behaviour of seven global real estate investment trusts (REITs) with respect to the appropriate distributional fit that captures tail and shape characteristics. The study adds to the knowledge of distributional properties of seven global REITs by using the generalised lambda distribution (GLD), which captures fairly well the higher moments of the returns.Design/methodology/approachThis is an empirical study with GLD through three rival methods of fitting tail and shape properties of seven REIT return data from January 2008 to November 2017. A post-Global Financial Crisis (GFC) (from July 2009) period fits from the same methods are juxtaposed for comparison.FindingsThe maximum likelihood estimates outperform the methods of moment matching and quantile matching in terms of goodness-of-fit in line with extant literature; for the post-GFC period as against the full-sample period. All three methods fit better in full-sample period than post-GFC period for all seven countries for the Region 4 support dynamics. Further, USA and Singapore possess the strongest and stronger infinite supports for both time regimes.Research limitations/implicationsThe REITs markets, however, developed, are of wide varied sizes. This makes comparison less than ideal. This is mitigated by a univariate analysis rather than multivariate one.Practical implicationsThis paper is a reminder of the inadequacy of the normal distribution, as well as the mean, variance, skewness and kurtosis measures, in describing distributions of asset returns. Investors and policymakers may look at the location and scale of GLD for decision-making about REITs.Originality/valueThe novelty of this work lies with the data used and the detailed analysis and for the post-GFC sample.

High Model Uncertainty and Approximating Combined Distributions

Some frameworks of model uncertainty combine a large number of densities. Computing the quantiles of the combined distribution might be practically infeasible when the number of densities is large. We introduce a numerical procedure that can reduce the computational burden. This consists of proving a sequential update formula that estimates different moments of the combined distribution and a finite moment problem that reconstructs the distribution from the moments. Based on an example on the determinants of long-run economic growth, we show that tracking the first four moments of the combined distribution along with generalized lambda distribution fit, provides a good approximation for the overall shape of the distributions.

A Runs Test for the Hypothesis of Symmetry with one Sided Alternative

We propose a trimmed runs test for the hypothesis of symmetry with one sided alternative, in samples coming from the Generalized Lambda Distribution (GLD) with unknown median. We provide a method to calculate the exact distribution, showing that it is symmetric around zero and we give arguments to justify the approximation by means of the normal distribution. The size of the proposed test is calibrated with four symmetrical cases of the GLD and the empirical power is compared with that of some other tests for the same hypothesis, using eight asymmetrical cases of the GLD. The results show that the proposed test is unbiased in the cases used for calibration, and that the empirical power of the proposed test overtakes the empirical power of all compared tests, excepting one of them in two specific cases. Some hints are given concerning how to optimize the empirical power according to the size of the tails of the sampled distributions.

A simple and efficient method for finding the closest generalized lambda distribution to a specific model

AbstractThe four-parameter Generalized Lambda distribution (GLD) can be used to approximate many probability distributions. We present a simple and efficient two-stage process for finding optimal G...