Search Results (16)

To address the low-voltage violations that may occur in distribution networks with high penetration of distributed photovoltaic (PV) during sunset and evening peak periods under extreme high-temperature conditions, this paper establishes a source–grid–load electro-thermal coupling model that accounts for load thermal accumulation, transient conductor thermal inertia, temperature-dependent line impedance, and PV thermal derating. Based on a soft safety lower bound and a risk-preference utility function, the probability of voltage violation, violation depth, and expected violation duration are introduced to construct node-level and system-level comprehensive risk factors. The cumulant method combined with the Cornish–Fisher expansion is used to reconstruct the probability distribution of nodal voltages, enabling analytical risk calculation. Simulation results on the IEEE 33-bus system at 45 °C show that the proposed method can quantitatively reflect the temporal variations of nodal voltage distributions, physical violation depth, dimensionless severity utility, and expected violation duration, and identify weak nodes in the later part of the evening peak, providing a reference for risk early warning in distribution networks under extreme heat. Full article

Suppose that we have a statistical model with q unknown parameters w, and an estimate $\widehat{w}$, based on a sample of size n. A basic question is: what is the covariance of the estimate? The covariance is needed for the Central Limit Theorem (CLT). This gives a first approximation for the distribution of $\widehat{w}$. But what if $q_n=n$ increases with n? How fast can it increase and the CLT still hold? An answer has so far only been given for the sample mean. The same is true for the Edgeworth expansions. These are expansions in powers of $n^{-1/2}$ for the density and distribution of $\widehat{w}$. For fixed q, these expansions are important, as they show how small n can be for the CLT to apply. When it does, they can greatly improve the accuracy of the CLT. I give conditions that allow for the Edgeworth expansions to remain valid when $q_n=q$ increases with n. Earlier Edgeworth expansions when $q_n=q$ increases, have only been done for a sample mean, and only for a 2nd order Edgeworth expansion. In contrast, I consider a very large class of estimates, the class of non-lattice standard estimates. An estimate is said to be a standard estimate if its mean converges to its true value as n increases, and for $r\ge 1$, its rth order cumulants have magnitude $n^{1-r}$ and can be expanded in powers of $n^{-1}$. For this class of estimates, I show that the Edgeworth expansions hold if $q_n$ grows as a power of n less than $1/6.$ That is, I give these expansions in powers of $n^{-1/2}{q}_n^3$. This large class of estimates has a huge range of potential applications, as estimates of high dimension are common in nearly all areas of applied statistics. The most important type of standard estimate is when $\widehat{w}$ is a smooth function of a sample mean, of dimension p say. When either or both $q_n=q$ and $p_n=p$ increase with n, I give conditions on their growth for the Edgeworth expansions for $\widehat{w}$ to remain valid: the eighth power of p times the sixth power of q cannot grow as fast as n. This holds for fixed $q=q_n$ if $p_n$ grows less than a power of n less than $1/8$. This appears to be the first time when Edgeworth expansions have been given when not one, but two dimensions, are allowed to increase to ∞ with n. This gives two different pathways for allowing an increase in dimensionality. When $q=1$, I give 5th order Edgeworth-Cornish-Fisher expansions for the standardized distribution and its quantiles of any smooth function of a sample mean of dimension $p_n$, when $p_n$ is a power of n less than $1/2$. However for the special case when this function is linear, there is no restriction whatever on how fast $p_n$ can increase! If also the components of the sample mean are independent, then these expansions are in powers of ${\left(np\right)}^{-1/2}$. I also give a method that greatly reduces the number of terms needed for the 2nd and 3rd order terms in the Edgeworth expansions, that is, for the 1st and 2nd order corrections to the CLTs. I also extend these results to the case where $\widehat{w}\in R^q$ is a function of several independent sample means, each of dimension increasing with n, with total dimension p. Full article

This paper gives expansions for the distribution, density and quantiles of any estimate that is a smooth function of the sample cross-moments of a stationary process. Three versions of these are given, depending on whether an exact, approximate, or asymptotic form is used for the variance or covariance of the estimate. Eight examples are provided, including sample autocovariances and autocorrelations. Their Central Limit Theorems extend those in the literature, such as Bartlett’s formula, by allowing for the effect of the mean and higher order cross-cumulants. Their distribution and quantiles are given to magnitude $n^{-r/2}$ up to r = 3, where n is the sample size. Full article

Conditioning is a very useful way of using correlated information to reduce the variability of an estimate. Conditioning an estimate on a correlated estimate, reduces its covariance, and so provides more precise inference than using an unconditioned estimate. Here we give expansions in powers of $n^{-1/2}$ for the conditional density and distribution of any multivariate standard estimate based on a sample of size n. Standard estimates include most estimates of interest, including smooth functions of sample means and other empirical estimates. We also show that a conditional estimate is not a standard estimate, so that Edgeworth-Cornish-Fisher expansions cannot be applied directly. Full article

This paper considers the problem of constructing confidence intervals (CIs) for nonlinear functions of parameters, particularly ratios of parameters a common issue in econometrics and statistics. Classical CIs (such as the Delta method and the Fieller method) often fail in small samples due to biased parameter estimators and skewed distributions. We extended the Delta method using the Edgeworth expansion to correct for skewness due to estimated parameters having non-normal and asymmetric distributions. The resulting bias-corrected confidence intervals are easy to compute and have a good coverage probability that converges to the nominal level at a rate of $O(n^{-1/2})$ where n is the sample size. We also propose bias-corrected estimators based on second-order Taylor expansions, aligning with the “almost unbiased ratio estimator” . We then correct the CIs according to the Delta method and the Edgeworth expansion. Thus, our new methods for constructing confidence intervals account for both the bias and the skewness of the distribution of the nonlinear functions of parameters. We conduct a simulation study to compare the confidence intervals of our new methods with the two classical methods. The methods evaluated include Fieller’s interval, Delta with and without the bias correction interval, and Edgeworth expansion with and without the bias correction interval. The results show that our new methods with bias correction generally have good performance in terms of controlling the coverage probabilities and average length intervals. They should be recommended for constructing confidence intervals for nonlinear functions of estimated parameters. Full article

Estimating skewness and kurtosis from real-world data remains a long-standing challenge in actuarial science and financial risk management, where these higher-order moments are critical for capturing asymmetry and tail risk. Traditional moment-based estimators are known to be highly sensitive to outliers and often fail when the assumption of normality is violated. Despite numerous extensions—from robust moment-based methods to quantile-based measures—being proposed over the decades, no universally satisfactory solution has been reported, and many existing methods exhibit limited effectiveness, particularly under challenging distributional shapes. In this paper we propose a novel method that jointly estimates skewness and kurtosis based on a regression adaptation of the Cornish–Fisher expansion. By modeling the empirical quantiles as a cubic polynomial of the standard normal variable, the proposed approach produces a reliable and efficient estimator that better captures distributional shape without strong parametric assumptions. Our comprehensive simulation studies show that the proposed method performs much better than existing estimators across a wide range of distributions, especially when the data are skewed or heavy-tailed, as is typical in actuarial and financial applications. Full article

Edgeworth–Cornish–Fisher expansions are hugely important, as they give the distribution, density and quantiles of any standard estimate. Here we show that the sample mean of a univariate or multivariate stationary process is a standard estimate, so that all the known results for standard estimates can be applied. We also show how to allow for missing data and weighted means. Full article

With the advent of large-scale electronic transportation, the construction of electric vehicle charging stations (EVCSs) has increased. The stochastic characteristic of the charging power of EVCSs leads to a risk of destabilization of the DC distribution network when there is a high degree of power electronification. Current deterministic stability analysis methods are too complicated to allow for brief descriptions of the effect of probabilistic characteristics of EVCSs on stability. This paper develops a probabilistic small-signal stability analysis method. Firstly, the probabilistic information of the system is obtained by combining the s-domain nodal impedance matrix based on the point estimation method. Then, the probability function of stability is fitted using the Cornish–Fisher expansion method. Finally, a comparison experiment using Monte Carlo simulation demonstrates that this method performs well in balancing accuracy and computational efficiency. The effects of line parameters and system control parameters on stability are investigated in the framework of probabilistic stability. This will provide a probabilistic perspective on the design of more complex power systems in the future. Full article

Recent trends in portfolio management emphasize the importance of reducing carbon footprints and aligning investments with sustainable practices. This paper introduces Sensitivity Value-at-Risk (SensitivityVaR), an advanced distortion risk measure that combines Value-at-Risk (VaR) and Expected Shortfall (ES) with the Cornish–Fisher expansion. SensitivityVaR provides a more robust framework for managing risk, particularly under extreme market conditions. By incorporating first- and second-order distorted stochastic dominance criteria, we enhance portfolio decarbonization strategies, aligning financial objectives with environmental targets such as the Paris Agreement’s goal of a 7% annual reduction in carbon intensity from 2019 to 2050. Our empirical analysis evaluates the impact of integrating carbon intensity data—including Scope 1, Scope 2, and Scope 3 emissions—on portfolio optimization, focusing on key sectors like technology, energy, and consumer goods. The results demonstrate the effectiveness of SensitivityVaR in managing both risk and environmental impact. The methodology led to significant reductions in carbon intensity across different portfolio configurations, while preserving competitive risk-adjusted returns. By optimizing tail risks and limiting exposure to carbon-intensive assets, this approach produced more balanced and efficient portfolios that aligned with both financial and sustainability goals. These findings offer valuable insights for institutional investors and asset managers aiming to integrate climate considerations into their investment strategies without compromising financial performance. Full article

The continuous increase in generation from renewable energy sources, marked by correlated forecast uncertainties, requires specific methodologies to support power system operators in security management. This paper proposes a probabilistic preventive control to ensure N-1 security in presence of correlated uncertainties of renewable sources and loads. By adopting a decoupled linear formulation of the AC load flow equations, the preventive control is decomposed into two subsequent linear programming problems, the former concerning the active power and the latter the voltage/reactive power-related issues. In particular, in the active control problem, the algorithm combines Third Order Polynomial Normal Transformation, Point Estimate Method, and Cornish–Fisher expansion to model the forecast uncertainties and characterize the chance constraints in the problem. The goal is to find the optimal phase shifting transformer tap setting, conventional generation redispatching, and renewable curtailment at the minimum cost to assure the probabilistic fulfillment of N and N-1 security constraints on branch active power flows. The second stage solves another linear programming problem, which aims to minimize the adjustments to generators’ set-point voltages to avoid violations at node voltages and branch-rated limits due to reactive power flows while meeting generator reactive power constraints. Simulations performed on an IEEE test system demonstrate the effectiveness of the proposed security control method in limiting the probability of violating security limits in N and N-1 state, including voltage/reactive power constraints, in presence of correlated uncertainties. Full article

This research examines the correlations between the return volatility of cryptocurrencies, global stock market indices, and the spillover effects of the COVID-19 pandemic. For this purpose, we employed a two-stage multivariate volatility exponential GARCH (EGARCH) model with an integrated dynamic conditional correlation (DCC) approach to measure the impact on the financial portfolio returns from 2019 to 2020. Moreover, we used value-at-risk (VaR) and value-at-risk measurements based on the Cornish–Fisher expansion (CFVaR). The empirical results show significant long- and short-term spillover effects. The two-stage multivariate EGARCH model’s results show that the conditional volatilities of both asset portfolios surge more after positive news and respond well to previous shocks. As a result, financial assets have low unconditional volatility and the lowest risk when there are no external interruptions. Despite the financial assets’ sensitivity to shocks, they exhibit some resistance to fluctuations in market confidence. The VaR performance comparison results with the assets portfolios differ. During the COVID-19 outbreak, the Dow (DJI) index reports VaR’s highest loss, followed by the S&P500. Conversely, the CFVaR reports negative risk results for the entire cryptocurrency portfolio during the pandemic, except for the Ethereum (ETH). Full article

We assess Value-at-Risk (VaR) and Expected Shortfall (ES) estimates assuming different models for the standardized returns: distributions based on polynomial expansions such as Cornish-Fisher and Gram-Charlier, and well-known parametric densities such as normal, skewed-t and Johnson. This paper aims to analyze whether models based on polynomial expansions outperform the parametric ones. We carry out the model performance comparison in two stages: first, with a backtesting analysis of VaR and ES; and second, using loss functions. Our backtesting results show that all distributions, except for normal ones, perform quite well in VaR and ES estimations. Regarding the loss function analysis, we conclude that polynomial expansions (specifically, the Cornish-Fisher one) usually outperform parametric densities in VaR estimation, but the latter (specifically, the Johnson density) slightly outperform the former in ES estimation; however, the gains of using one approach or the other are modest. Full article

In this paper, we propose control limits for monitoring the mean of a process variable based on a first and second order Cornish–Fisher expansion, which limits are inclusive of its skewness and kurtosis measures, respectively. These are shown to have better in-control error performance than other limits that were similarly derived from this expansion with smoothing functions, both when these measures are assumed to be known and estimated from sample data. The range of measure specifications where the underlying Cornish–Fisher function is monotonic is derived. Operating characteristic curves for select cases demonstrate the associated out-of-control error performance. The Cornish–Fisher limits are applied to a real-life dataset in developing a control chart for monitoring the mean lifetime of car brake pads, wherein they are compared to other limit approximations. Full article

In this paper, a comprehensive real-time dispatch model considering renewable uncertainty based on dynamic load flow (DLF) is proposed. Through DLF, the primary and secondary frequency regulation amount caused by the variation of renewable energy as well as the line flow when primary and secondary regulation are deployed can be obtained easily. Not only the frequency constraints, but also the regular constraints like generator production limits and line flow limits are respected under both primary and secondary frequency regulation. To solve the dispatch problem with renewable uncertainty, chance-constrained programming based on cumulants and Cornish-fisher expansions (CCP-CMCF) is adopted to get the probability of holding the chance constraints and then the real-time dispatch model can be transformed into a quadratic programming. The simulation results show that the dispatch model proposed in this paper can deal with both primary and secondary regulation well and has a fast computation speed. Full article

The traditional cumulant method (CM) for probabilistic optimal power flow (P-OPF) needs to perform linearization on the Karush–Kuhn–Tucker (KKT) first-order conditions, therefore requiring input variables (wind power or loads) varying within small ranges. To handle large fluctuations resulting from large-scale wind power and loads, a novel P-OPF method is proposed, where the correlations among input variables are also taken into account. Firstly, the inverse Nataf transformation and Cholesky decomposition are used to obtain samples of wind speeds and loads with a given correlation matrix. Then, the K-means algorithm is introduced to group the samples of wind power outputs and loads into a number of clusters, so that in each cluster samples of stochastic variables have small variances. In each cluster, the CM for P-OPF is conducted to obtain the cumulants of system variables. According to these cumulants, the moments of system variables corresponding to each cluster are computed. The moments of system variables for the total samples are obtained by combining the moments for all grouped clusters through the total probability formula. Then, the moments for the total samples are used to calculate the corresponding cumulants. Finally, Cornish–Fisher expansion is introduced to obtain the probability density functions (PDFs) of system variables. IEEE 9-bus and 118-bus test systems are modified to examine the proposed method. Study results show that the proposed method can produce more accurate results than traditional CM for P-OPF and is more efficient than Monte Carlo simulation (MCS). Full article