Accurate Efficient Evaluation of Cumulative or Exceedance Probability Distributions Directly from Characteristic Functions

1983 ◽  
Author(s):  
Albert H. Nuttall
Keyword(s):  
Probability Distributions ◽  
Characteristic Functions ◽  
Exceedance Probability

MATHEMATICAL MODELLING OF FORECASTING THE RESULTS OF KNOWLEDGE TESTING / ŽINIŲ TIKRINIMO REZULTATŲ PROGNOZĖS MATEMATINIS MODELIAVIMAS

2008 ◽  
Vol 14 (3) ◽  
pp. 388-401 ◽  
Author(s):  
Aleksandras Krylovas ◽  
Natalja Kosareva
Keyword(s):  
Item Response ◽  
Logistic Model ◽  
Probability Distributions ◽  
Characteristic Functions ◽  
Test Results ◽  
Test Items ◽  
Mathematics Test ◽  
Level Method ◽  
Selection For ◽  
Selection Of

In this paper a mathematical model for obtaining probability distribution of the knowledge testing results is proposed. Differences and similarities of this model and Item Response Theory (IRT) logistic model are discussed. Probability distributions of 10 items test results for low, middle and high ability populations selecting characteristic functions of the various difficulty items combinations are obtained. Entropy function values for these items combinations are counted. These results enable to formulate recomendations for test items selection for various testing groups according to their attainment level. Method of selection of a suitable item characteristic function based on the Kolmogorov compatibility test, is proposed. This method is illustrated by applying it to a discreet mathematics test item. Santrauka Straipsnyje pasiūlytas matematinis modelis žinių tikrinimo rezultatų tikimybiniam skirstiniui gauti. Aptarti šio modelio ir užduočių sprendimo teorijos (IRT) logistinio modelio skirtumai ir panašumai. Išnagrinėti 10 klausimų testo rezultatų tikimybiniai skirstiniai silpnai, vidutinei ir stipriai testuojamųjų populiacijoms parenkant įvairias testo klausimų sunkumo funkcijų kombinacijas. Apskaičiuotos entropijos funkcijos reikšmės. Gauti rezultatai leidžia formuluoti rekomendacijas testo klausimams parinkti skirtingoms testuojamųjų grupėms pagal jų žinių lygį. Pasiūlytas tinkamiausios klausimo charakteristinės funkcijos parinkimo būdas, grindžiamas Kolmogorovo kriterijumi. Ši procedūra iliustruojama taikant ją konkrečiam diskrečiosios matematikos testo klausimui.

A Method Based on Taking the Average of Probabilities to Compute the Flow Duration Curve

2000 ◽  
Vol 31 (3) ◽  
pp. 187-206 ◽  
Author(s):  
Hikmet Kerem Cigizoglu
Keyword(s):  
Probability Distributions ◽  
Maximum Flow ◽  
Exceedance Probability ◽  
Flow Duration Curve ◽  
Marginal Probability ◽  
Lognormal Distributions ◽  
Type Iii ◽  
Flow Duration ◽  
Pearson Type ◽  
Flow Duration Curves

In this study a method based on taking the average of the probabilities is presented to obtain flow duration curve. In this method the exceedance probability for each flow value is computed repeatedly for all time periods within a year. The final representing exceedance is just simply the average of all these probabilities. The applicability of the method to daily mean flows is tested assuming various marginal probability distributions like normal, Pearson type III, log-Pearson type III, 2-parameter lognormal and 3-parameter lognormal distributions. It is seen that the observed flow duration curves were quite well approximated by the 2-parameter lognormal average of probabilities curves. In that case the method requires the computation of the daily mean and standard deviation values of the observed flow data. The method curve enables extrapolation of the available data providing the exceedance probabilities for the flows higher than the observed maximum flow. The method is applied to the missing data and ungauged site problems and the results are quite satisfactory.

Estimating IDF-Relations consistently using a duration-dependent GEV with spatial covariates

2020 ◽  
Author(s):  
Jana Ulrich ◽  
Madlen Peter ◽  
Oscar E. Jurado ◽  
Henning W. Rust
Keyword(s):  
Extreme Precipitation ◽  
Probability Distributions ◽  
Skill Score ◽  
Exceedance Probability ◽  
Ungauged Sites ◽  
Extreme Precipitation Events ◽  
One Step ◽  
Idf Curves ◽  
Intensity Duration Frequency ◽  
A Site

<p>Intensity-Duration-Frequency (IDF) Curves are a popular tool in Hydrology for estimating the properties of extreme precipitation events. They describe the relationship between rainfall intensity and duration for a given non-exceedance probability (or frequency). For a site where precipitation measurements are available, these curves can be estimated consistently over durations using a duration-dependent GEV (d-GEV, after Koutsoyiannis et al. 1998). In this approach, the probability distributions are modeled simultaneously for all durations.</p><p>Additionally, we integrate covariates to describe the spatial variability of the d-GEV parameters so that we can model the distribution of extreme precipitation for a range of durations and locations in one step. Thus IDF Curves can be estimated even at ungauged sites. Further advantages are parameter reduction and more efficient use of the available data. We use the Quantile Skill Score to investigate under which conditions this method leads to an improved estimate compared to the single-site approach and to evaluate the performance at ungauged sites.</p>

Certain Fourier Transforms of Distributions

1951 ◽  
Vol 3 ◽  
pp. 140-144 ◽  
Author(s):  
Eugene Lukacs ◽  
Otto Szasz
Keyword(s):  
Probability Distribution ◽  
Characteristic Function ◽  
Fourier Transforms ◽  
Probability Distributions ◽  
Distribution Functions ◽  
Characteristic Functions

Fourier transforms of distribution functions are frequently studied in the theory of probability. In this connection they are called characteristic functions of probability distributions. It is often of interest to decide whether a given function φ(t) can be the characteristic function of a probability distribution, that is, whether it admits the representation

scholarly journals SINH-ACCELERATION: EFFICIENT EVALUATION OF PROBABILITY DISTRIBUTIONS, OPTION PRICING, AND MONTE CARLO SIMULATIONS

2019 ◽  
Vol 22 (03) ◽  
pp. 1950011 ◽  
Author(s):  
SVETLANA BOYARCHENKO ◽  
SERGEI LEVENDORSKIĬ
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Regime Switching ◽  
Probability Distributions ◽  
General Scheme ◽  
Characteristic Exponent ◽  
Heston Model ◽  
Characteristic Functions ◽  
Quadratic Term ◽  
Quadratic Term Structure

Characteristic functions of several popular classes of distributions and processes admit analytic continuation into unions of strips and open coni around [Formula: see text]. The Fourier transform techniques reduce calculation of probability distributions and option prices in the evaluation of integrals whose integrands are analytic in domains enjoying these properties. In the paper, we suggest to use changes of variables of the form [Formula: see text] and the simplified trapezoid rule to evaluate the integrals accurately and fast. We formulate the general scheme, and apply the scheme for calculation probability distributions and pricing European options in Lévy models, the Heston model, the CIR model, and a Lévy model with the CIR-subordinator. We outline applications to fast and accurate calibration procedures and Monte Carlo simulations in Lévy models, regime switching Lévy models that can account for stochastic drift, volatility and skewness, the Heston model, other affine models and quadratic term structure models. For calculation of quantiles in the tails using the Newton or bisection method, it suffices to precalculate several hundreds of values of the characteristic exponent at points on an appropriate grid (conformal principal components) and use these values in formulas for cpdf and pdf.

scholarly journals FORMULA OF INVERSION FOR RATIONAL CHARACTERISTIC FUNCTIONS OF PROBABILITY DISTRIBUTIONS

2018 ◽  
pp. 7-22
Author(s):  
Georgiy Aleksandrovich Popov
Keyword(s):  
Distribution Function ◽  
Probability Distributions ◽  
Rational Functions ◽  
Distribution Functions ◽  
Characteristic Functions ◽  
Arbitrary Distribution ◽  
Abscissa Axis ◽  
Entire Axis ◽  
Inversion Formulas ◽  
Gamma Distributions

The paper deals with the problem of clarifying the well-known inversion formulas for distribution functions, usually describing the increment of these functions. The validity of the corresponding inversion formulas for the distribution function π and their densities has been proved for the particular case of distributions with rational characteristic functions. The obtained formulas for distribution functions, which include additionally constant terms equal to 0.5, were not previously known. Functions of positively distributed random variables and quantities distributed over the entire axis have been considered separately. To test the hypothesis of fairness of the obtained treatment formula, including a previously unknown term equal to 0.5, in the general case there have been given examples of calculating distribution functions, whose characteristic functions are not considered as rational functions: for constant and uniform laws. The verification confirmed the objectiveness of the formulated hypothesis about the obtained validity of the inversion form for arbitrary distribution functions. It has also been shown that any distribution function and any density can be represented as a limit of a mixture of gamma distributions (distribution functions and densities), having shifts along the abscissa axis and, possibly, with altered signs of the arguments. The obtained result proves that the set of gamma distributions with shifted arguments is uniformly dense in the set of all distributions.

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