Estimating the distribution of a stochastic sum of IID random variables

2020 ◽  
Vol 70 (3) ◽  
pp. 759-774
Author(s):  
Viktor Witkovský ◽  
Gejza Wimmer ◽  
Tomas Duby
Keyword(s):  
Probability Distribution ◽  
Heavy Tails ◽  
Expert Knowledge ◽  
Parametric Method ◽  
Estimation Method ◽  
Random Variables ◽  
Parametric Estimation ◽  
Empirical Characteristic Function ◽  
Parametric Modelling ◽  
Non Parametric

AbstractSuggested is a non-parametric method and algorithm for estimating the probability distribution of a stochastic sum of independent identically distributed continuous random variables, based on combining and numerically inverting the associated empirical characteristic function (CF) derived from the observed data. This is motivated by classical problems in financial risk management, actuarial science, and hydrological modelling. This approach can be naturally generalized to more complex semi-parametric modelling and estimating approaches, e.g., by incorporating the generalized Pareto distribution fit for modelling heavy tails of the considered continuous random variables, or by considering the weighted mixture of the parametric CFs (used to incorporate the expert knowledge) and the empirical CFs (used to incorporate the knowledge based on the observed or historical data). The suggested numerical approach is based on combination of the Gil-Pelaez inversion formulae for deriving the probability distribution (PDF and CDF) from the associated CF and the trapezoidal quadrature rule used for the required numerical integration. The presented non-parametric estimation method is related to the bootstrap estimation approach, and thus, it shares similar properties. Applicability of the proposed estimation procedure is illustrated by estimating the aggregate loss distribution from the well-known Danish fire losses data.

scholarly journals Parameter estimation in a Black Scholes

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 117-122
Author(s):  
Mustafa Bayram ◽  
Buyukoz Orucova ◽  
Tugcem Partal
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Parametric Method ◽  
Estimation Method ◽  
Parametric Estimation ◽  
Likelihood Method ◽  
Observation Data ◽  
Unknown Parameters ◽  
Black Scholes ◽  
Non Parametric

In this paper we discuss parameter estimation in black scholes model. A non-parametric estimation method and well known maximum likelihood estimator are considered. Our aim is to estimate the unknown parameters for stochastic differential equation with discrete time observation data. In simulation study we compare the non-parametric method with maximum likelihood method using stochastic numerical scheme named with Euler Maruyama.

Which One is More Important Factor for Pricing Options, Skewness or Kurtosis?

2006 ◽  
Vol 14 (2) ◽  
pp. 25-50
Author(s):  
Sol Kim
Keyword(s):  
Parametric Method ◽  
Estimation Method ◽  
Relative Importance ◽  
Pricing Options ◽  
Pricing Errors ◽  
Risk Neutral ◽  
Definition Of ◽  
Skewness And Kurtosis ◽  
Non Parametric

This paper investigates the relative importance of the skewness and kurtosis of the risk neutral distribution for pricing KOSPI200 options. The skewness and kurtosis are estimated from non parametric method of Bakshi, Kapadia, and Madan (2003) and the parametric method of Corrado and Su (1996). We show that the skewness of the risk neutral distribution is more important factor than the kurtosis irrespective of the estimation method, the definition of pricing errors, the moneyness, the type of options and a period of time.

Parametric estimation of phase and group slownesses from sonic logging waveforms

Geophysics ◽  
1992 ◽  
Vol 57 (8) ◽  
pp. 978-985 ◽  
Author(s):  
Kai Hsu ◽  
Cengiz Esmersoy
Keyword(s):  
Parametric Method ◽  
Estimation Method ◽  
Parametric Estimation ◽  
Flexural Wave ◽  
Smooth Functions ◽  
Dispersive Waves ◽  
Prony Method ◽  
Receiver Array ◽  
Frequency Independent ◽  
Sonic Logging

Sonic logging waveforms consist of a mixture of nondispersive waves, such as the P‐ and S‐headwaves, and dispersive waves, such as the Stoneley and pseudo‐Rayleigh waves in monopole logging and the flexural wave in dipole logging. Conventionally, slowness dispersion curves of various waves are estimated at each frequency, independent of data at other frequencies. This approach does not account for the fact that slowness dispersion functions in sonic logging are continuous and, in most cases, smooth functions of frequency. We describe a parametric slowness estimation method that uses this property by locally approximating the wavenumber of each wave as a linear function of frequency. This provides a parametric model for the phase and group slownesses of the waves propagating across the receiver array. The estimation of phase and group slownesses is then carried out by minimizing the squared difference between the predicted and observed waveforms. The minimization problem is nonlinear and is solved by an iterative algorithm. Examples using synthetic and field data are shown and the results are compared with those obtained by the conventional Prony method. Based on the comparison, we conclude that the parametric method is better than the conventional Prony method in providing robust and stable slowness estimates.

Non-parametric estimation of copula parameters: testing for time-varying correlation

2015 ◽  
Vol 19 (1) ◽  
Author(s):  
Jinguo Gong ◽  
Weiou Wu ◽  
David McMillan ◽  
Daimin Shi
Keyword(s):  
Likelihood Function ◽  
Empirical Distribution ◽  
Estimation Method ◽  
Statistical Test ◽  
Parametric Estimation ◽  
Likelihood Method ◽  
Time Varying ◽  
Copula Parameter ◽  
Non Parametric Estimation ◽  
Non Parametric

AbstractThe correlation structure of financial assets is a key input with regard to portfolio and risk management. In this paper, we propose a non-parametric estimation method for the time-varying copula parameter. This is achieved in two steps: first, displaying the marginal distributions of financial asset returns by applying the empirical distribution function; second, by implementing the local likelihood method to estimate the copula parameters. The method for obtaining the optimal bandwidth through a maximum pseudo likelihood function and a statistical test on whether the copula parameter is time-varying are also introduced. A simulation study is conducted to show that our method is superior to its contender. Finally, we verify the proposed estimation methodology and time-varying statistical test by analysing the dynamic linkages between the Shanghai, Shenzhen and Hong Kong stock markets.

A note on the Probability that first component fails before the second component

2016 ◽  
Vol 5 (2) ◽  
Author(s):  
Ehtesham Husain ◽  
Masood ul Haq
Keyword(s):  
Probability Distribution ◽  
Failure Time ◽  
Random Variables ◽  
Life Time ◽  
Life Testing ◽  
Human Organs

<p><span>The reliability (unreliability) and life testing are important topics in the field of engineering, electronic, <span>medicine, economic and many more, where we are interested in, life of components, human organs, <span>subsystem and system. Statistically, a probability distribution failure time (life time) of a certain form is <span>usually assumed to give reliability of a component for a system for each time t. Some well known <span>parametric life time models (T ≥ 0) are Exponential, Weibull, Inverse Weibull, Gamma, Lognormal, <span>normal ( T&gt;0 ; left truncated ) etc. </span></span></span></span></span></span></p><p><span><span><span><span><span><span><span>In this paper we consider a system that, has two components with independent but non-identical life time <span>probabilities explained by two distinct random variables say T<span>1 <span>and T<span>2 <span>, where T<span>1 <span>has a constant hazard <span>rate and T<span>2 <span>has an increasing hazard respectively </span></span></span></span></span></span></span></span><br /><br class="Apple-interchange-newline" /></span></span></span></span></span></span></span></span></span></p>

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