Statistical Moments

Moments by Integrating the Moment-Generating Function ArXiv ID: 2410.23587 “View on arXiv” Authors: Unknown Abstract We introduce a novel method for obtaining a wide variety of moments of any random variable with a well-defined moment-generating function (MGF). We derive new expressions for fractional moments and fractional absolute moments, both central and non-central moments. The expressions are relatively simple integrals that involve the MGF, but do not require its derivatives. We label the new method CMGF because it uses a complex extension of the MGF and can be used to obtain complex moments. We illustrate the new method with three applications where the MGF is available in closed-form, while the corresponding densities and the derivatives of the MGF are either unavailable or very difficult to obtain. ...

Lower Bounds of Uncertainty of Observations of Macroeconomic Variables and Upper Limits on the Accuracy of Their Forecasts ArXiv ID: 2408.04644 “View on arXiv” Authors: Unknown Abstract This paper defines theoretical lower bounds of uncertainty of observations of macroeconomic variables that depend on statistical moments and correlations of random values and volumes of market trades. Any econometric assessments of macroeconomic variables have greater uncertainty. We consider macroeconomic variables as random that depend on random values and volumes of trades. To predict random macroeconomic variables, one should forecast their probabilities. Upper limits on the accuracy of the forecasts of probabilities of macroeconomic variables, prices, and returns depend on the number of predicted statistical moments. We consider economic obstacles that limit by the first two the number of predicted statistical moments. The accuracy of any forecasts of probabilities of random macroeconomic variables, prices, returns, and market trades doesn’t exceed the accuracy of Gaussian approximations. Any forecasts of macroeconomic variables have uncertainty higher than one determined by predictions of coefficients of variation of random values and volumes of trades. ...

Theoretical Economics as Successive Approximations of Statistical Moments ArXiv ID: 2310.05971 “View on arXiv” Authors: Unknown Abstract This paper studies the links between the descriptions of macroeconomic variables and statistical moments of market trade, price, and return. The randomness of market trade values and volumes during the averaging interval Δ results in the random properties of price and return. We describe how averages and volatilities of price and return depend on the averages, volatilities, and correlations of market trade values and volumes. The averages, volatilities, and correlations of market trade, price, and return can behave randomly during the long interval Δ2»Δ. To describe their statistical properties during the long interval Δ2, we introduce the secondary averaging procedure of trade, price, and return. We explain why, in the coming years, predictions of market-based probabilities of price and return will be limited by Gaussian distributions. We discuss the roots of the internal weakness of the commonly used hedging tool, Value-at-Risk, that cannot be solved and remains the source of additional risks and losses. One should consider theoretical economics as a set of successive approximations, each of which describes the next array of the n-th statistical moments of market trades, price, return, and macroeconomic variables, which are repeatedly averaged during the sequence of increasing time intervals. ...