A note on the degree of normal approximation to the distribution function of the mean of samples from finite populations

1964 ◽  
Vol 16 (1) ◽  
pp. 427-430 ◽  
Author(s):  
Yasushi Taga
Keyword(s):  
Distribution Function ◽  
Normal Approximation ◽  
Finite Populations ◽  
The Mean

scholarly journals On The Approximation of the Total Amount of Claims

1977 ◽  
Vol 9 (3) ◽  
pp. 281-289 ◽  
Author(s):  
T. Pentikäinen
Keyword(s):  
Distribution Function ◽  
Standard Deviation ◽  
Normal Approximation ◽  
Mean Value ◽  
Normal Distribution Function ◽  
Acceptable Accuracy ◽  
The Mean ◽  
The Moment ◽  
The Individual ◽  
Image Position

Several “short cut” methods exist to approximate the total amount of claims ( = χ) of an insurance collective. The classical one is the normal approximationwhere and σx are the mean value and standard deviation of x. Φ is the normal distribution function.It is well-known that the normal approximation gives acceptable accuracy only when the volume of risk business is fairly large and the distribution of the amounts of the individual claims is not “too dangerous”, i.e. not too heterogeneous (cf. fig. 2).One way to improve the normal approximation is the so called NP-method, which provides for the standardized variable a correction Δzwhereis the skewness of the distribution F(χ). Another variant (NP3) of the NP-method also makes use of the moment μ4, but, in the following, we limit our discussion mainly to the variant (2) (= NP2).If Δz is small, a simpler formulais available (cf. fig. 2).Another approximation was introduced by Bohman and Esscher (1963). It is based on the incomplete gamma functionwhere Experiments have been made with both formulae (2) and (4); they have been applied to various F functions, from which the exact (or at least controlled) values are otherwise known. It has been proved that the accuracy is satisfactory provided that the distribution F is not very “dangerous”.

scholarly journals On the Dynamics of the Solar Convection Zone

1980 ◽  
Vol 51 ◽  
pp. 15-16
Author(s):  
Bernard R. Durney ◽  
Hendrik C. Spruit
Keyword(s):  
Distribution Function ◽  
Energy Equation ◽  
Convection Zone ◽  
Turbulent Viscosity ◽  
Rotating Stars ◽  
Solar Convection Zone ◽  
Solar Convection ◽  
Convective Motions ◽  
The Mean ◽  
Convective Cells

AbstractWe derive expressions for the turbulent viscosity and turbulent conductivity applicable to convection zones of rotating stars. We assume that the dimensions of the convective cells are known and derive a simple distribution function for the turbulent convective velocities under the influence of rotation. From this distribution function (which includes, in particular, the stabilizing effect of rotation on convection) we calculate in the mixing-length approximation: i) the turbulent Reynolds stresstensor and ii) the expression for the heat flux in terms of the superadiabatic gradient. The contributions of the turbulent convective motions to the mean momentum and energy equation are treated consistently, and assumptions about the turbulent viscosity and heat transport are replaced by assumptions about the turbulent flow itself. The free parameters in our formalism are the relative cell sizes and their dependence on depth and latitude.

Testing for individual sphericity in heterogeneous panels

Biometrika ◽  
2019 ◽  
Vol 106 (3) ◽  
pp. 740-747
Author(s):  
Simon A Broda
Keyword(s):  
Distribution Function ◽  
Normal Approximation ◽  
Empirical Work ◽  
Null Distribution ◽  
Saddlepoint Approximation ◽  
Test Statistic ◽  
Cross Sectional ◽  
P Values ◽  
Cross Sectional Dimension ◽  
Heterogeneous Panels

Summary This manuscript considers locally best invariant tests for sphericity in heterogeneous panels. A new integral representation for the characteristic function of the test statistic under the null is presented, along with an algorithm for inverting it to obtain the distribution function. A saddlepoint approximation to the null distribution addresses the need to quickly compute approximate $p$-values in empirical work. The approximation shows substantial improvements over the normal approximation when the cross-sectional dimension is small.

NOTE ON THE VLASOV EQUATION FOR PLASMAS

1963 ◽  
Vol 41 (12) ◽  
pp. 1960-1966 ◽  
Author(s):  
Ta-You Wu ◽  
M. K. Sundaresan
Keyword(s):  
Distribution Function ◽  
Fourier Transform ◽  
Vlasov Equation ◽  
Hermite Polynomials ◽  
Infinite Matrix ◽  
Mean Square ◽  
Initial Value ◽  
Positive Ions ◽  
The Mean ◽  
The Fourier Transform

The linearized Vlasov equation is solved as an initial value problem by expanding (the Fourier components of) the distribution function in a series of Hermite polynomials in the momentum, with coefficients which are functions of time. The spectrum of frequencies is given by the eigenvalues of an infinite matrix. All the frequencies ω are real, extending from small values of order ω2 = k2(u22), where (u22) is the mean square velocity of the positive ions (of mass M), to [Formula: see text], where ω1, (u12) are the plasma frequency and mean square velocity of the electrons (of mass m). The classic work of Landau solves the Vlasov equation for (the Fourier transform of) the potential for which he obtains the "damping", whereas Van Kampen and the present writers solve the equation for (the Fourier transform of) the distribution function itself. While the present work gives results equivalent to those of Van Kampen, the method is simpler and in fact elementary.

scholarly journals Some Problems Connected with the Calculation of Stop Loss Premiums for Large Portfolios

1967 ◽  
Vol 4 (2) ◽  
pp. 170-174 ◽  
Author(s):  
Fredrik Esscher
Keyword(s):  
Distribution Function ◽  
Expected Number ◽  
Limit Values ◽  
Empirical Determination ◽  
The Mean ◽  
Premium Rates ◽  
Image Position ◽  
Stop Loss ◽  
The Given

When experience is insufficient to permit a direct empirical determination of the premium rates of a Stop Loss Cover, we have to fall back upon mathematical models from the theory of probability—especially the collective theory of risk—and upon such assumptions as may be considered reasonable.The paper deals with some problems connected with such calculations of Stop Loss premiums for a portfolio consisting of non-life insurances. The portfolio was so large that the values of the premium rates and other quantities required could be approximated by their limit values, obtained according to theory when the expected number of claims tends to infinity.The calculations were based on the following assumptions.Let F(x, t) denote the probability that the total amount of claims paid during a given period of time is ≤ x when the expected number of claims during the same period increases from o to t. The net premium II (x, t) for a Stop Loss reinsurance covering the amount by which the total amount of claims paid during this period may exceed x, is defined by the formula and the variance of the amount (z—x) to be paid on account of the Stop Loss Cover, by the formula As to the distribution function F(x, t) it is assumed that wherePn(t) is the probability that n claims have occurred during the given period, when the expected number of claims increases from o to t,V(x) is the distribution function of the claims, giving the conditioned probability that the amount of a claim is ≤ x when it is known that a claim has occurred, andVn*(x) is the nth convolution of the function V(x) with itself.V(x) is supposed to be normalized so that the mean = I.

Poiseuille Flow and Thermal Transpiration of a Rarefied Gas Between Parallel Plates: Effect of Nonuniform Surface Properties of the Plates in the Transverse Direction

2015 ◽  
Vol 137 (10) ◽  
Author(s):  
Toshiyuki Doi
Keyword(s):  
Distribution Function ◽  
Flow Rate ◽  
Poiseuille Flow ◽  
Transverse Direction ◽  
Rarefied Gas ◽  
Mean Free Path ◽  
Accommodation Coefficient ◽  
Parallel Plates ◽  
Nonuniform Surface ◽  
The Mean

Poiseuille flow and thermal transpiration of a rarefied gas between parallel plates with nonuniform surface properties in the transverse direction are studied based on kinetic theory. We considered a simplified model in which one wall is a diffuse reflection boundary and the other wall is a Maxwell-type boundary on which the accommodation coefficient varies periodically and smoothly in the transverse direction. The spatially two-dimensional (2D) problem in the cross section is studied numerically based on the linearized Bhatnagar–Gross–Krook–Welander (BGKW) model of the Boltzmann equation. The flow behavior, i.e., the macroscopic flow velocity and the mass flow rate of the gas as well as the velocity distribution function, is studied over a wide range of the mean free path of the gas and the parameters of the distribution of the accommodation coefficient. The mass flow rate of the gas is approximated by a simple formula consisting of the data of the spatially one-dimensional (1D) problems. When the mean free path is large, the distribution function assumes a wavy variation in the molecular velocity space due to the effect of a nonuniform surface property of the plate.

scholarly journals Self-distribution-function procedure in elastic incoherent neutron scattering for biosystems molecular motion characterization

2010 ◽  
Vol 24 (3-4) ◽  
pp. 387-391 ◽  
Author(s):  
Salvatore Magazù ◽  
Federica Migliardo ◽  
Antonio Benedetto ◽  
Miguel Gonzalez ◽  
Claudia Mondelli
Keyword(s):  
Distribution Function ◽  
Neutron Scattering ◽  
Mean Square Displacement ◽  
Aqueous Mixtures ◽  
Present Contribution ◽  
Incoherent Neutron Scattering ◽  
The Mean ◽  
Motion Characterization ◽  
Incoherent Neutron ◽  
Elastic Incoherent Neutron Scattering

In the present contribution we present a new procedure for the Mean Square Displacement (MSD) determination from Elastic Incoherent Neutron Scattering (EINS) where the connection between the Self-Distribution Function (SDF) and the measured EINS intensity profile is highlighted. We show how the SDF procedure allows both the total and the partial MSD evaluation, through the total and the partial SDFs. The procedure is applied on EINS data collected, by the IN13 backscattering spectrometer (ILL, Grenoble), on aqueous mixtures of sucrose and trehalose.

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