scholarly journals Credibility Approximations for Bayesian Prediction of Second Moments

1985 ◽  
Vol 15 (2) ◽  
pp. 103-121 ◽  
Author(s):  
William S. Jewell ◽  
Rene Schnieper
Keyword(s):  
Least Squares ◽  
Linear Functions ◽  
Quadratic Functions ◽  
Sample Variance ◽  
Credibility Theory ◽  
Sample Mean ◽  
Second Moment ◽  
Second Moments ◽  
Special Cases ◽  
The Mean

AbstractCredibility theory refers to the use of linear least-squares theory to approximate the Bayesian forecast of the mean of a future observation; families are known where the credibility formula is exact Bayesian. Second-moment forecasts are also of interest, for example, in assessing the precision of the mean estimate. For some of these same families, the second-moment forecast is exact in linear and quadratic functions of the sample mean. On the other hand, for the normal distribution with normal-gamma prior on the mean and variance, the exact forecast of the variance is a linear function of the sample variance and the squared deviation of the sample mean from the prior mean. Bühlmann has given a credibility approximation to the variance in terms of the sample mean and sample variance.In this paper, we present a unified approach to estimating both first and second moments of future observations using linear functions of the sample mean and two sample second moments; the resulting least-squares analysis requires the solution of a 3 × 3 linear system, using 11 prior moments from the collective and giving joint predictions of all moments of interest. Previously developed special cases follow immediately. For many analytic models of interest, 3-dimensional joint prediction is significantly better than independent forecasts using the “natural” statistics for each moment when the number of samples is small. However, the expected squared-errors of the forecasts become comparable as the sample size increases.

scholarly journals Properties of the Standard Deviation that are Rarely Mentioned in Classrooms

2016 ◽  
Vol 38 (3) ◽  
Author(s):  
Mohammad Fraiwan Al-Saleh ◽  
Adil Eltayeb Yousif
Keyword(s):  
Standard Deviation ◽  
Coefficient Of Variation ◽  
Sample Variance ◽  
Mean Absolute Deviation ◽  
Hidden Information ◽  
Absolute Deviation ◽  
Sample Mean ◽  
The Mean ◽  
Vague Concept ◽  
Chebyshev's Inequality

Unlike the mean, the standard deviation ¾ is a vague concept. In this paper, several properties of ¾ are highlighted. These properties include the minimum and the maximum of ¾, its relationship to the mean absolute deviation and the range of the data, its role in Chebyshev’s inequality and the coefficient of variation. The hidden information in the formula itself is extracted. The confusion about the denominator of the sample variance being n ¡ 1 is also addressed. Some properties of the sample mean and varianceof normal data are carefully explained. Pointing out these and other properties in classrooms may have significant effects on the understanding and the retention of the concept.

THE RECIPROCITY LAW FOR THE TWISTED SECOND MOMENT OF DIRICHLET -FUNCTIONS OVER RATIONAL FUNCTION FIELDS

2018 ◽  
Vol 98 (3) ◽  
pp. 383-388 ◽  
Author(s):  
GORAN DJANKOVIĆ
Keyword(s):  
Rational Function ◽  
Function Fields ◽  
Mean Square ◽  
Irreducible Polynomials ◽  
Second Moment ◽  
Reciprocity Law ◽  
Second Moments ◽  
The Mean

We prove the reciprocity law for the twisted second moments of Dirichlet $L$-functions over rational function fields, corresponding to two irreducible polynomials. This formula is the analogue of the formulas for Dirichlet $L$-functions over $\mathbb{Q}$ obtained by Conrey [‘The mean-square of Dirichlet $L$-functions’, arXiv:0708.2699 [math.NT] (2007)] and Young [‘The reciprocity law for the twisted second moment of Dirichlet $L$-functions’, Forum Math. 23(6) (2011), 1323–1337].

scholarly journals The Credible Distribution

1974 ◽  
Vol 7 (3) ◽  
pp. 237-269 ◽  
Author(s):  
William S. Jewell
Keyword(s):  
Least Squares ◽  
Conditional Distribution ◽  
The Other ◽  
Theoretical Development ◽  
Credibility Theory ◽  
Individual Risk ◽  
Detailed Knowledge ◽  
Computational Results ◽  
Claim Frequency ◽  
The Mean

AbstractCredibility theory is concerned with the problem of forecasting the mean performance (claim frequency, total losses, etc.) of an individual risk, selected from a collective of heterogeneous risks, based upon the statistics of the collective, and upon the individual's experience data. The classic results, derived by American actuaries in the 1920's, were further strengthened by Bailey and Mayerson in 1950 and 1965, who showed that these results were exact Bayesian for certain risk distributions. Bühlmann, in 1967, then showed that the credibility formulae were the best least-squares linearized approximation to the exact Bayesian forecast, for general risk distributions.This paper extends credibility theory to the problem of forecasting the distribution of individual risk, based upon collective statistics and individual experience data. Although the problem is, in principle, solved by finding a Bayesian conditional distribution, this approach requires a detailed knowledge of collective structure. The credible distribution, on the other hand, requires fewer prior statistics, and is also a best least-squares linearized approximation to the exact Bayesian distribution.Following the theoretical development, detailed computational results are given.

Decoration technique and surface physics

1978 ◽  
Vol 36 (1) ◽  
pp. 436-437 ◽  
Author(s):  
H. Bethge
Keyword(s):  
Diffusion Path ◽  
Special Importance ◽  
Surface Physics ◽  
Step Structure ◽  
Initial Stage ◽  
Special Cases ◽  
Atomic Surface ◽  
The Mean ◽  
Bulk Structure ◽  
Decoration Technique

Besides the atomic surface structure, diverging in special cases with respect to the bulk structure, the real structure of a surface Is determined by the step structure. Using the decoration technique /1/ it is possible to image step structures having step heights down to a single lattice plane distance electron-microscopically. For a number of problems the knowledge of the monatomic step structures is important, because numerous problems of surface physics are directly connected with processes taking place at these steps, e.g. crystal growth or evaporation, sorption and nucleatlon as initial stage of overgrowth of thin films.To demonstrate the decoration technique by means of evaporation of heavy metals Fig. 1 from our former investigations shows the monatomic step structure of an evaporated NaCI crystal. of special Importance Is the detection of the movement of steps during the growth or evaporation of a crystal. From the velocity of a step fundamental quantities for the molecular processes can be determined, e.g. the mean free diffusion path of molecules.

The Estimation of Blood Platelet Survival

1972 ◽  
Vol 28 (03) ◽  
pp. 447-456 ◽  
Author(s):  
E. A Murphy ◽  
M. E Francis ◽  
J. F Mustard
Keyword(s):  
Standard Deviation ◽  
Least Squares ◽  
Experimental Error ◽  
Blood Platelet ◽  
Least Squares Estimation ◽  
Systematic Relationship ◽  
Platelet Survival ◽  
The Mean ◽  
Normally Distributed

SummaryThe characteristics of experimental error in measurement of platelet radioactivity have been explored by blind replicate determinations on specimens taken on several days on each of three Walker hounds.Analysis suggests that it is not unreasonable to suppose that error for each sample is normally distributed ; and while there is evidence that the variance is heterogeneous, no systematic relationship has been discovered between the mean and the standard deviation of the determinations on individual samples. Thus, since it would be impracticable for investigators to do replicate determinations as a routine, no improvement over simple unweighted least squares estimation on untransformed data suggests itself.

scholarly journals Approximated least-squares solutions of a generalized Sylvester-transpose matrix equation via gradient-descent iterative algorithm

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Adisorn Kittisopaporn ◽  
Pattrawut Chansangiam
Keyword(s):  
Least Squares ◽  
Iterative Algorithm ◽  
Gradient Descent ◽  
Main Idea ◽  
Full Column Rank ◽  
Minimum Error ◽  
Matrix Coefficients ◽  
Special Cases ◽  
Efficiency And Effectiveness ◽  
Associated Matrix

AbstractThis paper proposes an effective gradient-descent iterative algorithm for solving a generalized Sylvester-transpose equation with rectangular matrix coefficients. The algorithm is applicable for the equation and its interesting special cases when the associated matrix has full column-rank. The main idea of the algorithm is to have a minimum error at each iteration. The algorithm produces a sequence of approximated solutions converging to either the unique solution, or the unique least-squares solution when the problem has no solution. The convergence analysis points out that the algorithm converges fast for a small condition number of the associated matrix. Numerical examples demonstrate the efficiency and effectiveness of the algorithm compared to renowned and recent iterative methods.

The Initial Return and Its Conditional Return Volatility: Evidence from the Chinese IPO Market

2014 ◽  
Vol 17 (04) ◽  
pp. 1450022 ◽  
Author(s):  
M. Monica Hussein ◽  
Zhong-Guo Zhou
Keyword(s):  
Time Series ◽  
Linear Functions ◽  
Return Volatility ◽  
Cross Sectional ◽  
Variance Model ◽  
Initial Return ◽  
Traditional Market ◽  
Mean And Variance ◽  
The Mean ◽  
Sectional Analysis

This paper investigates the monthly initial return and its conditional return volatility for Chinese IPOs. We find that the mean initial return (IR) and cross-sectional return volatility are highly auto- and cross-correlated, and time-varying. We propose a system of two simultaneous equations: a GARCH-in-mean (GARCH-M) process with an ARMA(1,1) adjustment in the residuals for the IR and an EGARCH process for the conditional return volatility, assuming that the IR and its conditional return volatility are linear functions of the same market, firm- and offer-specific characteristics. We find that the model captures both time-series and cross-sectional correlations at the mean and variance levels. Our findings suggest that the conditional return volatility affects the IR positively and significantly, in addition to the traditional market, firm- and offer-specific characteristics. IPOs with higher conditional return volatility, as a proxy for information asymmetry, tend to be underpriced more. The paper demonstrates the merit of using a conditional variance model, along with time series and cross-sectional analysis to price Chinese IPOs.

Comparison of plasma progesterone profiles in cyclic, pregnant, pseudopregnant and hysterectomized pigs between 8 and 27 days after oestrus

1988 ◽  
Vol 119 (1) ◽  
pp. 111-116 ◽  
Author(s):  
G. J. King ◽  
R. Rajamahendran
Keyword(s):  
Least Squares ◽  
Analysis Of Variance ◽  
The Other ◽  
Corpora Lutea ◽  
Plasma Progesterone ◽  
The Third ◽  
Secretory Potential ◽  
The Mean

ABSTRACT Plasma progesterone concentrations were compared in cyclic (n = 12), pregnant (n =12), oestradiol-induced pseudopregnant (n=12) and hysterectomized gilts (n=10) between days 8 and 27 after oestrus. The results were grouped into periods covering days 8–13, 14–20 and 21–27 and analysed by least-squares analysis of variance. Plasma progesterone concentrations were significantly (P<0·001) higher in hysterectomized compared with other groups between days 8 and 13. Progesterone concentrations declined rapidly after day 14 in cyclic females and gradually in the other groups. Throughout the third and fourth weeks the mean progesterone concentrations for hysterectomized animals were consistently higher than for pseudopregnant animals (P<0·05). The pregnant group means were below but not significantly different from the hysterectomized means in both of the last two periods. The greater progesterone concentrations in hysterectomized gilts indicated that secretion is high without any conceptus-produced or -mediated luteotrophin, and corpora lutea in cyclic, pregnant or pseudopregnant gilts may never reach full secretory potential. J. Endocr. (1988) 119, 111–116

The Regression of the Sample Variance on the Sample Mean

1946 ◽  
Vol s1-21 (1) ◽  
pp. 22-28 ◽  
Author(s):  
M. C. K. Tweedie
Keyword(s):  
Sample Variance ◽  
Sample Mean

scholarly journals Thermodynamic and Elastic Properties of Interstitial Alloy FeC with BCC Structure at Zero Pressure

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Bui Duc Tinh ◽  
Nguyen Quang Hoc ◽  
Dinh Quang Vinh ◽  
Tran Dinh Cuong ◽  
Nguyen Duc Hien
Keyword(s):  
Nearest Neighbor ◽  
Young Modulus ◽  
Thermodynamic Quantities ◽  
Atom Concentration ◽  
Main Metal ◽  
Special Cases ◽  
Analytic Expressions ◽  
Rigidity Modulus ◽  
Bcc Structure ◽  
The Mean

The analytic expressions for the thermodynamic and elastic quantities such as the mean nearest neighbor distance, the free energy, the isothermal compressibility, the thermal expansion coefficient, the heat capacities at constant volume and at constant pressure, the Young modulus, the bulk modulus, the rigidity modulus, and the elastic constants of binary interstitial alloy with body-centered cubic (BCC) structure, and the small concentration of interstitial atoms (below 5%) are derived by the statistical moment method. The theoretical results are applied to interstitial alloy FeC in the interval of temperature from 100 to 1000 K and in the interval of interstitial atom concentration from 0 to 5%. In special cases, we obtain the thermodynamic quantities of main metal Fe with BCC structure. Our calculated results for some thermodynamic and elastic quantities of main metal Fe and alloy FeC are compared with experiments.

Sign in / Sign up

Export Citation Format

Share Document