Distribution Of Students Research Articles

Geometrically convergent sequences of upper and lower bounds on the Wallis ratio and related expressions

Sequences of algebraic upper and lower bounds on the Wallis ratio ( x + 1)/( x + 1 ) are given with the relative errors that converge to 0 geometrically and uniformly on any interval of the form (x0,1) for x0 > 1 ; moreover, the relative and absolute errors converge to 0 as x ! 1. These conclusions are based on corresponding results for the digamma function := 0 /. Relations with other relevant results are discussed, as well as the corresponding computational aspects. This work was motivated by studies of exact bounds involving the Student probability distribution. AMS 2000 subject classifications: Primary 33B15, 26D07, 26D15, 41A17; secondary 33F05, 65D20, 60E15, 62E15, 62E17. Keywords and phrases: Gamma function, digamma function, Wallis ra- tio, upper bounds, lower bounds, exact bounds, inequalities in approxima- tion, Student's distribution, Student's statistic, probability inequalities.

Exact bounds on the closeness between the Student and standard normal distributions

Upper bounds on the Kolmogorov distance (and, equivalently in this case, on the total variation distance) between the Student distribu- tion with p degrees of freedom (SDp) and the standard normal distribution are obtained. These bounds are in a certain sense best possible, and the corresponding relative errors are small even for moderate values of p. The same bounds hold on the closeness between SDp and SDq with q > p. AMS 2000 subject classifications: Primary 62E17; secondary 60E15, 62E20, 62E15. Keywords and phrases: Student's distribution, standard normal distri- bution, Kolmogorov distance, total variation distance, probability inequal- ities.

Parametric Methods for Estimating the Level of Risk in Finance

Abstract The present paper deals with quantifying a wide range of risks through techniques known as Value at Risk. A given group of methods is discussed, whether human of financial practice or financial theorists from academia, more than twenty years, and it is therefore logical that there are currently exists a lot of approaches and algorithms risk quantification. We will be invented to a group called parametric methods for estimating the level of risk. This is a procedure where the theoretical probability distribution of losses is known, but we do not know its parameters. We must then be suitably approximated with some theoretical probability distributions and thus used for the distribution of individual risk rate. In our contribution we real data to approximate Normal, Student's and Laplace distribution. Percentage approximation we checked on the basis Likelihood logarithmic functions. Theoretical aspects of selected theoretical distributions will be applied to real data. Specifically, the shares of Oracle, Coca Cola and Pfizer term in January 2012 to February 2015. The price development is on a daily basis.

Realized Non-linear Stochastic Volatility Models with Asymmetric Effects and Generalized Student's <i>t</i>-Distributions

This study proposes a class of realized non-linear stochastic volatility models with asymmetric effects and generalized Student's t-error distributions by applying three families of power transformation—exponential, modulus, and Yeo-Johnson—to lagged log volatility. The proposed class encompasses a raw version of the realized stochastic volatility model. In the Markov chain Monte Carlo algorithm, an efficient Hamiltonian Monte Carlo (HMC) method is developed to update the latent log volatility and transformation parameter, whereas the other parameters that could not be sampled directly are updated by an efficient Riemann manifold HMC method. Empirical studies on daily returns and four realized volatility estimators of the Tokyo Stock Price Index (TOPIX) over 4-year and 8-year periods demonstrate statistical evidence supporting the incorporation of skew distribution into the error density in the returns and the use of power transformations of lagged log volatility.

Reviews

Reviews

Characterization of the void size distribution in fluidized beds using statistics of pressure fluctuations

Experiments carried out in fluidized beds of Geldart A and B powders revealed that in-bed pressure fluctuations typically have power-law statistics, which are well described by the Student's distribution. We show that the pressure statistics encode information about the bubble size distribution, and propose a novel method that allows an easy determination of the approximate shape of the bubble size distribution from the statistics of pressure fluctuations.

03.6.1 The Central Limit Theorem for Student's Distribution—Solution

03.6.1 The Central Limit Theorem for Student's Distribution—Solution

Investigating Extreme Dependences: Concepts and Tools

We investigate the relative information content of six measures of dependence between two random variables X and Y for large or extreme events for several models of interest for financial time series. The six measures of dependence are respectively the linear correlation and Spearman's rho conditioned on signed exceedance of one variable above the threshold, or on both variables, the linear correlation conditioned on absolute value exceedance (or large volatility) of one variable, the so-called asymptotic tail-dependence and a probability-weighted tail dependence coefficient. The models are the bivariate Gaussian distribution, the bivariate Student's distribution, and the factor model for various distributions of the factor. We offer explicit analytical formulas as well as numerical estimations for these six measures of dependence in the limit exploring the extreme tails. This provides a quantitative proof that conditioning on exceedance leads to conditional correlation coefficients that may be very different from the unconditional correlation and gives a straightforward mechanism for fluctuations or changes of correlations, based on fluctuations of volatility or changes of trends. Moreover, these various measures of dependence exhibit different and sometimes opposite behaviors, suggesting that, somewhat similarly to risks whose adequate characterization requires an extension beyond the restricted one-dimensional measure in terms of the variance (volatility) to include all higher order cumulants or more generally the knowledge of the full distribution, tail-dependence has also a multidimensional character.

Student's simplest distribution

Summary. In this paper, I note the pedagogical advantages of an existing, but largely ignored, simple symmetric distribution on the real line: Student's t-distribution on 2 degrees of freedom. The t2-distribution has particularly simple forms for its distribution and quantile functions, and these lead to simply calculated values for a whole host of basic properties of the distribution. In addition, distributions of order statistics from the t2-distribution lead to a recently introduced skew t family of distributions.

Confidence intervals for the activation energy estimated by few experiments

Arrhenius parameters are usually estimated by substituting data of 3–5 experiments into a linearized kinetic equation. The use of both small population and linearization invalidates the normal hypothesis. The study focuses on estimating valid confidence intervals for the activation energy computed by the isoconversional method. For this method, a comparison of confidence intervals estimated by using Student's distribution and a nonparametric (distribution-free) method has shown that Student's estimates tend to be oversized. Realistic 95% confidence intervals can be constructed by using corrected Student's percentiles ϑ n−2, 0.975 of 4.0, 2.5, and 2.3 for 3, 4, and 5 heating rates experiments, respectively.

A Monte Carlo Study on the Fit of a Cost Function by a Fourier Flexible Functional Form

We use the simulation model of Chalfallt and Gallant (1985) to investigate the fit of a Fourier flexible form to a cost function when interest is to estimate elasticities of substitution. The data is subject to errors in variables and the estimation method is seemingly unrelated regressions. We use two approximating Fourier functional forms with 13 (FFF13) and 22 (FFF22) parameters. The biases in point estimation are negligible for FFF13. The classical t statistics do not follow, in general, the Student's distribution. Even when estimates are properly centered and scaled there are cases where the normal approximation does not hold.

Approximating the Characteristic Function of the Student's Distribution

Approximating the Characteristic Function of the Student's Distribution

A comparison of the box-cox transformation method and nonparametric methods for estimating quantiles in clinical data with repeated measures

In this paper we studied the problem of estimating the quantiles of a distribution when the observations are not independent. The type of data we considered was repeated measurements on independent units. Four methods of quantile estimation were compared, one parametric and three nonparametric. Box- Cox power transformations were used to transform a set of repeated measures to multivariate normality and a quantile estimate was obtained from the inverse transformation of the quantile on the transformed scale. The first nonparametric estimate was obtained by taking a weighted average of two order statistics from a sample consisting of one observation per independent unit, the second nonparametric estimate was a bootstrap of the first estimate, and the third was a bootstrapped version of the Harrell-Davis estimate. Comparisons were made with simulated data from Gaussian, Student's Tlog-normal, chisquared, contaminated normal, and mixed distributions of the first, fifth, ninety-fifth and ninety-ninth quantile...

Comparing the exact distribution of the t-statistic to student's distribution when its constituent normal and chi-squared variables are dependent

The ubiquitous t-statistic is well known to follow Student's distribution in a variety of settings. Even when a statistical model is such that a derived t-statistic is not distributed exactly according to Student's distribution, it is common practice to use critical values of Student's distribution as opposed to critical values of the normal distribution which are often appropriate under asymptotic considerations. This paper examines the effect which dependence between the normal and chi-square constituent variables of the t-statistic has in respect to Student's distribution. Situations where the aforementioned dependence occurs, often arise in, for example, econometric practice, especially in respect of analysis of t- statistics obtained from simultaneous equation models and dynamic regression models.

Modern Business Statistics

The relationship between sampling and statistics displaying sample data descriptive sample statistics probability, populations and random variables some useful discrete and continuous distributions estimation (one sample) hypothesis testing two related examples (matched pairs) estimation and hypothesis testing with two independent samples contingency tables correlation regression time series analysis formulating general linear models for fitting data multiple linear regression analysis of variance of one-factor experiments analysis of variance of two-factor experiments other useful topics in experiment design decision theory under uncertainty decision theory with sample information tables: binomial distribution, normal distribution, student's distribution, chi-squared distribution and the f distribution.

A new simple straightforward and practical asymptotic expansion for the incomplete beta function

It is shown that by using elementary transformations, the Incomplete Beta Function can be turned into a form suggesting that a close approximation by a normal integral can be obtained. By using standard methods this approximation is then developed in the form of an asymptotic series consisting of four exhibited correction terms to the main normal integral. An alternative means of deriving the third of these terms is shown confirming the correctness of the calculations (paradoxically the fourth correction term is easily derived and needs no such check). From this asymptotic series an asymptotic series for a corresponding standard normal variable is derived. The excellence of these approximations is exhibited by studying two limiting cases: that of equal indices (equivalent to a Student's distribution) and that of one index becoming infinite so that we have in place of an Incomplete Beta Function an Incomplete Gamma Function. Some similar recent results in the literature are commented on. Tables to show how excellent the expansion is, even down to parameter values as low as unity, are provided for a selection of parameter values and percentage points.

Comparing the Means of Two Independent Samples

SUMMARY A practical procedure is given for assessing a P value for the hypothesis that two samples come from populations having the same mean, for the case when the two variances cannot be assumed equal. 1. The Problem, and a Procedure that sometimes Solves it Given two samples xl,x2,. . ., xm, and Yi ,Y2,** .,Yn, supposed normal with means g, A + 8, and variances a2, p2 a2, respectively, standard texts are curiously silent on the question of testing whether 8 = 0 (or any other prescribed value) when it cannot be taken as known that the variances are equal, p2 = 1. A few texts give approximations which are valid for moderately large samples, and a few refer to the Tables for Welch's test given in Biometrika Tables, Volume I (Table 11, 3rd edition). Fewer still refer to Fisher and Yates' Statistical Tables (Table VI, 6th edition) which give critical values for the Behrens-Fisher test. And yet it is easy to see that if p can be taken to have any known value, the quantity t =x6 - -_ ) ({ (I /m) + (p2/In)} { (m -1)s2 + (n - 1 s/p2 }/(m + n- /2)ll2( (with the usual sample mean and variance notation) has Student's distribution with m + n -2 degrees of freedom. It is not difficult to program a hand calculator to take in p2 , S2y m, n, and (j7- - 6) and to calculate this t. Then if a range of p2 values is considered plausible, p2 can be varied over this range to see what effect this variation has on the value of t. It will often happen (especially when m and n are nearly equal, and moderately large, and sX and sy do not differ by much) that t will remain nearly constant over a wide range of p2 values, and in this case the P value associated with the common t value can be taken as indicating the credibility or otherwise of the value of 6.

A compilation study of the ϕ mass

The review of particle properties (1) (RPP) determines the mass and the width averages and their errors with two different procedures. The standard procedure--the weighted means with scaled confidence intervals--assum es that the measurements are normally distributed around the true value. However, it is well krmwn that the distribution has longer tails. To get more realistic errors the confidence intervals are broadened by scaling with a scale factor S. This is justified only if the errors of all the measurements are underestimated by the same factor S. The empirical distribution of residuals has been studied for some particles and it has turned out that they are well described by student's distribution Slo(h/1.11 ) (2). This can be taken into account by using the maximum likelihood me~hod. The second procedure for RPP masses is based on this result. However, neither of the methods can take into account the fact that different experiments use different fitting procedures to their data. finis may lead to systematic differences. Besides, there are many experiments whose mass determinations are not included in the RPP averages. Generally the systematic errors of these measurements are not evaluated or the resolution function has not been studied. The purpose of the present study is to determine the ~ mass M~ from the data not used in the RPP averages. To eliminate systematic errors all the data are analysed by the same method; a Breith-Wigner function is smeared by a student resolution function Ss(t). However, this resolution function need not correspond to the true resohttion of every experiment. To eliminate possible bias due to this choice, the masses are extrapolated to zero resolution. The same method has previously been used to study the co mass (3).

図書館と統計的方法

Needs for applications of statistical techniques to studies on library activities are growing rapidly, because the library management should make a sound plan based on correct data assembled in the recent severe circumstances. The techniques are powerful tools, for example, to acquire precise data about and to analyse relations between cost and productivity of library services.The author first explains basic concepts and methods of statistical inference, test, sampling (simple random and proportional sampling), and errors entailed on inference and test. Then he describes applications of such methods as inference, test, linear regression, correlation, analysis of variance to actual library situations. The standard normal distribution, student's t-distribution, chi-square distribution, and F-distribution are picked up here for exemplifying inference and/or test methods.

Estimating Sample Size in Computing Simulation Experiments

A method is described for estimating and collecting the sample size needed to estimate the mean of a process (with a specified level of statistical precision) in a simulation experiment. Steps are also discussed for incorporating the determination and collection of the sample size into a computer library routine that can be called by the ongoing simulation program. We present the underlying probability model that enables us to denote the variance of the sample mean as a function of the autoregressive representation of the process under study and describe the estimation and testing of the parameters of the autoregressive representation in a way that can easily be “built into” a computer program. Several reliability criteria are discussed for use in determining sample size. Since these criteria assume that the variance of the sample mean is known, an adjustment is necessary to account for the substitution of an estimate for this variance. It is suggested that Student's distribution be used as the sampling distribution, with “equivalent degrees of freedom” determined by analogy with a sequence of independent observations. A bias adjustment is described that can be applied to the beginning of the collected data to reduce the influence of initial conditions on events in the experiment. Four examples are presented using these techniques, and comparisons are made with known theoretical solutions. One unfortunate shortcoming of the proposed procedure is that its performance is directly linked to the initially chosen sample size. Our results show that as this sample size increases, the procedure gives results which agree more closely with predicted results.