DOCSLIB.ORG

Moments and Measures of Skewness and Kurtosis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Moments and Measures of Skewness and Kurtosis Moments and Measures of Skewness and Kurtosis Moments The term moment has been taken from physics. The term moment in statistical use is analogous to moments of forces in physics. In statistics the values measure something relative to the center of the values. Moments are the constants of a population, as mean, variance, etc are. These constants help in deciding the characteristics of the population. Moments help in finding Arithmetic Mean, Standard Deviation and Variance of the population directly and they help, in knowing the graphic shapes of the population. There two types Raw Moments and Central Moments. Ref: (link) Central Moment First (s=1) The 1st moment 1 1 1 1 = (x1 + x2 + x3 + . + xn )/n = (x1 + x2 + x3 + . + xn)/n This formula is identical to the formula, to find the sample mean. You just add up all of the values and divide by the number of items in your data set. Second (s=2) The 2nd moment around the mean 2 = Σ(xi – μx) The second is the Variance. Third (s=3) The 3rd moment 3 3 3 3 = (x1 + x2 + x3 + . + xn )/n The third is skewness. Fourth (s=4) The 4th moment 4 4 4 4 = (x1 + x2 + x3 + . + xn )/n The fourth is kurtosis. Skewness Some distributions of data, such as the bell curve are symmetric. This means that the right and the left of the distribution are perfect mirror images of one another. Not every distribution of data is symmetric. Sets of data that are not symmetric are said to be asymmetric. The measure of how asymmetric a distribution can be is called skewness. The mean, median and mode are all measures of the center of a set of data. The skewness of the data can be determined by how these quantities are related to one another. Skewed to the Right Data that are skewed to the right have a long tail that extends to the right. An alternate way of talking about a data set skewed to the right is to say that it is positively skewed. In this situation the mean and the median are both greater than the mode. As a general rule, most of the time for data skewed to the right, the mean will be greater than the median. In summary, for a data set skewed to the right: Always: mean greater than mode Always: median greater than mode Most of the time: mean greater than median Skewed to the Left The situation reverses itself when we deal with data skewed to the left. Data that are skewed to the left have a long tail that extends to the left. An alternate way of talking about a data set skewed to the left is to say that it is negatively skewed. In this situation the mean and the median are both less than the mode. As a general rule, most of the time for data skewed to the left, the mean will be less than the median. In summary, for a data set skewed to the left: Always: mean less than mode Always: median less than mode Most of the time: mean less than median Measures of Skewness It’s one thing to look at two set of data and determine that one is symmetric while the other is asymmetric. It’s another to look at two sets of asymmetric data and say that one is more skewed than the other. It can be very subjective to determine which is more skewed by simply looking at the graph of the distribution. This is why there are ways to numerically calculate the measure of skewness. One measure of skewness, called Pearson’s first coefficient of skewness, is to subtract the mean from the mode, and then divide this difference by the standard deviation of the data. The reason for dividing the difference is so that we have a dimensionless quantity. This explains why data skewed to the right has positive skewness. If the data set is skewed to the right, the mean is greater than the mode, and so subtracting the mode from the mean gives a positive number. A similar argument explains why data skewed to the left has negative skewness. Pearson’s second coefficient of skewness is also used to measure the asymmetry of a data set. For this quantity we subtract the mode from the median, multiply this number by three and then divide by the standard deviation. Kurtosis This Greek word has the meaning "arched" or "bulging," making it an apt description of the concept known as kurtosis. Distributions of Data and Probability Distributions are not all the same shape. Some are asymmetric and skewed to the left or to the right. Other distributions are bimodal, and have two peaks. The degree of flatness or peakedness is measured by kurtosis. It tells us about the extent to which the distribution is flat or peak vis-a-vis the normal curve. The kurtosis of a distribution is in one of three categories of classification: Mesokurtic Leptokurtic Platykurtic Mesokurtic Kurtosis is typically measured with respect to the normal distribution. A distribution that has tails shaped in roughly the same way as any normal distribution. The normal curve is called Mesokurtic curve.The kurtosis of a mesokurtic distribution is neither high nor low, rather it is considered to be a baseline for the two other classifications. Leptokurtic A leptokurtic distribution is one that has kurtosis greater than a mesokurtic distribution. If the curve of a distribution is more peaked than a normal or mesokurtic curve then it is referred to as a Leptokurtic curve. Leptokurtic distributions are sometimes identified by peaks that are thin and tall. The tails of these distributions, to both the right and the left, are thick and heavy. Leptokurtic distributions are named by the prefix "lepto" meaning "skinny." Platykurtic The third classification for kurtosis is platykurtic. Platykurtic distributions are those that have slender tails. Many times they possess a peak lower than a mesokurtic distribution. If a curve is less peaked than a normal curve, it is called as a platykurtic curve. The meaning of the prefix "platy" is "broad". Relation between Raw and Central Moments r Recall mr = (1/n) ( xi - x´ ) , for r = 0, 1, 2, … You can apply binomial theorem and then expand R. H. S. of above relation. Then use the definition of raw moments. You will get following results. These are the relationships in which central moments are expressed in terms of raw moments of lower order. Proof of the following results is simple. 1) m1 = 0 always ’ ’ 2 2) m2 = m 2 – (m 1 ) ’ ’ ’ ’ 3 3) m3 = m 3 – 3m 2 m 1 + 2(m 1 ) ’ ’ ’ ’ ’ 2 ’ 4 4) m4 = m 4 – 4m 3 m 1 + 6m 2 (m 1 ) -3(m 1 ) Thus, if raw moments are known then the central moments can be obtained (and conversely). Properties of Central Moments 1. The central moments are invariant to the change of origin. Let U = X – A, then since that X = A + U , rth central moment of U, denoted by r r mr (u) = (1/n) (ui - u ) = (1/n) (xi – A + A - x ) = mr (x) 2. Effect of change of scale Let U = X / h then X = h U . Hence rth central moment of U, denoted by r mr (u) = (1/n) (ui - u ) r = (1/n) [(xi / h) – ( x / h)] = (1/ hr )mr(x) 3. You can combine above two properties and apply it while obtaining moments. We have th r defined r central moment of a random variable X as m r = (1/n) (xi – x ) . Define U = (X – A )/h, then X = A + h U and you will get mr (X) =hr [mr (U)], where th mr (U) and mr (X) denotes r central moments of U and X respectively. Coefficient of skewness We can specify several indexes of skewness. We list them below. 1. Karl Pearson’s Coefficient of skewness S1 = (A.M. – Mode)/ S. D. This is based on measures of central tendency. But if mode is not uniquely defined then this measure is also not well defined. In this case you can use the next measure, 2. Karl Pearson’s measure of Coefficient of skewness S2 = 3(A.M. – Me) / S.D. 3. Bowley’s coefficient of skewness, SkB, (based on quartiles) Where Q1 and Q3 are respectively lower and upper quartiles. Below Q1 , 25% observations lie and above Q 3, 25 % observations lie. Determination of Q1 and Q3 is easy. It is done similar to Q2 (which is same as median).It is particularly useful if you have open-ended classes (such as less than or greater than a particular value) 4. Coefficient of skewness (based on moments) Y1 = √ β 1 Coefficient of skewness is a pure number (no units) and is independent of change of origin and scale.
Load more

Recommended publications
  • On Moments of Folded and Truncated Multivariate Normal Distributions
    On Moments of Folded and Truncated Multivariate Normal Distributions Raymond Kan Rotman School of Management, University of Toronto 105 St. George Street, Toronto, Ontario M5S 3E6, Canada (E-mail: [email protected]) and Cesare Robotti Terry College of Business, University of Georgia 310 Herty Drive, Athens, Georgia 30602, USA (E-mail: [email protected]) April 3, 2017 Abstract Recurrence relations for integrals that involve the density of multivariate normal distributions are developed. These recursions allow fast computation of the moments of folded and truncated multivariate normal distributions. Besides being numerically efficient, the proposed recursions also allow us to obtain explicit expressions of low order moments of folded and truncated multivariate normal distributions. Keywords: Multivariate normal distribution; Folded normal distribution; Truncated normal distribution. 1 Introduction T Suppose X = (X1;:::;Xn) follows a multivariate normal distribution with mean µ and k kn positive definite covariance matrix Σ. We are interested in computing E(jX1j 1 · · · jXnj ) k1 kn and E(X1 ··· Xn j ai < Xi < bi; i = 1; : : : ; n) for nonnegative integer values ki = 0; 1; 2;:::. The first expression is the moment of a folded multivariate normal distribution T jXj = (jX1j;:::; jXnj) . The second expression is the moment of a truncated multivariate normal distribution, with Xi truncated at the lower limit ai and upper limit bi. In the second expression, some of the ai's can be −∞ and some of the bi's can be 1. When all the bi's are 1, the distribution is called the lower truncated multivariate normal, and when all the ai's are −∞, the distribution is called the upper truncated multivariate normal.
    [Show full text]
  • 5. the Student T Distribution
    Virtual Laboratories > 4. Special Distributions > 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5. The Student t Distribution In this section we will study a distribution that has special importance in statistics. In particular, this distribution will arise in the study of a standardized version of the sample mean when the underlying distribution is normal. The Probability Density Function Suppose that Z has the standard normal distribution, V has the chi-squared distribution with n degrees of freedom, and that Z and V are independent. Let Z T= √V/n In the following exercise, you will show that T has probability density function given by −(n +1) /2 Γ((n + 1) / 2) t2 f(t)= 1 + , t∈ℝ ( n ) √n π Γ(n / 2) 1. Show that T has the given probability density function by using the following steps. n a. Show first that the conditional distribution of T given V=v is normal with mean 0 a nd variance v . b. Use (a) to find the joint probability density function of (T,V). c. Integrate the joint probability density function in (b) with respect to v to find the probability density function of T. The distribution of T is known as the Student t distribution with n degree of freedom. The distribution is well defined for any n > 0, but in practice, only positive integer values of n are of interest. This distribution was first studied by William Gosset, who published under the pseudonym Student. In addition to supplying the proof, Exercise 1 provides a good way of thinking of the t distribution: the t distribution arises when the variance of a mean 0 normal distribution is randomized in a certain way.
    [Show full text]
  • A Study of Non-Central Skew T Distributions and Their Applications in Data Analysis and Change Point Detection
    A STUDY OF NON-CENTRAL SKEW T DISTRIBUTIONS AND THEIR APPLICATIONS IN DATA ANALYSIS AND CHANGE POINT DETECTION Abeer M. Hasan A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 2013 Committee: Arjun K. Gupta, Co-advisor Wei Ning, Advisor Mark Earley, Graduate Faculty Representative Junfeng Shang. Copyright c August 2013 Abeer M. Hasan All rights reserved iii ABSTRACT Arjun K. Gupta, Co-advisor Wei Ning, Advisor Over the past three decades there has been a growing interest in searching for distribution families that are suitable to analyze skewed data with excess kurtosis. The search started by numerous papers on the skew normal distribution. Multivariate t distributions started to catch attention shortly after the development of the multivariate skew normal distribution. Many researchers proposed alternative methods to generalize the univariate t distribution to the multivariate case. Recently, skew t distribution started to become popular in research. Skew t distributions provide more flexibility and better ability to accommodate long-tailed data than skew normal distributions. In this dissertation, a new non-central skew t distribution is studied and its theoretical properties are explored. Applications of the proposed non-central skew t distribution in data analysis and model comparisons are studied. An extension of our distribution to the multivariate case is presented and properties of the multivariate non-central skew t distri- bution are discussed. We also discuss the distribution of quadratic forms of the non-central skew t distribution. In the last chapter, the change point problem of the non-central skew t distribution is discussed under different settings.
    [Show full text]
  • Chapter 8 Fundamental Sampling Distributions And
    CHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS 8.1 Random Sampling pling procedure, it is desirable to choose a random sample in the sense that the observations are made The basic idea of the statistical inference is that we independently and at random. are allowed to draw inferences or conclusions about a Random Sample population based on the statistics computed from the sample data so that we could infer something about Let X1;X2;:::;Xn be n independent random variables, the parameters and obtain more information about the each having the same probability distribution f (x). population. Thus we must make sure that the samples Define X1;X2;:::;Xn to be a random sample of size must be good representatives of the population and n from the population f (x) and write its joint proba- pay attention on the sampling bias and variability to bility distribution as ensure the validity of statistical inference. f (x1;x2;:::;xn) = f (x1) f (x2) f (xn): ··· 8.2 Some Important Statistics It is important to measure the center and the variabil- ity of the population. For the purpose of the inference, we study the following measures regarding to the cen- ter and the variability. 8.2.1 Location Measures of a Sample The most commonly used statistics for measuring the center of a set of data, arranged in order of mag- nitude, are the sample mean, sample median, and sample mode. Let X1;X2;:::;Xn represent n random variables. Sample Mean To calculate the average, or mean, add all values, then Bias divide by the number of individuals.
    [Show full text]
  • Chapter 4: Central Tendency and Dispersion
    Central Tendency and Dispersion 4 In this chapter, you can learn • how the values of the cases on a single variable can be summarized using measures of central tendency and measures of dispersion; • how the central tendency can be described using statistics such as the mode, median, and mean; • how the dispersion of scores on a variable can be described using statistics such as a percent distribution, minimum, maximum, range, and standard deviation along with a few others; and • how a variable’s level of measurement determines what measures of central tendency and dispersion to use. Schooling, Politics, and Life After Death Once again, we will use some questions about 1980 GSS young adults as opportunities to explain and demonstrate the statistics introduced in the chapter: • Among the 1980 GSS young adults, are there both believers and nonbelievers in a life after death? Which is the more common view? • On a seven-attribute political party allegiance variable anchored at one end by “strong Democrat” and at the other by “strong Republican,” what was the most Democratic attribute used by any of the 1980 GSS young adults? The most Republican attribute? If we put all 1980 95 96 PART II :: DESCRIPTIVE STATISTICS GSS young adults in order from the strongest Democrat to the strongest Republican, what is the political party affiliation of the person in the middle? • What was the average number of years of schooling completed at the time of the survey by 1980 GSS young adults? Were most of these twentysomethings pretty close to the average on
    [Show full text]
  • Notes Mean, Median, Mode & Range
    Notes Mean, Median, Mode & Range How Do You Use Mode, Median, Mean, and Range to Describe Data? There are many ways to describe the characteristics of a set of data. The mode, median, and mean are all called measures of central tendency. These measures of central tendency and range are described in the table below. The mode of a set of data Use the mode to show which describes which value occurs value in a set of data occurs most frequently. If two or more most often. For the set numbers occur the same number {1, 1, 2, 3, 5, 6, 10}, of times and occur more often the mode is 1 because it occurs Mode than all the other numbers in the most frequently. set, those numbers are all modes for the data set. If each number in the set occurs the same number of times, the set of data has no mode. The median of a set of data Use the median to show which describes what value is in the number in a set of data is in the middle if the set is ordered from middle when the numbers are greatest to least or from least to listed in order. greatest. If there are an even For the set {1, 1, 2, 3, 5, 6, 10}, number of values, the median is the median is 3 because it is in the average of the two middle the middle when the numbers are Median values. Half of the values are listed in order. greater than the median, and half of the values are less than the median.
    [Show full text]
  • The Normal Moment Generating Function
    MSc. Econ: MATHEMATICAL STATISTICS, 1996 The Moment Generating Function of the Normal Distribution Recall that the probability density function of a normally distributed random variable x with a mean of E(x)=and a variance of V (x)=2 is 2 1 1 (x)2/2 (1) N(x; , )=p e 2 . (22) Our object is to nd the moment generating function which corresponds to this distribution. To begin, let us consider the case where = 0 and 2 =1. Then we have a standard normal, denoted by N(z;0,1), and the corresponding moment generating function is dened by Z zt zt 1 1 z2 Mz(t)=E(e )= e √ e 2 dz (2) 2 1 t2 = e 2 . To demonstate this result, the exponential terms may be gathered and rear- ranged to give exp zt exp 1 z2 = exp 1 z2 + zt (3) 2 2 1 2 1 2 = exp 2 (z t) exp 2 t . Then Z 1t2 1 1(zt)2 Mz(t)=e2 √ e 2 dz (4) 2 1 t2 = e 2 , where the nal equality follows from the fact that the expression under the integral is the N(z; = t, 2 = 1) probability density function which integrates to unity. Now consider the moment generating function of the Normal N(x; , 2) distribution when and 2 have arbitrary values. This is given by Z xt xt 1 1 (x)2/2 (5) Mx(t)=E(e )= e p e 2 dx (22) Dene x (6) z = , which implies x = z + . 1 MSc. Econ: MATHEMATICAL STATISTICS: BRIEF NOTES, 1996 Then, using the change-of-variable technique, we get Z 1 1 2 dx t zt p 2 z Mx(t)=e e e dz 2 dz Z (2 ) (7) t zt 1 1 z2 = e e √ e 2 dz 2 t 1 2t2 = e e 2 , Here, to establish the rst equality, we have used dx/dz = .
    [Show full text]
  • Volatility Modeling Using the Student's T Distribution
    Volatility Modeling Using the Student’s t Distribution Maria S. Heracleous Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Economics Aris Spanos, Chair Richard Ashley Raman Kumar Anya McGuirk Dennis Yang August 29, 2003 Blacksburg, Virginia Keywords: Student’s t Distribution, Multivariate GARCH, VAR, Exchange Rates Copyright 2003, Maria S. Heracleous Volatility Modeling Using the Student’s t Distribution Maria S. Heracleous (ABSTRACT) Over the last twenty years or so the Dynamic Volatility literature has produced a wealth of uni- variateandmultivariateGARCHtypemodels.Whiletheunivariatemodelshavebeenrelatively successful in empirical studies, they suffer from a number of weaknesses, such as unverifiable param- eter restrictions, existence of moment conditions and the retention of Normality. These problems are naturally more acute in the multivariate GARCH type models, which in addition have the problem of overparameterization. This dissertation uses the Student’s t distribution and follows the Probabilistic Reduction (PR) methodology to modify and extend the univariate and multivariate volatility models viewed as alternative to the GARCH models. Its most important advantage is that it gives rise to internally consistent statistical models that do not require ad hoc parameter restrictions unlike the GARCH formulations. Chapters 1 and 2 provide an overview of my dissertation and recent developments in the volatil- ity literature. In Chapter 3 we provide an empirical illustration of the PR approach for modeling univariate volatility. Estimation results suggest that the Student’s t AR model is a parsimonious and statistically adequate representation of exchange rate returns and Dow Jones returns data.
    [Show full text]
  • The Value at the Mode in Multivariate T Distributions: a Curiosity Or Not?
    The value at the mode in multivariate t distributions: a curiosity or not? Christophe Ley and Anouk Neven Universit´eLibre de Bruxelles, ECARES and D´epartement de Math´ematique,Campus Plaine, boulevard du Triomphe, CP210, B-1050, Brussels, Belgium Universit´edu Luxembourg, UR en math´ematiques,Campus Kirchberg, 6, rue Richard Coudenhove-Kalergi, L-1359, Luxembourg, Grand-Duchy of Luxembourg e-mail: [email protected]; [email protected] Abstract: It is a well-known fact that multivariate Student t distributions converge to multivariate Gaussian distributions as the number of degrees of freedom ν tends to infinity, irrespective of the dimension k ≥ 1. In particular, the Student's value at the mode (that is, the normalizing ν+k Γ( 2 ) constant obtained by evaluating the density at the center) cν;k = k=2 ν converges towards (πν) Γ( 2 ) the Gaussian value at the mode c = 1 . In this note, we prove a curious fact: c tends k (2π)k=2 ν;k monotonically to ck for each k, but the monotonicity changes from increasing in dimension k = 1 to decreasing in dimensions k ≥ 3 whilst being constant in dimension k = 2. A brief discussion raises the question whether this a priori curious finding is a curiosity, in fine. AMS 2000 subject classifications: Primary 62E10; secondary 60E05. Keywords and phrases: Gamma function, Gaussian distribution, value at the mode, Student t distribution, tail weight. 1. Foreword. One of the first things we learn about Student t distributions is the fact that, when the degrees of freedom ν tend to infinity, we retrieve the Gaussian distribution, which has the lightest tails in the Student family of distributions.
    [Show full text]
  • The Multivariate Normal Distribution=1See Last Slide For
    Moment-generating Functions Definition Properties χ2 and t distributions The Multivariate Normal Distribution1 STA 302 Fall 2017 1See last slide for copyright information. 1 / 40 Moment-generating Functions Definition Properties χ2 and t distributions Overview 1 Moment-generating Functions 2 Definition 3 Properties 4 χ2 and t distributions 2 / 40 Moment-generating Functions Definition Properties χ2 and t distributions Joint moment-generating function Of a p-dimensional random vector x t0x Mx(t) = E e x t +x t +x t For example, M (t1; t2; t3) = E e 1 1 2 2 3 3 (x1;x2;x3) Just write M(t) if there is no ambiguity. Section 4.3 of Linear models in statistics has some material on moment-generating functions (optional). 3 / 40 Moment-generating Functions Definition Properties χ2 and t distributions Uniqueness Proof omitted Joint moment-generating functions correspond uniquely to joint probability distributions. M(t) is a function of F (x). Step One: f(x) = @ ··· @ F (x). @x1 @xp @ @ R x2 R x1 For example, f(y1; y2) dy1dy2 @x1 @x2 −∞ −∞ 0 Step Two: M(t) = R ··· R et xf(x) dx Could write M(t) = g (F (x)). Uniqueness says the function g is one-to-one, so that F (x) = g−1 (M(t)). 4 / 40 Moment-generating Functions Definition Properties χ2 and t distributions g−1 (M(t)) = F (x) A two-variable example g−1 (M(t)) = F (x) −1 R 1 R 1 x1t1+x2t2 R x2 R x1 g −∞ −∞ e f(x1; x2) dx1dx2 = −∞ −∞ f(y1; y2) dy1dy2 5 / 40 Moment-generating Functions Definition Properties χ2 and t distributions Theorem Two random vectors x1 and x2 are independent if and only if the moment-generating function of their joint distribution is the product of their moment-generating functions.
    [Show full text]
  • Statistics, Measures of Central Tendency I
    Statistics, Measures of Central TendencyI We are considering a random variable X with a probability distribution which has some parameters. We want to get an idea what these parameters are. We perfom an experiment n times and record the outcome. This means we have X1;:::; Xn i.i.d. random variables, with probability distribution same as X . We want to use the outcome to infer what the parameters are. Mean The outcomes are x1;:::; xn. The Sample Mean is x1+···+xn x := n . Also sometimes called the average. The expected value of X , EX , is also called the mean of X . Often denoted by µ. Sometimes called population mean. Median The number so that half the values are below, half above. If the sample is of even size, you take the average of the middle terms. Mode The number that occurs most frequently. There could be several modes, or no mode. Dan Barbasch Math 1105 Chapter 9 Week of September 25 1 / 24 Statistics, Measures of Central TendencyII Example You have a coin for which you know that P(H) = p and P(T ) = 1 − p: You would like to estimate p. You toss it n times. You count the number of heads. The sample mean should be an estimate of p: EX = p, and E(X1 + ··· + Xn) = np: So X + ··· + X E 1 n = p: n Dan Barbasch Math 1105 Chapter 9 Week of September 25 2 / 24 Descriptive StatisticsI Frequency Distribution Divide into a number of equal disjoint intervals. For each interval count the number of elements in the sample occuring.
    [Show full text]
  • Strategic Information Transmission and Stochastic Orders
    A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Szalay, Dezsö Working Paper Strategic information transmission and stochastic orders SFB/TR 15 Discussion Paper, No. 386 Provided in Cooperation with: Free University of Berlin, Humboldt University of Berlin, University of Bonn, University of Mannheim, University of Munich, Collaborative Research Center Transregio 15: Governance and the Efficiency of Economic Systems Suggested Citation: Szalay, Dezsö (2012) : Strategic information transmission and stochastic orders, SFB/TR 15 Discussion Paper, No. 386, Sonderforschungsbereich/Transregio 15 - Governance and the Efficiency of Economic Systems (GESY), München, http://nbn-resolving.de/urn:nbn:de:bvb:19-epub-14011-6 This Version is available at: http://hdl.handle.net/10419/93951 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen die in der dort Content Licence (especially Creative Commons Licences), you genannten Lizenz gewährten Nutzungsrechte. may exercise further usage rights as specified in the indicated licence.
    [Show full text]