Sample Skewness Research Articles

Skewness, kurtosis, and black-scholes option mispricing

The Black-Scholes option pricing model assumes that (instantaneous) common stock returns are normally distributed. However, the observed distribution exhibits deviations from normality; in particular skewness and kurtosis. We attribute these deviations to gross data errors. Using options' transactions data, we establish that the sample standard deviation, sample skewness, and sample kurtosis contribute to the Black-Scholes model's observed mispricing of a sample from the Berkeley Options Data Base of 2323 call options written on 88 common stocks paying no dividends during the options'life. Following Huber's statement that the primary case for robust statistics is when the shape of the observed distribution deviates slightly from the assumed distribution (usually the Gaussian), we show that robust volatility estimators eliminate the mispricing with respect to sample skewness and sample kurtosis, and significantly improve the Black-Scholes model's pricing performance with respect to estimated volatility.

Notes on the Influence Functions of Standardized Moments

The third and the fourth standardized sample moments are frequently used in practice as measures of skewness and kurtosis for assessing departure of the data from normality. In this article, it is shown that the influence functions of the functionals corresponding to the sample skewness and kurtosis at the standard normal distribution are identical with the third- and the fourth-degree Hermite polynomials. The influence functions of mth-order (m ≥ 5) standardized moments are also obtained. When m ≥ 5, it is revealed that the influence functions are not identical with the Hermite polynomials. Therefore, we determine moment-based statistics whose influence functions are given by Hermite polynomials. For bivariate distributions, we also calculate influence functions of product moments and that of bivariate kurtosis.

Heavy-mineral reconnaissance off the coast of the Apalachicola river delta, northwest Florida: A summary and new interpretations

Two-hundred-and-fifty sediment samples were collected for heavy-mineral and textural analysis along the northwest Florida coastline from approximately 24 km offshore of Apalachee Bay to the same distance offshore of Pensacola Bay. The heavy-mineral suite characterizing sediments within this region consists of opaque minerals, kyanite, staurolite, tourmaline, zircon and rutile. Minor constituents of this suite include epidote, sphene, amphibole, sillimanite, garnet and leucoxene. The average heavy-mineral concentration within these sediments is approximately 0.12 wt.%. Specifically, the 2 to 3∅ grain-size interval contains an average of 0.51 wt.%, whereas the 3 to 4∅ interval contains an average of 4.39 wt.% heavy minerals. Note that the 3 to 4∅ interval typically represents only 4% of the sample volume. There is a general westward increase in heavy-mineral concentrations throughout the study area. Superimposed on this regional trend, areas of maximum heavy-mineral concentration occur within sediments offshore of St. George and Santa Rosa islands. The primary source of sediments in the region is the crystalline rocks of the southern Appalachians. Granulometric analyses of these sediments reveal a westward increase in values of sample mean grain size, and decrease in standard deviation, and percent fines. It is postulated from these data, in addition to the interpretation of sample grain-size distributions, skewness, and kurtosis, that these inner continental shelf sediments are primarily fluvial in origin. These sediments have been transported to the shelf by the Apalachicola and surrounding major rivers during Pleistocene low sea-level stands. Data also indicate evidence of reworking by coastal or marine offshore wave processes.

Algebraic bounds on standardized sample moments

Best possible bounds are given for standardized sample (or finite population) moments and absolute moments of arbitrary order, generalizing those given by Cramér (1945) and Kirby (1974) on sample skewness and kurtosis.

The statistical analysis of the sand grain size distribution of Al- Ubay-lah barchan dunes, Northwestern Ar-Rub-Alkhali desert, Saudi Arabia

The paper analyses the statistical parameters of the sand grain size distribution of Al-Ubaylah barchan dunes applying the sieving technique and Folk and Ward, and Trask's equation. The results of the sieving analysis divulged that nearly over 90% of the sand particles fall in the size ranging between 1.50o and 3.o. Insignificant amount of sand grains also fall in the size ranging from 0.0o to 1.50o and from 3.0o to 4.0o respectively. The statistical analysis reveals that the sand is characteristically fine grained, moderately to moderately well sorted with an average diameter of 2.4o and a standard deviation of 0.64o. The sand distribution is positively fine skewed and mesokurtic to leptokurtic in nature. No significant variation in the statistical parameters (median, mean, sorting, skewness and kurtosis) of the sand grains samples used in this research is detected. The sand grain size distribution of all the samples can also be considered symmetrical and unimodal in nature. Those abovementioned characteristics are found to be nearly similar to that of other Saudi dune deserts. The miscroscopic investigation undertaken by this researcher displays that quartz represents often more than 99% of the samples. Felsic, mafic and calcium carbonates minerals represent less than 1% of the mineralogical constituents. The sand grains are subrounded to subangular and iron and calcium carbonates coated.

Sand movement and vegetation in the Apostle Islands, Lake Superior

Recent sand movement was inferred from grain morphology, modal grain size, and grain size distributional skewness of samples from the surfaces of beach–dune landscapes. Negative skewness indicated recent net deflation, and either positive skewness or small modal grain size indicated recent net aeolian accretion. Large grains that were shiny and angular indicated surfaces that could not have been deposited by wind. Net deflation occurred where vegetational cover was below 10%, in foredunes and an interdunal blowout. All other sites in dunal, interdunal, or forested zones showed evidence of net accretion or a balance of accretion and deflation. Net accretion was evident in locations with as little as 15% cover. Relationships between sand movement and vegetational composition could not be separated from relationships with total cover. The sandy foreland of one island was bulldozed during the 1950s. This apparently brought coarser Holocene beach deposits near the surface, which limited the severity of blowout by providing a deflation armor. Ammophila breviligulata has not recolonized the interdunal zone of this island, though it persists in this zone on the other islands. The study of grains in cores may reveal the history of vegetational cover at a site.

A continuously adaptive nonparametric two–sample test

A two–sample test statistic for detecting shifts in location is developed for a broad range of underlying distributions using adaptive techniques. The test statistic is a linear rank statistics which uses a simple modification of the Wilcoxon test; the scores are Winsorized ranks where the upper and lower Winsorinzing proportions are estimated in the first stage of the adaptive procedure using sample the first stage of the adaptive procedure using sample measures of the distribution's skewness and tailweight. An empirical relationship between the Winsorizing proportions and the sample skewness and tailweight allows for a ‘continuous’ adaptation of the test statistic to the data. The test has good asymptotic properties, and the small sample results are compared with other populatr parametric, nonparametric, and two–stage tests using Monte Carlo methods. Based on these results, this proposed test procedure is recommended for moderate and larger sample sizes.

Goodness-of-Fit Test-Statistics on Gaussian and Exponential Reliability Data

The paper presents three statisics for testing s-normality and one statistic for testing exponentiality in system reliability data. The distributions of these statistics were approximated by using Monte Carlo simulation. The power of these statistics is investigated with respect to several alternatives. For testing small sample s- normality, two of them, namely, the newly proposed Lilliefors type statistic and sample skewness coefficient have very good power.

Estimating Log Pearson Parameters by Mixed Moments

When applying the method of moments, parameters of log Pearson type 3 distribution (LP) can be estimated either from the moments of real data or from the moments of the logarithmic data, or by mixing the moments of real and log data. Mixed moments methods eliminate the use of sample skewness coefficient which is generally biased. By a previous study, the mixed moments method (MXM1) which uses the mean and variance of real data and the mean of log data was found to possess superior statistical properties, but the estimating procedure was tedious. This paper describes simplified approaches by which LP parameters, through the MXM1 method, can be obtained even by ordinary pocket type calculators. Practical examples are presented to illustrate the application of eight LP fitting methods.

Log Pearson Type 3 Distribution: Method of Mixed Moments

Two new methods of moments (MO) which do not use the sample skewness coefficient (CS) are evolved for fitting log Pearson type 3 distribution (LP). The two current MO methods which use CS are: (Method 1) real data are fit to LP; or (Method 2) logarithmic data are fit to P (Pearson type 3 distribution). By virtue of the combined properties of P and LP, it is shown that: (Method 3) skewness coefficient (γ) of LP can be estimated from the mean of logarithmic data (Y¯); or (Method 4) skewness of coefficient of P (γ) can be estimated from the mean of real data (X¯). The values (Y¯) and (X¯) are unbiased estimates while CS is generally a biased estimate. Monto Carlo experiments have indicated that, as sample size becomes small, the estimates for quantiles of LP are: (1) Systematically biased for Method 1; (2) highly positive biased when (γ) is negative, and unbiased to somewhat positively biased when (γ) is positive for Method 2; (3) unbiased to least biased of the four methods when (γ) is negative, and somewhat negatively biased when (γ) is positive for Method 3; and (4) negatively biased for Method 4.

An Asymptotically Distribution-Free Test for Symmetry versus Asymmetry

Abstract An asymptotically distribution-free procedure is proposed for testing whether a univariate population is symmetric about some unknown value versus a broad class of asymmetric distribution alternatives. The consistency class of the test is discussed and two competing tests are described, one based on the sample skewness, and the other on Gupta's nonparametric procedure. A Monte Carlo study shows that the proposed test is superior to either competitor since it maintains the designated α levels fairly accurately even for sample sizes as small as 20, while displaying good power for detecting asymmetric distributions.

An Asymptotically Distribution-Free Test for Symmetry Versus Asymmetry

Abstract An asymptotically distribution-free procedure is proposed for testing whether a univariate population is symmetric about some unknown value versus a broad class of asymmetric distribution alternatives. The consistency class of the test is discussed and two competing tests are described, one based on the sample skewness, and the other on Gupta's nonparametric procedure. A Monte Carlo study shows that the proposed test is superior to either competitor since it maintains the designated α levels fairly accurately even for sample sizes as small as 20, while displaying good power for detecting asymmetric distributions.

Weather Modification from Cooling Towers: A Test Based on the Distributional Properties of Rainfall

Abstract A statistical technique for the treatment of data from weather modification experiments is presented. This work, a part of the Meteorological Effects of Thermal Energy Releases (METER) Program, is aimed at determining the potential precipitation modification effects of the Bowen Electric Generating Plant near Cartersville, Georgia. For that purpose a network of 49 recording raingages and four recording windsets situated on a square (42 km side) grid was installed in early 1978. The statistical technique utilizes a design which resembles the crossover statistical design used in cloud seeding studies and employs the distributional properties of the sample skewness and kurtosis. Based on the storm data from the above network for the period February 1978-August 1978 the technique demonstrates an anomaly in the target area rainfall with a 90% confidence level. This anomaly is interpreted as a plant-induced modification of the spatial variability of rainfall volume.

Bounds on the Sample Skewness and Kurtosis

For any sample of size N, the absolute value of the sample skewness is bounded by (N – I), and the sample kurtosis is bounded by N. Although these bounds are easily derived, they are not widely known.

Bounds on the Sample Skewness and Kurtosis

Bounds on the Sample Skewness and Kurtosis

An Evaluation of Two Model Specification Techniques for a Lognormal Distribution

Two frequently used statistical estimation techniques are applied to the lognormal distribution with unit shape-parameter. The first technique involves comparing sample estimates of skewness and kurtosis with their corresponding population values in order to determine the suitability of the distribution as a model for a set of empirical data. The sample skewness and kurtosis for samples of moderate size provide biased downward estimates of the population values. This bias can be considered if this technique is applied to a data set by using results presented in this paper. The second technique is probability plotting. The recommended plotting position is (i -0.5)/n. An evaluation of the graphical estimators shows that they are substantially inferior to the best linear unbiased estimators and Blom's estimators. The results are applied to a data set consisting of automotive emissions.

The use of U-statistics for testing normality against nonsymmetric alternatives

SUMMARY We propose a class of tests for normality and investigate a subclass of these tests for testing against nonsymmetric alternatives. Attention is devoted to a convenient linear combination of the order statistics divided by the sample standard deviation and to a second statistic. The null distribution of the first statistic is for most purposes satisfactorily approximated by a normal distribution if the sample size is 20 or greater, and the same is true of the second statistic for sample sizes of 50 or greater. Monte Carlo critical points are also provided. A Monte Carlo power study shows that the two statistics have good power properties relative to the sample skewness coefficient.

Goodness-of-Fit Tests for Correlated Data

Goodness-of-fit tests for stationary processes are a problem of practical importance, e.g. in the analysis of electroencephalographic data. The distribution of the chi-squared statistic under the normal hypothesis is studied by simulation; power is investigated by an inverse filtering procedure for processes which can be well represented by an autoregressive-moving average model. For a second model, consisting of a Gaussian or non-Gaussian signal plus Gaussian noise, sample skewness and kurtosis are suggested as test statistics. The asymptotic normality and the asymptotic variance of these statistics are derived, as well as the behaviour for a broad class of alternatives. The second model is of primary interest in E.E.G.-analysis.

Discrimination among some parametric models

SUMMARY Linear models, with errors that follow the distribution of the logarithm of an F statistic, are shown to include a number of common statistical models as special cases. The error model is transformed and reparameterized to induce regular estimation on the boundary with one or both degrees of freedom infinite. This leads to bivariate score tests for normal, extreme value and logistic special cases as well as an evaluation of these models within a more general framework. In particular, the test for normality is found to reduce to the usual tests based on sample skewness and kurtosis. Sample sizes are given for pairwise discrimination among some specific models. Applications are indicated.

Goodness-of-fit tests for correlated data

SUMMARY Goodness-of-fit tests for stationary processes are a problem of practical importance, e.g. in the analysis of electroencephalographic data. The distribution of the chi-squared statistic under the normal hypothesis is studied by simulation; power is investigated by an inverse filtering procedure for processes which can be well represented by an autoregressive-moving average model. For a second model, consisting of a Gaussian or non-Gaussian signal plus Gaussian noise, sample skewness and kurtosis are suggested as test statistics. The asymptotic normality and the asymptotic variance of these statistics are derived, as well as the behaviour for a broad class of alternatives. The second model is of primary interest in E.E.G..analysis.