Bucket plot: A visual tool for skewness and kurtosis comparisons
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This study introduces the bucket plot, a visual tool to detect skewness and kurtosis in a continuously distributed random variable. The plot can be applied to both moment and centile skewness and kurtosis. The bucket plot is used to detect skewness and kurtosis either in a response variable, or in the residuals from a fitted model as a diagnostic tool by which to assess the adequacy of a fitted distribution to the response variable regarding skewness and kurtosis. We demonstrate the bucket plot in nine simulated skewness and kurtosis scenarios, and the usefulness of the plot is shown in a real-data situation.Keywords:
Kurtosis
Plot (graphics)
Central moment
This study introduces the bucket plot, a visual tool to detect skewness and kurtosis in a continuously distributed random variable. The plot can be applied to both moment and centile skewness and kurtosis. The bucket plot is used to detect skewness and kurtosis either in a response variable, or in the residuals from a fitted model as a diagnostic tool by which to assess the adequacy of a fitted distribution to the response variable regarding skewness and kurtosis. We demonstrate the bucket plot in nine simulated skewness and kurtosis scenarios, and the usefulness of the plot is shown in a real-data situation.
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Central moment
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Rayleigh–Ritz method
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Many financial portfolios are not mean-variance-skewness-kurtosis efficient. We recommend tilting these portfolios in a direction that increases their estimated mean and third central moment and decreases their variance and fourth central moment. The advantages of tilting come at the cost of deviation from the initial optimality criterion. In this paper, we show the usefulness of portfolio tilting applied to the equally-weighted, equal-risk-contribution and maximum diversification portfolios in a UCITS-compliant asset allocation setting.
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In this study, in order to research the effect of skewness on statistical power, four different distributions are handled in power function of Fleishman in which the kurtosis value is 0.00 and the skewness values are 0.75, 0.50, 0.25 and 0.00. In the study, Kolmogorov-Smirnov two sample tests are taken benefit of and the importance level of ? is taken as 0.05. The sample sizes used in the study are the equal and small sample sizes from (2, 2) to (20, 20); additionally, in this study the mean ratio of the samples are taken as 0:0.5, 0:1, 0:1.5, 0:2, 0:2.5 and 0:3. As per the results obtained from the study, when the kurtosis coefficient in the same sample size is maintained fixed and taken as 0, and when the skewness coefficient is increased from 0 to 0.25, big scale of changes do not occur in the statistical power. When the skewness ratio is increased from 0.25 to 0.50; while the ratio of the means are 0:0.5, 0:1 and 0:1.5; decrease is viewed in the statistical power, and when the ratio of the means are 0:2, 0:2.5 and 0:3; increase is viewed in the statistical power. And when the kurtosis coefficient is increased from 0.50 to 0.75 it is viewed that almost in all of the ratios mean the statistical power is increasing. It is concluded that the statistical power increases as the ratio of the means is increasing for all of the sample sizes.
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In the second-order reliability method (SORM), the failure probability is generally estimated based on parabolic approximation of the performance function. In the present paper, the first four moments (i.e., the mean, standard deviation, skewness, and kurtosis) of the second-order approximation of performance functions are obtained using the definition of the probability moment. Based on the recently developed fourth-moment standardization function, an explicit second-order fourth-moment reliability index is proposed for the estimation of failure probability corresponding to both the simple and general parabolic approximations. The simplicity and accuracy of the second-order fourth-moment reliability index is demonstrated using numerical examples. It can be concluded that the proposed method is applicable to the second-order approximation of performance functions with strong nonnormality and is accurate enough to improve the existing SORM in structural reliability analysis with minimal additional computational effort.
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First-order reliability method
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Grain size parameters of 270 seabed sediment samples from the Rongcheng Bay region, Shandong Province, China, were calculated using graphic and moment methods. A comparison between the graphic moment derived parameters (including mean grain size, sorting coefficient, skewness and kurtosis) shows that the mean grain sizes and the sorting coefficients are almost identical. However, the skewness values vary considerably, although there is a significant linear relationship between the two data sets. No clear relationship between graphic and moment kurtosis values is found. The differences in the skewness values are caused by the fact that graphic measures reflect the characteristics of a main part of a sample, whilst moment measures provide an overall pattern of the sample. Hence, the differences in the two methods must be taken into account in sedimentary environment and sediment dynamic studies and historical data comparisons.
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Abstract Studies of kurtosis often concentrate on only symmetric distributions. This paper identifies a process through which the standardized measure of kurtosis based on the fourth moment about the mean can be written in terms of two parts: (i) an irreducible component, about L 4 , which can be seen to occur naturally in the analysis of fourth moments; (ii) terms that depend only on moments of lower order, in particular including the effects of asymmetry attached to the third moment about the mean. This separation of the effect of skewness allows definition of an improved measure of kurtosis. This paper calculates and discusses examples of the new measure of kurtosis for a range of standard distributions.
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It is well known that product moment ratio estimators of the coefficient of variation C ν , skewness γ, and kurtosis κ exhibit substantial bias and variance for the small ( n ≤ 100) samples normally encountered in hydrologic applications. Consequently, L moment ratio estimators, termed L coefficient of variation τ 2 , L skewness τ 3 , and L kurtosis τ 4 are now advocated because they are nearly unbiased for all underlying distributions. The advantages of L moment ratio estimators over product moment ratio estimators are not limited to small samples. Monte Carlo experiments reveal that product moment estimators of C ν and γ are also remarkably biased for extremely large samples ( n ≥ 1000) from highly skewed distributions. A case study using large samples ( n ≥ 5000) of average daily streamflow in Massachusetts reveals that conventional moment diagrams based on estimates of product moments C ν , γ, and κ reveal almost no information about the distributional properties of daily streamflow, whereas L moment diagrams based on estimators of τ 2 , τ 3 , and τ 4 enabled us to discriminate among alternate distributional hypotheses.
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Abstract Taylor series in the sample size are set up for the first four moments of the standard deviation, skewness, kurtosis, and coefficient of variation, the populations being x2 (gamma, Pearson Type III). These moments being out of reach of purely mathematical development, the study proceeds along two independent lines. For the one, simulation methods are used, an attempt being made to fix a cycle length to ensure some stability -this cycle length is pivoted on the fourth moment of the kurtosis, an expression involving sixteenth powers of the basic x2 - random variable. The second line of attack uses the Taylor moment series (which are taken out to at most sixty terms in the total derivatives). An algorithm is used to derive the expectation of a product of powers of elements which consist of non-central sample deviates; there are four of these involved in the kurtosis, three in the skewness, and two in the standard deviation. There is an added parameter for sample size. This expectation of products of powers of sample deviates generates a set of coefficients, each coefficient multiplied by a power of n1 the larger the moment product, the greater is the span of the powers of n1 . If a final moment series is desired to include all contributions up to ns , then at least 2s terms will be required in the Taylor expansion; moreover the series turn out to be divergent, as far as can be judged by the behavior of the terms computed. At this point, since the series are not seen to be onesigned, and since divergence is not too chaotic (as fast as the triple factorial, say), rational fraction sequences are set up to dilute divergence (or accelerate apparent convergence); the approach is often successful but there are problems with small sample sizes and large skewness of the population sampled. Lastly, gross errors in relying on basic asymptotes are noted. The study brings out unusual confluences - computer oriented numerical analysis, distributional theory and approximation, and the power of rational fractions as divergency reducing tools. Keywords: continued fractionsdivergent seriesmomentsmultivariate taylor seriespadt sequencessample momentssimulation cycle
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