LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUpZXF1YWxpdHlHRjQvJStleGVjdXRhYmxlR0Y0Ri8= Joint Cumulants of polykays E. Di Nardo* elvira.dinardo@unito.it https://www.elviradinardo.it Tel: +390116702862 Fax: +390116702878G. Guarino** giuseppe.guarino@rete.basilicata.it * Mathematics Department \342\200\234G. Peano\342\200\235, University of Turin, Turin, Italy**Local Health Authority of Potenza, Italy

Abstract: Given a multivariate random sample, this set of functions returns the joint cumulants of multivariate polykays in terms of joint cumulants of the multivariate population underlying the sample. Multivariate polykays are unbiased estimators with minimum variance of joint cumulant products. These estimators are usually indexed by a list of integer vectors: each vector corresponds to a joint cumulant. When this list reduces to a list of single integers, the multivariate polykay reduces to a univariate polykay. When this list reduces to a single integer vector, the multivariate polykay reduces to a multivariate k-statistic. When this list contains a single integer vector with only one element, the multivariate k-statistic reduces to a univariate k-statistic. Application Areas/Subject: Combinatorics & algebraic methods in statistics.Keyword: umbral calculus, symmetric polynomial, set partition, multiset, cumulant, k-statistic, polykay.See Also: Maple algorithm [1] and [2]Remark: k-statistics, polykays and their multivariate generalization are commonly given in terms of power 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 n the sample size.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi9RMCZBcHBseUZ1bmN0aW9uO0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHRkIvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVS8lKWVxdWFsaXR5R0ZCLUkobWZlbmNlZEdGJDYkLUYjNistRiw2JVEpY29tYmluYXRGJ0Y2RjktRj02LlEiLEYnRkNGRS9GSEY4RklGS0ZNRk9GUUZTL0ZXUSwwLjMzMzMzMzNlbUYnRlgtRiw2JVEqcGFydGl0aW9uRidGNkY5RlxvLUYsNiVRLG11bHRpbm9taWFsRidGNkY5RlxvLUYsNiVRKnN0aXJsaW5nMkYnRjZGOUZARkNGQ0ZARkNGK0ZARkNGK0ZARkM=

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The polyk function handles four different algorithms. Depending on the input parameters, univariate or multivariate polykays as well as univariate or multivariate k-statistics are expressed in terms of power sums. Specifically, if the input is a single vector containing a single integer, univariate k-statistics are returned; if the input is a single vector containing more than one integer, multivariate k-statistics are returned; if the input is a list of vectors each containing a single integer, then univariate polykays are returned; if the input is a list of vectors, each containing more than one integer, multivariate polykays are returned. All these functions are in the MAPLE worksheet [1] and have been included here for user convenience. For the multivariate case, the algorithm listing all subdivisions of a multiset has also been included. This algorithm is given and discussed in the MAPLE worksheet [2]. For details on the procedures and mathematical backgroud see [3 - 6].

The n-th k-statistic LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYmLUYsNiZRIm5GJ0YvRjJGNUYyLyUwZm9udF9zdHlsZV9uYW1lR1ErMkR+Q29tbWVudEYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn is the unique symmetric unbiased estimator of the n-th cumulant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEoJmthcHBhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYlUSJuRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjU= of a given statistical distribution, that is E[LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRIm5GJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ==] = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEoJmthcHBhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYlUSJuRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjU=where E denotes the expectation. They are expressed in terms of power sums involving the random variables of a random sample. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 Exampe: The k-statistic of order 3 is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEja3NGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1JI21uR0YkNiRRIjNGJy9GM1Enbm9ybWFsRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJ0Y+Rj5GQEY+

KiosKComSSJuRzYiIiIjJkkiU0dGJjYjIiIkIiIiRiwqKEYlRiwmRik2I0YsRiwmRik2I0YnRiwhIiQqJEYuRitGJ0YsRiUhIiIsJkYlRixGNEYsRjQsJkYlRiwhIiNGLEY0

The symmetric statistic 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 is an unbiased estimator of the cumulant product LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEoJmthcHBhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiYtRi82JlEickYnL0YzUSV0cnVlRidGNS9GOFEnaXRhbGljRidGNS8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGJ0Y3LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYuUTEmSW52aXNpYmxlVGltZXM7RidGNy8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmduLyUpZXF1YWxpdHlHRjQtRiw2JUYuLUYjNiYtRi82JlEic0YnRj9GNUZBRjVGQ0Y3RkZGNUZDRjc=..., that is E[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] = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JJW1zdWJHRiQ2JS1GLDYlUSgma2FwcGE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUYsNiVRInJGJy9GOFEldHJ1ZUYnL0Y7USdpdGFsaWNGJ0Y6LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYuUTEmSW52aXNpYmxlVGltZXM7RidGOi8lJmZlbmNlR0Y5LyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmduLyUpZXF1YWxpdHlHRjktRjI2JUY0LUYjNiQtRiw2JVEic0YnRkJGREY6RkZGOkYrRjo=...where 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 is the r-th cumulant. These statistics are called polykays or generalized k-statistics. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZELUkjbWlHRiQ2JVEqbWFrZUNUUl9zRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEjOj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC8lKWVxdWFsaXR5R0Y9LUY2NjBRJXByb2NGJy8lJWJvbGRHRjEvRjNRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRldGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZlbkZPLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEiTkYnRi9GMkY5RjktRjY2LlEifkYnRjlGO0Y+RkBGQkZERkZGSEZaRmZuRk9GX28tSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGZW4vJSZkZXB0aEdGZ28vJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRmNvNiZGZW9GaG9Gam8vRl1wUSVhdXRvRictRjY2MFEmbG9jYWxGJ0ZURlZGWEY7Rj5GQEZCRkRGRkZIRlpGZm5GT0Zfby1GLDYlUSJpRidGL0YyLUY2Ni5RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkhGWi9GTlEsMC4zMzMzMzMzZW1GJ0ZPLUYsNiVRInZGJ0YvRjJGaXAtRiw2JVEieEYnRi9GMkZpcC1GLDYlUSJyRidGL0YyLUY2Ni5RIjtGJ0Y5RjtGXHFGQEZCRkRGRkZIRlpGTUZPLUZjbzYmRmVvL0Zpb1EmMC41ZW1GJ0Zqby9GXXBRMWZpcnN0cHJvY25ld2xpbmVGJy1GaG42Ji1GIzYmLUYsNiNRIUYnLUYjNiYtRiw2JVEkc2VxRidGL0YyLUY2Ni9RMCZBcHBseUZ1bmN0aW9uO0YnLyUrZXhlY3V0YWJsZUdGPUY5RjtGPkZARkJGREZGRkhGWkZmbkZPLUZobjYkLUYjNi0tSSVtc3ViR0YkNiUtRiw2JVEia0YnRi9GMi1GIzYkRmZwRjkvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GNjYuUSI9RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUZPLUYsNiVRJGFkZEYnRi9GMi1GaG42JC1GIzYpLUkmbWZyYWNHRiQ2KC1GIzYsLUYsNiVRJG11bEYnRi9GMi1GaG42JC1GIzYoLUZnczYlLUYsNiVRJSZtdTtGJy9GMEY9RjktRiM2JEZlcUY5Rl50RmlwRmVxRmF0Rl9xRjlGOS1GNjYuUScmc2RvdDtGJ0Y5RjtGPkZARkJGREZGRkhGWkZmbkZPLUklbXN1cEdGJDYlLUZobjYkLUYjNiUtRjY2LlEoJm1pbnVzO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZdd0ZPLUkjbW5HRiQ2JFEiMUYnRjlGOUY5LUYjNictRiw2JVElbm9wc0YnRi9GMi1GaG42JC1GIzYkRl9xRjlGOUZpdkZfd0Y5LyUxc3VwZXJzY3JpcHRzaGlmdEdGYHRGX3YtRmhuNiQtRiM2J0Zld0Zody1GNjYuUSp+Jm1pbnVzO35GJ0Y5RjtGPkZARkJGREZGRkhGWkZmbkZPRl93RjlGOS1GNjYuUSIhRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnL0ZORml4Rk9GZnBGZXhGOS1GIzYmRmB1LUZobjYkLUYjNi0tRmhuNiQtRiM2JEZicUY5RjktRmN2NiVGZXgtRiM2JS1GLDYlUSpudW1ib2NjdXJGJ0YvRjItRmhuNiQtRiM2JkZfcUZpcEZicUY5RjlGOUZceEZfdkZpeUZcekZleEZpcEZicUZhdC1GaG42Ji1GIzYmRmVyLUYjNiYtRiw2JVEjb3BGJ0YvRjJGXXNGaHdGOUZlckY5RjkvJSVvcGVuR1EifGZyRicvJSZjbG9zZUdRInxockYnRjlGOUZlckY5LyUubGluZXRoaWNrbmVzc0dGYncvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGY1tsLyUpYmV2ZWxsZWRHRj1GaXBGX3FGYXQtRiw2JVEqcGFydGl0aW9uRidGL0YyLUZobjYkRlx0RjlGOUY5RmlwRmZwRmF0Rl93LUY2Ni5RIy4uRidGOUY7Rj5GQEZCRkRGRkZIRlx3RmZuRk9GXG9GOUY5RjlGZXJGOUY5L0ZqelEiW0YnL0ZdW2xRIl1GJ0Zlci1GY282JkZlb0ZdckZqby9GXXBRNmRlY3JlYXNlaW5kZW50bmV3bGluZUYnLUY2NjBRJGVuZEYnRlRGVkZYRjtGPkZARkJGREZGRkhGWkZmbkZPRl9vRlEtRjY2LlEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk1GT0Zib0Zib0Zfb0Zib0ZfcEZlckZgc0Y5LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZlcC1JI21pR0YkNiVRJ21ha2VNdUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIzo9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwvJSllcXVhbGl0eUdGPS1GNjYwUSVwcm9jRicvJSVib2xkR0YxL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0ZXRjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GZW5GTy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiNRIUYnLyUrZXhlY3V0YWJsZUdGPUY5RjlGXG8tSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuNWVtRicvJSZkZXB0aEdGZm8vJSpsaW5lYnJlYWtHUTFmaXJzdHByb2NuZXdsaW5lRictRjY2MFEmbG9jYWxGJ0ZURlZGWEY7Rj5GQEZCRkRGRkZIRlpGZm5GTy1GNjYuUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRlpGZm5GTy1GLDYlUSJ1RidGL0YyLUY2Ni5RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkhGWi9GTlEsMC4zMzMzMzMzZW1GJ0ZPLUYsNiVRInZGJ0YvRjJGaHAtRiw2JVEiTkYnRi9GMkZocC1GLDYlUSNldUYnRi9GMkZocC1GLDYlUSJpRidGL0YyRmhwLUYsNiVRInhGJ0YvRjJGaHAtRiw2JVEia0YnRi9GMi1GNjYuUSI7RidGOUY7RltxRkBGQkZERkZGSEZaRk1GT0Zhb0ZhcUY1LUYsNiVRJGFkZEYnRi9GMi1GaG42JC1GIzYoRmdxRmhwLUYjNidGZ3EtRjY2LlEiPUYnRjlGO0Y+RkBGQkZERkZGSEZKRk1GTy1GLDYlUSVhcmdzRidGL0YyRl9vRjlGXG9GX29GOUY5RmByRmFvRmRxRjUtRmhuNiYtRiM2J0Zcby1GIzYnLUYsNiVRJHNlcUYnRi9GMi1GNjYvUTAmQXBwbHlGdW5jdGlvbjtGJ0Zfb0Y5RjtGPkZARkJGREZGRkhGWkZmbkZPLUZobjYkLUYjNi0tSSVtc3ViR0YkNiUtRiw2JVEnJiM5NTY7RicvRjBGPUY5LUYjNiVGZ3FGL0YyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGXHMtSSNtbkdGJDYkUSIxRidGOUZocEZncUZcc0ZedS1GNjYuUSMuLkYnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJ0ZmbkZPRmFxRl9vRjlGOUZfb0Y5RlxvRl9vRjlGOS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZgckZhby1GNjYwUSNpZkYnRlRGVkZYRjtGPkZARkJGREZGRkhGWkZmbkZPLUZibzYmRmRvRmdvRmpvL0ZdcFElYXV0b0YnLUYsNiVRJm5hcmdzRidGL0YyRlxzRl51RmB2LUY2NjBRJXRoZW5GJ0ZURlZGWEY7Rj5GQEZCRkRGRkZIRlpGZm5GTy1GYm82JkZkb0Znb0Zqby9GXXBRNmluY3JlYXNlaW5kZW50bmV3bGluZUYnRmVwRjUtRmhuNiYtRiM2J0Zcby1GIzYnRmhzRlt0LUZobjYkLUYjNiotRmhuNiYtRiM2JS1GaG42Ji1GIzYlRmpxRl9vRjlGOUZndUZqdUZfb0Y5RjlGZ3VGanVGaHBGanFGXHMtRiw2JVEqcGFydGl0aW9uRidGL0YyLUZobjYkLUYjNiUtRmN0NiVGX3MtRiM2JUZedUYvRjJGW3VGX29GOUY5Rl9vRjlGOUZfb0Y5RlxvRl9vRjlGOUZndUZqdUZcby1GYm82JkZkb0Znb0Zqby9GXXBRNmRlY3JlYXNlaW5kZW50bmV3bGluZUYnLUY2NjBRJWVsc2VGJ0ZURlZGWEY7Rj5GQEZCRkRGRkZIRlpGZm5GT0ZqdkZlcEY1LUZobjYmLUYjNidGXG8tRiM2J0Zoc0ZbdC1GaG42JC1GIzYoLUZobjYmLUYjNidGXG8tRiM2J0Zoc0ZbdC1GaG42JC1GIzYqRmpxRmhwRmpxRlxzRmB4LUZobjYkLUYjNiVGZ3FGX29GOUY5Rl9vRjlGOUZfb0Y5RlxvRl9vRjlGOUZndUZqdUZocEZqckZcb0Zfb0Y5RjlGX29GOUZcb0Zfb0Y5RjlGZ3VGanVGXG9GW3ktRjY2MFEnZW5kfmlmRidGVEZWRlhGO0Y+RkBGQkZERkZGSEZaRmZuRk9GYHJGYW9GZXAtRjY2LlEqJmNvbG9uZXE7RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUZPLUYsNiVRI29wRidGL0YyLUZobjYkLUYjNidGXG8tRiM2J0ZjckZbdC1GaG42JC1GIzYxLUYsNiVRJG11bEYnRi9GMi1GaG42JC1GIzYyLUklbXN1cEdGJDYlLUZobjYkLUYjNiYtRjY2LlEoJiM4NzIyO0YnRjlGO0Y+RkBGQkZERkZGSEZldS9GTkZmdUZPRl51Rl9vRjlGOS1GIzYoLUYsNiVRJW5vcHNGJ0YvRjItRmhuNiRGaXRGOS1GNjYuUSgmbWludXM7RidGOUY7Rj5GQEZCRkRGRkZIRmV1Rl5dbEZPRl51Ri9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHRl11LUY2Ni5RJyZzZG90O0YnRjlGO0Y+RkBGQkZERkZGSEZaRmZuRk8tRmhuNiQtRiM2J0Zcby1GIzYnRlxvLUYjNidGYV1sRlt0RmZ6Rl9vRjktRjY2LlEqfiYjODcyMjt+RidGOUY7Rj5GQEZCRkRGRkZIRlpGZm5GT0ZedUY5RlxvRl9vRjlGOS1GNjYuUSIhRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnL0ZORl1fbEZPRltebEZdXGwtRmhuNiQtRiM2KS1GY3Q2JUZldC1GIzYlRl1yRi9GMkZbdUZocEZdckZcc0ZncUZfb0Y5RjlGW15sLUYsNiVRJ2NvdW50UEYnRi9GMkZmekZocEZncUZcc0ZecUZfb0Y5RjlGW15sLUYsNiVRI2ZkRidGL0YyLUZobjYkLUYjNilGXG8tRiM2JkZhcUZmXmxGXnVGOUZocC1GIzYnRmNyRlt0LUZobjYkLUYjNilGXG8tRiM2J0ZhXWxGW3QtRmhuNiQtRiM2JUZdckZfb0Y5RjlGX29GOUZocC1GIzYnRl1yRlxzRl5xRl9vRjlGXG9GX29GOUY5Rl9vRjlGXG9GX29GOUY5RltebEZnX2wtRmVcbDYlLUZobjYkLUYjNiUtRmhuNiYtRiM2JkZocy1GaG42JC1GIzYqRmBbbEZmekZocEZncUZcc0ZecUZfb0Y5RjlGX29GOUY5Rmd1Rmp1Rl9vRjlGOS1GIzYmLUY2Ni5RKiZ1bWludXMwO0YnRjlGO0Y+RkBGQkZERkZGSEZldUZeXWxGT0ZedUYvRjJGaV1sRmhwRl5xRlxzLUYsNiVRJWBpZmBGJ0YvRjItRmhuNiQtRiM2KkZcby1GIzYnRmR2RlxzRl51Rl9vRjlGaHAtRiM2Jy1GLDYlUSVjb21iRidGL0YyRlt0LUZobjYkLUYjNilGZXBGaHBGXnVGaHAtRmhuNiZGam5GOUZndUZqdUZfb0Y5RjlGX29GOUZocC1GaG42Ji1GIzYnRlxvRl1jbEZcb0Zfb0Y5RjlGZ3VGanVGX29GOUY5Rl9vRjlGOUZfb0Y5RlxvRl9vRjlGOUZgckZicC1GYm82JkZkby9GaG9GZW5Gam8vRl1wUShuZXdsaW5lRidGYnBGYnBGYnBGYnBGYnBGYnBGYnBGaHMtRmhuNiQtRiM2LS1GLDYlUSlzaW1wbGlmeUYnRi9GMi1GaG42JC1GIzYlLUkmbWZyYWNHRiQ2KEZecS1GIzYmLUYsNiVRJWV2YWxGJ0YvRjItRmhuNiQtRiM2J0ZecUZocEZkcUYvRjJGOUYvRjIvJS5saW5ldGhpY2tuZXNzR0ZhdS8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZcZmwvJSliZXZlbGxlZEdGPUZfb0Y5RjlGXHNGYWVsLUZobjYkLUYjNidGXnFGaHBGZHFGX29GOUY5RmhwRl5xRlxzRmVwRl9vRjlGOUZcb0ZcZGxGYnAtRjY2MFEkZW5kRidGVEZWRlhGO0Y+RkBGQkZERkZGSEZaRmZuRk9GYnBGUS1GNjYuUSI6RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUZPRlxkbEZcZGxGXGRsRmJwRlxkbC1GYm82JkZkb0ZeZGxGam9GYnZGXG9GX29GOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZPLUkjbWlHRiQ2JVElY29tYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIzo9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwvJSllcXVhbGl0eUdGPS1GNjYwUSVwcm9jRicvJSVib2xkR0YxL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0ZXRjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GZW5GTy1JKG1mZW5jZWRHRiQ2JC1GIzYpLUYsNiVRIlZGJ0YvRjItRjY2LlEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSEZaL0ZOUSwwLjMzMzMzMzNlbUYnRk8tRiw2JVEkcHRyRidGL0YyRl9vLUYsNiVRIllGJ0YvRjIvJStleGVjdXRhYmxlR0Y9RjlGOS1GLDYjUSFGJy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC41ZW1GJy8lJmRlcHRoR0ZlcC8lKmxpbmVicmVha0dRMWZpcnN0cHJvY25ld2xpbmVGJy1GNjYwUSZsb2NhbEYnRlRGVkZYRjtGPkZARkJGREZGRkhGWkZmbkZPLUY2Ni5RIn5GJ0Y5RjtGPkZARkJGREZGRkhGWkZmbkZPLUYsNiVRIkxGJ0YvRjItRjY2LlEiO0YnRjlGO0Zib0ZARkJGREZGRkhGWkZNRk8tRjY2MFEjaWZGJ0ZURlZGWEY7Rj5GQEZCRkRGRkZIRlpGZm5GT0ZhcUZlby1GNjYuUSI9RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUZPLUYsNiVRJW5vcHNGJ0YvRjItRmhuNiQtRiM2JUZcb0ZbcEY5RjktRjY2LlEiK0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZbc0ZPLUkjbW5HRiQ2JFEiMUYnRjktRmFwNiZGY3BGZnBGaXAvRlxxUSVhdXRvRictRjY2MFEldGhlbkYnRlRGVkZYRjtGPkZARkJGREZGRkhGWkZmbkZPLUZhcDYmRmNwRmZwRmlwL0ZccVE2aW5jcmVhc2VpbmRlbnRuZXdsaW5lRictRjY2MFEncmV0dXJuRidGVEZWRlhGO0Y+RkBGQkZERkZGSEZaRmZuRk9GYXNGaG8tRmFwNiZGY3BGZnBGaXAvRlxxUTZkZWNyZWFzZWluZGVudG5ld2xpbmVGJy1GNjYwUSdlbmR+aWZGJ0ZURlZGWEY7Rj5GQEZCRkRGRkZIRlpGZm5GT0ZncUZgcC1GLDYlUSRzZXFGJ0YvRjItRmhuNiQtRiM2KkYrLUZobjYkLUYjNilGXG9GX28tRiM2JkZlb0ZnckZdc0Y5Rl9vLUZobjYmLUYjNihGXXAtRiM2Jy1GLDYlUSNvcEYnRi9GMi1GNjYvUTAmQXBwbHlGdW5jdGlvbjtGJ0ZbcEY5RjtGPkZARkJGREZGRkhGWkZmbkZPLUZobjYkLUYjNiVGaG9GW3BGOUY5RltwRjlGX29GZHFGW3BGOUY5LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRltwRjlGOUZfb0ZkcUZdci1JJW1zdWJHRiQ2JUZcby1GIzYlRmVvRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRltwRjlGOUZdcEZfdC1GNjYwUSRlbmRGJ0ZURlZGWEY7Rj5GQEZCRkRGRkZIRlpGZm5GT0ZhcUZRLUY2Ni5RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNRk8tRmFwNiZGY3AvRmdwRmVuRmlwL0ZccVEobmV3bGluZUYnRmd3Rmd3RmFxRmd3LUZhcDYmRmNwRml3RmlwRmNzRl1wRltwRjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZHLUkjbWlHRiQ2JVEnY291bnRQRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwvJSllcXVhbGl0eUdGPS1GNjYuUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORlVGTy1GNjYwUSVwcm9jRicvJSVib2xkR0YxL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0ZnbkY7Rj5GQEZCRkRGRkZIRlRGVkZPLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEidUYnRi9GMi8lK2V4ZWN1dGFibGVHRj1GOUY5LUYsNiNRIUYnLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHRlUvJSZkZXB0aEdGXHAvJSpsaW5lYnJlYWtHUShuZXdsaW5lRidGUUZRRlFGUUZRRlFGUUZRRlEtRjY2MFEmbG9jYWxGJ0ZaRmZuRmhuRjtGPkZARkJGREZGRkhGVEZWRk9GUS1GLDYlUSJ4RidGL0YyLUY2Ni5RIjtGJ0Y5RjsvRj9GMUZARkJGREZGRkhGVEZNRk9GZ29GUUZRRlFGUUZRRlFGUUZRLUkmbWZyYWNHRiQ2KC1GIzYpRmRvLUZobzYmRmpvL0ZecFEmMC41ZW1GJ0ZfcC9GYnBRMWZpcnN0cHJvY25ld2xpbmVGJy1GLDYlUSRhZGRGJ0YvRjItRltvNiQtRiM2KEZncC1GNjYuUSIsRidGOUY7Rl1xRkBGQkZERkZGSEZUL0ZOUSwwLjMzMzMzMzNlbUYnRk8tRiM2J0ZncC1GNjYuUSI9RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUZPRl9vRmJvRjlGZG9GYm9GOUY5LUY2Ni5RIiFGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMTExMTExMWVtRicvRk5GXnNGT0Zib0Y5LUYjNiYtRiw2JVEkbXVsRidGL0YyLUZbbzYkLUYjNi0tSSVtc3VwR0YkNiVGZ3AtRiM2Ji1GLDYlUSpudW1ib2NjdXJGJ0YvRjItRltvNiQtRiM2J0Zfb0ZgckZncEYvRjJGOUYvRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY2Ni5RJyZzZG90O0YnRjlGO0Y+RkBGQkZERkZGSEZURlZGT0ZedEZhdEZqckZgckZncEZnci1GW282Ji1GIzYmLUYsNiVRI29wRidGL0YyLUZbbzYkLUYjNiRGX29GOUY5RmRvRjlGOS8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRidGL0YyRjlGL0YyLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zhdi8lKWJldmVsbGVkR0Y9RmRvRmdvLUZobzYmRmpvRl1wRl9wL0ZicFElYXV0b0YnLUY2NjBRKWVuZH5wcm9jRidGWkZmbkZobkY7Rj5GQEZCRkRGRkZIRlRGVkZPLUY2Ni5RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNRk9GYm9GOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZnbi1JI21pR0YkNiNRIUYnLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjBlbUYnLyUmZGVwdGhHRjQvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRjA2JkYyRjVGOC9GO1ElYXV0b0YnRi8tSSNtb0dGJDYuUSJ+RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZKLyUpc3RyZXRjaHlHRkovJSpzeW1tZXRyaWNHRkovJShsYXJnZW9wR0ZKLyUubW92YWJsZWxpbWl0c0dGSi8lJ2FjY2VudEdGSi8lJ2xzcGFjZUdGNy8lJ3JzcGFjZUdGNy8lKWVxdWFsaXR5R0ZKLUYsNiVRI3BzRicvJSdpdGFsaWNHUSV0cnVlRicvRkZRJ2l0YWxpY0YnLUZCNi5RKiZjb2xvbmVxO0YnRkVGSEZLRk1GT0ZRRlNGVS9GWFEsMC4yNzc3Nzc4ZW1GJy9GWkZjb0ZlbkZBLUZCNjBRJXByb2NGJy8lJWJvbGRHRlxvL0ZGUSVib2xkRicvJStmb250d2VpZ2h0R0ZbcEZIRktGTUZPRlFGU0ZVRldGWUZlbi1JKG1mZW5jZWRHRiQ2JC1GIzYlRisvJStleGVjdXRhYmxlR0ZKRkVGRUYrRi8tRjA2JkYyL0Y2USYwLjVlbUYnRjgvRjtRMWZpcnN0cHJvY25ld2xpbmVGJy1GQjYwUSZsb2NhbEYnRmhvRmpvRlxwRkhGS0ZNRk9GUUZTRlVGV0ZZRmVuRkEtRiw2JVEidUYnRmpuRl1vLUZCNi5RIixGJ0ZFRkgvRkxGXG9GTUZPRlFGU0ZVRlcvRlpRLDAuMzMzMzMzM2VtRidGZW4tRiw2JVEidkYnRmpuRl1vRmFxLUYsNiVRIk5GJ0ZqbkZdb0ZhcS1GLDYlUSJ4RidGam5GXW8tRkI2LlEiO0YnRkVGSEZkcUZNRk9GUUZTRlVGV0Zkb0ZlbkZlcEZqcS1GQjYuUSM6PUYnRkVGSEZLRk1GT0ZRRlNGVUZib0Zkb0Zlbi1GLDYlUSRhZGRGJ0ZqbkZdby1GX3A2JC1GIzYoRl1yRmFxLUYjNidGXXItRkI2LlEiPUYnRkVGSEZLRk1GT0ZRRlNGVUZib0Zkb0Zlbi1GLDYlUSVhcmdzRidGam5GXW9GY3BGRUYrRmNwRkVGRUZgckZlcEZecUZjci1GLDYlUSdleHBhbmRGJ0ZqbkZdby1GX3A2JC1GIzYnRistRiM2Jy1GLDYlUSVldmFsRidGam5GXW8tRkI2L1EwJkFwcGx5RnVuY3Rpb247RidGY3BGRUZIRktGTUZPRlFGU0ZVRldGWUZlbi1GX3A2JC1GIzYpRistRiM2Jy1GLDYlUSttYWtlVGFiX3NzRidGam5GXW9GYXQtRl9wNiQtRiM2JUZqcUZjcEZFRkVGY3BGRUZhcS1GIzYnLUYsNiVRKm1ha2VDVFJfc0YnRmpuRl1vRmF0Rl11RmNwRkVGK0ZjcEZFRkVGY3BGRUYrRmNwRkVGRUZgckZlcEZncUZjckZlcy1GX3A2JC1GIzYnRistRiM2J0ZedEZhdC1GX3A2JC1GIzYnRl5xRmFxLUZfcDYmLUYjNilGKy1GIzYnLUYsNiVRJ21ha2VNdUYnRmpuRl1vRmF0LUZfcDYkLUYjNiVGYnNGY3BGRUZFRmNwRkVGYXEtRiM2Jy1GLDYlUScmIzk1NjtGJy9GW29GSkZFRl9zLUkjbW5HRiQ2JFEiMEYnRkVGY3BGRUYrRmNwRkVGRS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZjcEZFRkVGY3BGRUYrRmNwRkVGRUZgckZlcC1JJm1mcmFjR0YkNigtRiM2JkZmci1GX3A2JC1GIzYrLUklbXN1YkdGJDYlRl1yLUYjNiUtRmR3NiRRIjFGJ0ZFRmpuRl1vLyUvc3Vic2NyaXB0c2hpZnRHRmZ3LUZCNi5RJyZzZG90O0YnRkVGSEZLRk1GT0ZRRlNGVUZXRllGZW4tRmd4NiVGXXItRiM2JS1GZHc2JFEiMkYnRkVGam5GXW9GXnlGYXFGXXJGX3NGZ3FGY3BGRUZFRmNwRkUtRiM2J0YrLUYjNiUtRiw2JVEkbXVsRidGam5GXW8tRl9wNiQtRiM2Ky1GLDYlUSJuRidGam5GXW8tRkI2LlEqfiZtaW51czt+RidGRUZIRktGTUZPRlFGU0ZVRldGWUZlbkZdckZhcUZdckZfcy1GIzYnRmN3LUZCNi5RIy4uRidGRUZIRktGTUZPRlFGU0ZVL0ZYUSwwLjIyMjIyMjJlbUYnRllGZW4tRiM2JkZqcUZoekZbeUZFRitGRUYrRkVGRUZFRitGam5GXW8vJS5saW5ldGhpY2tuZXNzR0ZdeS8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZoW2wvJSliZXZlbGxlZEdGSkYrRi9GPUYvRkEtRkI2MFEkZW5kRidGaG9Gam9GXHBGSEZLRk1GT0ZRRlNGVUZXRllGZW5GQUZlby1GQjYuUSI6RidGRUZIRktGTUZPRlFGU0ZVRmJvRmRvRmVuRi9GQUYvRj1GQUZjcEZF 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KiosKCokJkkiU0c2IjYjIiIiIiIkISIiKigsJkYpRilJIm5HRidGKUYpRiVGKSZGJjYjIiIjRilGKSomRi5GKSZGJjYjRipGKUYrRilGLkYrLCZGLkYpRitGKUYrLCZGLkYpISIjRilGKw==

The makeTab function lists all subdivisions of a multiset (for a thorough discussion see [2]). With multivariate k-statistics and/or multivariate polykays, this function speeds up the overall procedure (see [7]). For details on the procedure and mathematical background see also [3] and [4]. 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The multivariate k-statistic is the unique symmetric unbiased estimator of the n-th joint cumulant 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 of a given multivariate statistical distribution, that is E[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] = 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 where E denotes the expectation. They are expressed in terms of multivariate power sums involving the random vectors of a random sample. 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is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEja21GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Jy1JI21uR0YkNiRRIjJGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYuUSIsRidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy8lKWVxdWFsaXR5R0ZGLUY7NiRRIjFGJ0Y+LyUrZXhlY3V0YWJsZUdGRkY+Rj4tRkE2LlEiO0YnRj5GREZHRklGS0ZNRk9GUUZTL0ZXUSwwLjI3Nzc3NzhlbUYnRllGaG5GPg==

KiosKiomSSJuRzYiIiIjJkkiU0dGJjYkRiciIiJGK0YrKihGJUYrJkYpNiQiIiFGK0YrJkYpNiRGJ0YvRishIiIqKEYlRismRik2JEYrRi9GKyZGKTYkRitGK0YrISIjKiZGLUYrRjRGJ0YnRitGJUYyLCZGJUYrRjJGK0YyLCZGJUYrRjhGK0Yy

Note that the power sums in (3.4.1) are: 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 where (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) refers to the population LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn and (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) refers to the population LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2L1EiLkYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTi8lKWVxdWFsaXR5R0Y0LUY5Ni9RIn5GJ0YyRjxGPkZARkJGREZGRkhGSkZMRk9GUUYvRjJGNQ==

The multivariate polykay 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 is an unbiased estimator of the cumulant product 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..., that is E[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] = 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... where E denotes the expectation and 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 are joint cumulants of order (r,s,...) and (i,j,..) respectively. 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Note that the vectors in the input list must be the same length: if any vector is "smaller," the function adds 0 until it reaches the correct length.

This function allows us to recall all functions for generate k-statistics, polykays and their multivariate generalizzations. The input is the following:- for generate k-statistics LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInJGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== the parameter is: [ r ]- for generate polykays LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRJHI7c0YnRjJGNS9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y9 the parameter is: [ r ], [ s ] - for generate multivariate k-statistics 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 the parameter is: [ r, s ] - for generate multivariate polykays LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRKXIsczt+dSx2RidGMkY1L0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj0= the parameter is: [ r, s ], [ u, v] 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 Esample: The k-statistic of order 3 is 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

KiosKComSSJuRzYiIiIjJkkiU0dGJjYjIiIkIiIiRiwqKEYlRiwmRik2I0YsRiwmRik2I0YnRiwhIiQqJEYuRitGJ0YsRiUhIiIsJkYlRixGNEYsRjQsJkYlRiwhIiNGLEY0

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

KiosKCokJkkiU0c2IjYjIiIiIiIkISIiKigsJkYpRilJIm5HRidGKUYpRiVGKSZGJjYjIiIjRilGKSomRi5GKSZGJjYjRipGKUYrRilGLkYrLCZGLkYpRitGKUYrLCZGLkYpISIjRilGKw==

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is 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

KiosKiomSSJuRzYiIiIjJkkiU0dGJjYkRiciIiJGK0YrKihGJUYrJkYpNiQiIiFGK0YrJkYpNiRGJ0YvRishIiIqKEYlRismRik2JEYrRi9GKyZGKTYkRitGK0YrISIjKiZGLUYrRjRGJ0YnRitGJUYyLCZGJUYrRjJGK0YyLCZGJUYrRjhGK0Yy

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

KiosKiooSSJuRzYiIiIiJkkiU0dGJjYkRiciIiFGJyZGKTYkRidGJ0YnRicqJiZGKTYkRitGJ0YnRigiIiMhIiIqJkYlRicmRik2JEYxRidGJ0YyKiZGL0YnJkYpNiRGMUYrRidGJ0YnRiVGMiwmRiVGJ0YyRidGMiwmRiVGJyEiI0YnRjI=

The following functions allow the random variables of a random sample to be replaced with corresponding numerical observations. The powS function returns the sum of the r-th powers of the observations. 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 Example 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

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKCZJIlhHRic2JEkiaUdGJyIiIiIiJiZGKzYkRi0iIiMiIiQmRis2JEYtRjNGLi9GLTtGLkkibkdGJw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVElcG93U0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUkjbW5HRiQ2JFEiOUYnL0YzUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnRj5GPi1JI21vR0YkNi5RIjtGJ0Y+LyUmZmVuY2VHRkIvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSllcXVhbGl0eUdGQkZARj4=

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJCZJIlhHRic2I0kiaUdGJyIiKi9GLTsiIiJJIm5HRic=

The LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlEnbnBvbHlrRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=computes a numerical value of a k-statistic or a polykay (including the multivariate cases), replacing the random variables of the random sample with the corresponding numerical observations. The parameters are the following:- for generate k-statistics LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInJGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== the parameters are: [ r ], [ [ n1, n2, ...] ] - for generate polykays LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRJHI7c0YnRjJGNS9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y9 the parameters are: [ [ r ], [ s ] ], [ [ n1, n2, ...] ] - for generate multivariate k-statistics 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 the parameters are: [ [ r , s ] ], [ [ n1a, n2a], [ n1b, n2b] , ... ] - for generate multivariate polykays LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRKXIsczt+dSx2RidGMkY1L0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj0= the parameters are: [ [ r , s ], [ u , v] ], [ [ n1a, n2a], [ n1b, n2b] , ... ] 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 Examples: suppose to have the following (univariate) numerical sample 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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVkYXRhR0YoNyM3QCQiJU07ISIjJCIldzVGMiQiJSU9IkYyJCIlYjhGMiQiJSZlIkYyJCIlPz1GMiQiJF4oRjIkIiVBNUYyJCIlXzdGMiQiJW85RjIkIiUzO0YyJCIlVj5GMiQiJDcpRjIkIiU/NkYyJCIlJkgiRjIkIiV4OUYyJCIlJG8iRjIkIiUhKT5GMiQiJGIpRjIkIiVlNkYyJCIlNTdGMiQiJS06RjJGUSQiJSlwIkYyJCIlIyo+RjIkIiRaKkYyJCIlbzZGMiQiJVQ4RjIkIiVOOkYyJCIlNj5GMjcjRi4=

An estimation of the mean (k-statistic of first order) is 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

JCIrbm07LTkhIik=

An estimation of the variance (k-statistic of second order) is 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

JCIrYXArbDchIik=

An estimation of the skewness is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== / LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUkobXN1YnN1cEdGJDYnLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJ0ZDLUZANiRRIjNGJ0ZDLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lL3N1YnNjcmlwdHNoaWZ0R0ZKRkNGQw== 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

JCErLkFCO0shIzY=

An estimation of the kurtosis is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiNEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== / LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JKG1zdWJzdXBHRiQ2Jy1GLDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSIyRicvRjtRJ25vcm1hbEYnRkNGPy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGR0ZDRitGQw== 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

JCErLEYkSCYpKSEjNQ==

The estimation of the product 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 is 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

JCErQDE0YzohIik=

Examples: suppose to have the following (bivariate) numerical sample 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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVkYXRhR0YoNy03JCQiJEomISIjJCIlOzZGMjckJCIkRSRGMkY2NyQkIiROI0YyRjk3JCQiJEspRjIkIiVNOUYyNyQkIiVbOEYyJCIlWFxGMjckJCIkRCdGMiQiJTA6RjI3JCQiJCwoRjJGSzckJCIkXylGMkZONyQkIiNYRjJGUTckJCIlMzdGMkZUNyQkIiVSPkYyJCIlVTVGMjcjRi4=

An estimation of the joint cumulant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEoJmthcHBhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1JI21uR0YkNiRRIjJGJ0Y1LUkjbW9HRiQ2LlEiLEYnRjUvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy8lKWVxdWFsaXR5R0Y0LUY7NiRRIjFGJ0Y1RjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1 is 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

JCErLi56dEIhIik=

An estimation of the product of the joint cumulant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEoJmthcHBhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1JI21uR0YkNiRRIjJGJ0Y1LUkjbW9HRiQ2LlEiLEYnRjUvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy8lKWVxdWFsaXR5R0Y0LUY7NiRRIjFGJ0Y1RjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEoJmthcHBhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1JI21uR0YkNiRRIjFGJ0Y1LUkjbW9HRiQ2LlEiLEYnRjUvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy8lKWVxdWFsaXR5R0Y0LUY7NiRRIjBGJ0Y1RjUvJS9zdWJzY3JpcHRzaGlmdEdGZW4tRj82LlEifkYnRjVGQi9GRUY0RkdGSUZLRk1GT0ZRL0ZVRlNGV0ZobkY1is 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

JCIrQHdsVUghIig=

An estimation of the product of the joint cumulant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEoJmthcHBhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1JI21uR0YkNiRRIjJGJ0Y1LUkjbW9HRiQ2LlEiLEYnRjUvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy8lKWVxdWFsaXR5R0Y0LUY7NiRRIjFGJ0Y1RjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEoJmthcHBhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1JI21uR0YkNiRRIjJGJ0Y1LUkjbW9HRiQ2LlEiLEYnRjUvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy8lKWVxdWFsaXR5R0Y0LUY7NiRRIjBGJ0Y1RjUvJS9zdWJzY3JpcHRzaGlmdEdGZW5GNQ==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

JCIrVHYlcEIiISIm

The expression produced by the following set of functions corresponds to formula (4.10) pp. 96 in [8]. This formula expresses the cumulants of multivariate polykays in terms of joint cumulants of the population underlying the multivariate sample. This choice is motivated by the reduction of the overall final result complexity. Referring to the simplest case of k-statistics, the procedure is as follows: 1) the cumulants of the k-statistics are expressed in terms of their moments using the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkY3RyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRi9GMg== ; 2) the moments of the k-statistics are expressed in terms of the moments of the population underlying the sample using the functions pSS and E; 3) the moments of the population underlying the sample are expressed in terms of its cumulants using the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkcmN0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRi9GMg==.The same procedure works for multivariate polykays. In the following details are given for each step.

The ctr function (see [1]) expresses cumulants of a random vector in terms of joint moments 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. Details on the procedure and mathematical background are given in [1] and [3] Example: The joint cumulant 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in terms of joint moments is 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

LCoqJiZJI211RzYiNiQiIiEiIiJGKSZGJTYkRilGKCIiI0YsKiZGJEYpJkYlNiRGLEYoRikhIiIqJkYqRikmRiU2JEYpRilGKSEiIyZGJTYkRixGKUYp

The rct function expresses joint moments of a random vector in terms of joint cumulants 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. Details on the procedure and mathematical background are given in [3]. 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 Example: The joint moment 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in terms of joint cumulants is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEkcnRjRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictSSNtbkdGJDYkUSIyRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LlEiLEYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRicvJSllcXVhbGl0eUdGRi1GOzYkUSIxRidGPi8lK2V4ZWN1dGFibGVHRkZGPkY+RmhuRj4=

LCoqJiZJJmthcHBhRzYiNiQiIiEiIiJGKSZGJTYkRilGKCIiI0YpKiZGJEYpJkYlNiRGLEYoRilGKSomRipGKSZGJTYkRilGKUYpRiwmRiU2JEYsRilGKQ==

Products of multivariate polykays return products of power sums. For example the expectation of LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYmLUkjbW5HRiQ2JVEiMkYnRjIvRjZRJ25vcm1hbEYnRjIvJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RidGPy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtSSNtbkdGJDYlUSIxRidGNS9GOVEnbm9ybWFsRidGNUZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRi82I1EhRidGNUZB is: 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[LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYmLUkjbW5HRiQ2JVEiMkYnRjIvRjZRJ25vcm1hbEYnRjIvJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RidGPy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtSSNtbkdGJDYlUSIxRidGNS9GOVEnbm9ybWFsRidGNUZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRi82I1EhRidGNUZB] = LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2J1EiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==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 where 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and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2J1EiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2KC1GIzYlLUkobXN1YnN1cEdGJDYnLUYsNidRIlNGJ0YvRjJGNUY4LUkjbW5HRiQ2JlEiMUYnRjJGNS9GOVEnbm9ybWFsRictRiM2JS1GRzYmUSIzRidGMkY1RkpGMkZKLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lL3N1YnNjcmlwdHNoaWZ0R0ZTRjJGSkYyRjVGSi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0YyRko= in terms of moments, a fundamental result of estimation might be used if these products are transformed in augmented symmetric functions, see [9]. For univariate sample, an augmented symmetric function involves products of variables with different indexes, that is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZYLUkjbW9HRiQ2L1EifkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRS8lKWVxdWFsaXR5R0YxRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGKy1JJW1zdWJHRiQ2JS1JI21pR0YkNiZRIkFGJy8lJ2l0YWxpY0dRJXRydWVGJ0YvL0YzUSdpdGFsaWNGJy1GIzYnLUkobWZlbmNlZEdGJDYnLUYjNi4tRk42JlEiaUYnRlFGL0ZULUYsNi9RIixGJ0YvRjJGNS9GOEZTRjlGO0Y9Rj9GQUZDL0ZHUSwwLjMzMzMzMzNlbUYnRkgtRk42JlEiakYnRlFGL0ZURmpuLUZONiZRImtGJ0ZRRi9GVEZqbi1GLDYvUSMuLkYnRi9GMkY1RjdGOUY7Rj1GP0ZBL0ZEUSwwLjIyMjIyMjJlbUYnRkZGSC1GLDYvUSIuRidGL0YyRjVGN0Y5RjtGPUY/RkFGQ0ZGRkgvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRidGLy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnRjJGL0YyLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRl5wRi9GYXBGMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRi9GMg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW9HRiQ2L1EiPUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGRS8lKWVxdWFsaXR5R0YxLUkrbXVuZGVyb3ZlckdGJDYnLUYsNjBRJiZTdW07RidGL0YyL0krbXNlbWFudGljc0dGJFEmaW5lcnRGJ0Y1RjcvRjpRJXRydWVGJ0Y7L0Y+RlQvRkBGVEZBL0ZEUSYwLjBlbUYnL0ZHUSwwLjE2NjY2NjdlbUYnRkgtRiM2MC1JI21pR0YkNiZRImFGJy8lJ2l0YWxpY0dGVEYvL0YzUSdpdGFsaWNGJy1GLDYvUSsmTm90RXF1YWw7RidGL0YyRjVGN0Y5RjtGPUY/RkFGQ0ZGRkgtRiw2L1EifkYnRi9GMkY1RjdGOUY7Rj1GP0ZBRlcvRkdGWEZILUZobjYmUSJiRidGW29GL0Zdb0Zfb0Ziby1GaG42JlEiY0YnRltvRi9GXW9GYm9GX29GYm8tRiw2L1EjLi5GJ0YvRjJGNUY3RjlGO0Y9Rj9GQS9GRFEsMC4yMjIyMjIyZW1GJ0Zlb0ZILUYsNi9RIi5GJ0YvRjJGNUY3RjlGO0Y9Rj9GQUZXRmVvRkhGL0YyLUZobjYjUSFGJ0ZBLyUsYWNjZW50dW5kZXJHRjFGYm8tSShtc3Vic3VwR0YkNictRmhuNidRIlhGJ0Zbb0YvLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0Zdby1GIzYlLUZobjYnRmpuRltvRi9GX3FGXW9GL0YyLUYjNiUtRmhuNidRImlGJ0Zbb0YvRl9xRl1vRi9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGXXItRmpwNidGXHEtRiM2JS1GaG42J0Zob0Zbb0YvRl9xRl1vRi9GMi1GIzYlLUZobjYnUSJqRidGW29GL0ZfcUZdb0YvRjJGW3JGXnJGL0YyLUkobXN1YnN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYlLUYsNiZRImNGJ0YvRjJGNUYyL0Y2USdub3JtYWxGJy1GOTYlLUYsNiZRImtGJ0YvRjJGNUYyRj4vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRkc=... The pSS function carries out this task. 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 Example: product of power sums 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 in terms of augmented symmetric functions: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkcFNTRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNistRjY2Ji1GIzYlLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnRkJGQi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJy1JI21vR0YkNi5RIixGJ0ZCLyUmZmVuY2VHRkYvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRicvJSllcXVhbGl0eUdGRi1GNjYmLUYjNiUtRj82JFEiMkYnRkJGREZCRkJGR0ZKRk1GYW9GTUZhb0ZERkJGQi1GLDYjUSFGJ0ZERkI=

LDAmSSJBRzYiNiM3JiIiIiIiI0YpRilGKCZGJDYjNyVGKUYpIiIkRi0mRiQ2IzclRihGKSIiJUYtJkYkNiM3JEYtRjFGLSZGJDYjNyRGKSIiJkYtJkYkNiM3JEYoIiInRigmRiQ2IzcjIiIoRig=

Example: product of power sums 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 in terms of multivariate augmented symmetric functions: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkcFNTRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiktRjY2Ji1GIzYlLUY2NiYtRiM2Jy1JI21uR0YkNiRRIjJGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYuUSIsRidGRi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZOLyUqc3ltbWV0cmljR0ZOLyUobGFyZ2VvcEdGTi8lLm1vdmFibGVsaW1pdHNHRk4vJSdhY2NlbnRHRk4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy8lKWVxdWFsaXR5R0ZOLUZDNiRRIjFGJ0ZGLyUrZXhlY3V0YWJsZUdGTkZGRkYvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGYG9GRkZGRmJvRmVvRkgtRjY2Ji1GIzYlLUY2NiYtRiM2J0Zdb0ZIRl1vRmBvRkZGRkZib0Zlb0Zgb0ZGRkZGYm9GZW9GSC1GNjYmLUYjNiUtRjY2Ji1GIzYnRkJGSC1GQzYkUSIzRidGRkZgb0ZGRkZGYm9GZW9GYG9GRkZGRmJvRmVvRmBvRkZGRi1GLDYjUSFGJ0Zgb0ZG

LCwmSSJBRzYiNiM3JTckIiIiRik3JCIiI0YpNyRGKyIiJEYpJkYkNiM3JEYsNyRGLUYrRikmRiQ2IzckRig3JCIiJUY2RikmRiQ2IzckRio3JEYtRjZGKSZGJDYjNyM3JCIiJkY/Rik=

The following is a test explaing how to transform the power sum product 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 with in augmented symmetric functions with n=3: 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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGMUdGKCw0KiYmSSJ4R0YoNiMiIiIiIiMmSSJ5R0YoRjIiIiRGMyoqRjBGMyZGMTYjRjRGM0Y1RjQmRjZGOkYzRjMqKkYwRjNGOUYzRjVGM0Y7RjRGMyoqRjBGMyZGMTYjRjdGM0Y1RjQmRjZGP0YzRjMqKkYwRjNGPkYzRjVGM0ZARjRGMyomRjlGNEY7RjdGMyoqRjlGM0Y+RjNGO0Y0RkBGM0YzKipGOUYzRj5GM0Y7RjNGQEY0RjMqJkY+RjRGQEY3RjM3I0Yu

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

LCYmSSJBRzYiNiM3JDckIiIiRik3JEYpIiIjRikmRiQ2IzcjNyRGKyIiJEYp

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGMkdGKCw0KiYmSSJ4R0YoNiMiIiIiIiMmSSJ5R0YoRjIiIiRGMyoqRjBGMyZGMTYjRjRGM0Y1RjQmRjZGOkYzRjMqKkYwRjNGOUYzRjVGM0Y7RjRGMyoqRjBGMyZGMTYjRjdGM0Y1RjQmRjZGP0YzRjMqKkYwRjNGPkYzRjVGM0ZARjRGMyomRjlGNEY7RjdGMyoqRjlGM0Y+RjNGO0Y0RkBGM0YzKipGOUYzRj5GM0Y7RjNGQEY0RjMqJkY+RjRGQEY3RjM3I0Yu

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

SSV0cnVlRyUqcHJvdGVjdGVkRw==

The fundamental result of estimation we refer to allows to express expectation of augmented symmetric functions in terms of moments [10], that is 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 = n(n-1)...(n-s+1)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 where s denotes the number of elements in the list 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 product n(n-1)...(n-s+1) is the lower factorial LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUklbXJvd0dGJDYlLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjgvRjxRJ25vcm1hbEYnRjhGPi1GLzYlLUYyNiZRInNGJ0Y1RjhGO0Y4Rj4vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw==. The df function computes the lower factorial. This result can be suitably generalized to the multivariate case. 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 Example: the decreasing factorial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkobWZlbmNlZEdGJDYlLUYjNiYtSSNtaUdGJDYmUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGOi8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RicvRj5RJ25vcm1hbEYnRjpGQy1GIzYmLUkjbW5HRiQ2JVEiM0YnRjpGQ0Y6RkBGQy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjpGQEZD LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjZGZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1JI21uR0YkNiRRIjNGJy9GM1Enbm9ybWFsRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJ0Y+Rj5GQEY+

KihJIm5HNiIiIiIsJkYjRiUhIiJGJUYlLCZGI0YlISIjRiVGJQ==

The E function computes the expectation of augmented symmetric functions. 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 Example: the expectation of 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 is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUY2NiYtRiM2Ji1JI21uR0YkNiRRIjFGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYuUSIsRidGQi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZKLyUqc3ltbWV0cmljR0ZKLyUobGFyZ2VvcEdGSi8lLm1vdmFibGVsaW1pdHNHRkovJSdhY2NlbnRHRkovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy8lKWVxdWFsaXR5R0ZKRj5GQkZCLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRkQtRjY2Ji1GIzYmLUY/NiRRIjJGJ0ZCRkRGPkZCRkJGaW5GXG9GQkZCLUYsNiNRIUYnLUYjNiYtRkU2LlEiK0YnRkJGSC9GTEZKRk1GT0ZRRlNGVS9GWFEsMC4yMjIyMjIyZW1GJy9GZW5GYHBGZ24tRiM2JUYrLUY2NiQtRiM2JS1GNjYmLUYjNiYtRj82JFEiM0YnRkJGREZjb0ZCRkJGaW5GXG9GZm9GQkZCRkJGZm9GQkZmby8lK2V4ZWN1dGFibGVHRkpGQg==

LCYqKkkibkc2IiIiIiwmRiRGJiEiIkYmRiYmSSNtdUdGJTYkRiZGJkYmJkYqNiQiIiNGJkYmRiYqJkYkRiYmRio2JCIiJEYuRiZGLg==

The fE function repeatedly executes the E function on an expression. 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 JSFH Example: the expectation of 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

KigsMiomSSJuRzYiIiIkJkkjbXVHRiY2IyIiIkYnISIiKihGJUYnRihGKyZGKTYjIiIjRitGKyomRiVGMEYoRidGJyooRiVGMEYoRitGLkYrISIlKiZGJUYrRihGJyEiIyomRiVGMCZGKTYjRidGK0YrKihGJUYrRihGK0YuRitGJyomRiVGK0Y3RitGLEYrRiVGNSwmRiVGK0YsRitGLA==

Example: the expectation of 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

LDQqKEkibkc2IiIiJCZJI211R0YlNiMiIiMiIiImRig2I0YmRipGKyooRiRGKkYnRitGLEYqISIkKihGJEYqRidGKyZGKDYjIiInRitGKyooRiRGKkYsRismRig2IyIiJkYrRioqKEYkRitGJ0YrRixGKkYqKihGJEYrRidGK0YxRishIiIqKEYkRitGLEYrRjVGKyEiIyomRiRGKyZGKDYjIiIlRitGKyomRiRGKyZGKDYjIiIpRitGKw==

The jcks function returns the joint cumulants of multivariate polykays according to the procedure described in the previous section. 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEoJmthcHBhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiUtSSNtbkdGJDYlUSIxRidGNUY3RjVGNy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JS1GLzYmUSJrRicvRjNRJXRydWVGJ0Y1L0Y4USdpdGFsaWNGJy1GIzYlLUY9NiVRIjJGJ0Y1RjdGNUY3RkBGNUY3RjVGN0Y1Rjc=is the first cumulant of 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

Jkkma2FwcGFHNiI2IyIiIw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEoJmthcHBhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiUtSSNtbkdGJDYlUSIyRidGNUY3RjVGNy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JS1GLzYmUSJrRicvRjNRJXRydWVGJ0Y1L0Y4USdpdGFsaWNGJ0Y6RkBGNUY3RjVGN0Y1Rjc=is the second cumulant of 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

LCYqJiZJJmthcHBhRzYiNiMiIiUiIiJJIm5HRiYhIiJGKSomJkYlNiMiIiNGLywmRipGKUYrRilGK0Yv

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEoJmthcHBhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiUtSSNtbkdGJDYlUSIzRidGNUY3RjVGNy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JS1GLzYmUSJrRicvRjNRJXRydWVGJ0Y1L0Y4USdpdGFsaWNGJy1GIzYlLUY9NiVRIjJGJ0Y1RjdGNUY3RkBGNUY3RjVGN0Y1Rjc=is the third cumulant of 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

LCoqJiZJJmthcHBhRzYiNiMiIiciIiJJIm5HRiYhIiNGKSomJkYlNiMiIiMiIiQsJkYqRikhIiJGKUYrIiIpKiomRiU2I0YwRi8sJkYqIiIlISIpRilGKUYqRjJGMUYrRikqKkYtRikmRiU2I0Y4RilGKkYyRjFGMiIjNw==

In the following the variances of the first few k-statistics are computed, see [11]. Note that 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

Ly1JJHZhckc2IjYjJkkia0dGJTYjIiIiKiYmSSZrYXBwYUdGJTYjIiIjRipJIm5HRiUhIiI=

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

Ly1JJHZhckc2IjYjJkkia0dGJTYjIiIjLCYqJiZJJmthcHBhR0YlNiMiIiUiIiJJIm5HRiUhIiJGMSomJkYuRilGKiwmRjJGMUYzRjFGM0Yq

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

Ly1JJHZhckc2IjYjJkkia0dGJTYjIiIkLCoqJiZJJmthcHBhR0YlNiMiIiciIiJJIm5HRiUhIiJGMSoqJkYuNiMiIiNGKkYyRjEsJkYyRjFGM0YxRjMsJkYyRjEhIiNGMUYzRjAqJiZGLkYpRjdGOEYzIiIqKihGNUYxJkYuNiMiIiVGMUY4RjNGPQ==

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

Ly1JJHZhckc2IjYjJkkia0dGJTYjIiIlLDAqJiZJJmthcHBhR0YlNiMiIikiIiJJIm5HRiUhIiJGMSouJkYuNiMiIiNGKkYyRjEsJkYxRjFGMkYxRjEsJkYyRjFGM0YxRjMsJkYyRjEhIiNGMUYzLCZGMkYxISIkRjFGMyIjQyomJkYuRilGN0Y5RjMiI00qLEY1RjEmRi42IyIiJEY3RjJGMUY5RjNGOkYzIiRXIiooRjVGMSZGLjYjIiInRjFGOUYzIiM7KixGNUY3RkBGMUYyRjFGOUYzRjpGMyIjcyooRkNGMSZGLjYjIiImRjFGOUYzIiNb

More examples are given in [10], pag. 265. The joint cumulant 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 of order (2, 1) of (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) is 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

LCYqJiZJJmthcHBhRzYiNiMiIiYiIiJJIm5HRiYhIiNGKSoqJkYlNiMiIiNGKSZGJTYjIiIkRilGKiEiIiwmRipGKUYzRilGMyIiJQ==

The joint cumulant 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 of order (2, 2) of 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

LCgqJiZJJmthcHBhRzYiNiMiIiciIiJJIm5HRiYhIiRGKSooJkYlNiMiIiQiIiNGKiEiIywmRipGKSEiIkYpRjMiIiUqKiZGJTYjRjBGKSZGJTYjRjRGKUYqRjFGMkYzRjQ=

The joint cumulant 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 of order (1, 1) of 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

LCYqJiZJJmthcHBhRzYiNiMiIiYiIiJJIm5HRiYhIiJGKSooJkYlNiMiIiNGKSZGJTYjIiIkRiksJkYqRilGK0YpRisiIic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The joint cumulant 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 of order (1,2) of 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiYtRiM2Ky1GLDYnUSRzZWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJStleGVjdXRhYmxlR0Y5LyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRjA2KC1GIzYlLUkjbW5HRiQ2JlEjMTFGJ0Y6RjxGP0Y6Rj9GOkY8Rj8vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictSSNtb0dGJDYwUSJ+RidGOkY8Rj8vJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0Zeby8lKWVxdWFsaXR5R0Y5LUYsNidRJHBhZ0YnRjdGOkY8Rj8tRlE2MFEiLkYnRjpGPEY/RlRGVkZYRlpGZm5GaG5Gam5GXG9GX29GYW9GUC1GRzYmUSQyNzJGJ0Y6RjxGP0Y6Rj9GOkY8Rj9GOkY8Rj8= 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

LC4qJiZJJmthcHBhRzYiNiMiIikiIiJJIm5HRiYhIiNGKSoqJkYlNiMiIiUiIiMsJkYqIiNHISNTRilGKUYqISIiLCZGKkYpRjRGKUYrRikqKCZGJTYjRjBGKSZGJTYjIiIkRjBGNUYrIiQ/IioqRjdGKSZGJTYjIiInRilGKkY0RjVGNCIjPyooRjdGMEYtRilGNUYrIiMhKSosRjlGKSZGJTYjIiImRiksJkYqIiNTISNjRilGKUYqRjRGNUYrRik=

The joint cumulant of order (1,1) of 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(see[8] pag.94) 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

LCgqJiZJJmthcHBhRzYiNiYiIiJGKEYoRihGKEkibkdGJiEiIkYoKigmRiU2JkYoIiIhRi5GKEYoJkYlNiZGLkYoRihGLkYoLCZGKUYoRipGKEYqRigqKCZGJTYmRihGLkYoRi5GKCZGJTYmRi5GKEYuRihGKEYxRipGKA==

The joint cumulant of order (1,1,1) of 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(see[8] pag.94) 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEoJmthcHBhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiUtSSNtbkdGJDYlUSIyRidGNUY3RjVGNy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JS1GLzYmUSJrRicvRjNRJXRydWVGJ0Y1L0Y4USdpdGFsaWNGJy1GIzYlLUY9NiVRIjNGJ0Y1RjdGNUY3RkBGNUY3RjVGN0Y1Rjc=is the second cumulant of 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElamNrc0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiNRIUYnLUY2NiYtRiM2JS1GNjYmLUYjNiQtSSNtbkdGJDYkUSIzRicvRjNRJ25vcm1hbEYnRklGSS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y6RklGSUZLRk4tSSNtb0dGJDYuUSIsRidGSS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZXLyUqc3ltbWV0cmljR0ZXLyUobGFyZ2VvcEdGVy8lLm1vdmFibGVsaW1pdHNHRlcvJSdhY2NlbnRHRlcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy8lKWVxdWFsaXR5R0ZXLUY2NiYtRiM2JC1GRjYkUSIyRidGSUZJRklGS0ZORklGSS8lK2V4ZWN1dGFibGVHRldGSQ==

LCoqJiZJJmthcHBhRzYiNiMiIiciIiJJIm5HRiYhIiJGKSoqJkYlNiMiIiMiIiRGKkYpLCZGKkYpRitGKUYrLCZGKkYpISIjRilGK0YoKiYmRiU2I0YwRi9GMUYrIiIqKihGLUYpJkYlNiMiIiVGKUYxRitGNw==

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 the second cumulant of 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

LCgqJiZJJmthcHBhRzYiNiQiIiNGKCIiIkkibkdGJiEiIkYpKiYmRiU2JEYpRilGKCwmRipGKUYrRilGK0YpKigmRiU2JEYoIiIhRikmRiU2JEYzRihGKUYvRitGKQ==

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

KiQmSSZrYXBwYUc2IjYkIiIiRiciIiM=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Given a random numerical sample, the njcks function allows estimating the joint cumulants of multivariate polykays. The function invokes the nployk function in the background section. The first two parameters are the same as in the njcks function, while the data vector must be entered as third parameter. The procedure consists in expressing the joint cumulant of a multivariate polykay in terms of joint cumulants of the underlying population and then replacing the occurrences of these joint cumulants and/or their products with the corresponding multivariate numerical polykays. 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 Example: suppose to have the following (univariate) numerical sample 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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZkYXRhMUdGKDcjN0AkIiVNOyEiIyQiJXc1RjIkIiUlPSJGMiQiJWI4RjIkIiUmZSJGMiQiJT89RjIkIiReKEYyJCIlQTVGMiQiJV83RjIkIiVvOUYyJCIlMztGMiQiJVY+RjIkIiQ3KUYyJCIlPzZGMiQiJSZIIkYyJCIleDlGMiQiJSRvIkYyJCIlISk+RjIkIiRiKUYyJCIlZTZGMiQiJTU3RjIkIiUtOkYyRlEkIiUpcCJGMiQiJSMqPkYyJCIkWipGMiQiJW82RjIkIiVUOEYyJCIlTjpGMiQiJTY+RjI3I0Yu

The estimation of the joint cumulant of order (2,2) of 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

JCIrQHpYPj4hIzY=

Example: suppose to have the following (bivariate) numerical sample 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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZkYXRhMkdGKDctNyQkIiRKJiEiIyQiJTs2RjI3JCQiJEUkRjJGNjckJCIkTiNGMkY5NyQkIiRLKUYyJCIlTTlGMjckJCIlWzhGMiQiJVhcRjI3JCQiJEQnRjIkIiUwOkYyNyQkIiQsKEYyRks3JCQiJF8pRjJGTjckJCIjWEYyRlE3JCQiJTM3RjJGVDckJCIlUj5GMiQiJVU1RjI3I0Yu

An estimation of the joint cumulant of order (1,1) of 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

JCIrQid6XSM9ISIm

JSFH

The jcks function computes the cumulants of multivariate polykays. These routines are particularly useful for assessing the goodness of the estimate obtained through these estimators. For example, the variance of first- and fourth-order k-statistics provides insights into the goodness-of-fit estimation of the mean, variance, skeweness, and kurtosis, respectively. Further examples are provided in the last section of this worksheet.

[1] (2009) Di Nardo E., Guarino G., Senato D. Fast algorithms for k-statistics, polykays and their multivariate generalizations. (Worksheet Maple Software: available at https://www.maplesoft.com/Applications/Detail.aspx?id=33041)[2] (2009) Di Nardo E., Guarino G., Senato D. Multiset subdivisions. (Worksheet Maple Software: : available at https://www.maplesoft.com/Applications/Detail.aspx?id=33039). [3] (2015) Di Nardo E. Symbolic calculus in mathematical statistics: a review. Seminaire Lotharingien de Combinatoire Vol. 67 (B67a), pp. 72[4] (2009) Di Nardo E., Guarino G., Senato D. A new method for fast computing unbiased estimators of cumulants. Statistics and Computing 19, 155--165.[5] (2008) Di Nardo E., Guarino G., Senato D. A unifying framework for k-statistics, polykays and their multivariate generalizations. Bernoulli 14, 440--468.[6] (2006) Di Nardo E., Senato D. A symbolic method for k-statistics. Applied Mathematics Letters 19, 968--975.[7] (2011) Di Nardo E., Guarino G., Senato D. A new algorithm for computing the multivariate Faa di Bruno's formula. Applied Mathematics and Computation 217, 6286--6295 [8] P. McCullagh, Tensor Methods in Statistics, Monographs on Statistics and Applied Probability, Taylor & Francis Group, 22 December 2017[9] (2008) Di Nardo E., Guarino G., Senato D. Symbolic computation of moments of sampling distributions. Computational Statistics and Data Analysis 52, 4909--4922.[10] Stuart, A., Ord, J.K., 1987. Kendall\342\200\231s Advanced Theory of Statistics 1. Charles Griffin and Company Limited, London.[11] Weisstein, Eric W. "k-Statistic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/k-Statistic.html Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=