{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "L-moments are only defined for distributions with finite mean. If a formula for the L-moment is not specifically implemented for the chosen distribution, SciPy will attempt to compute the moment via a generic method, which may yield a finite result where none exists. This is not a critical bug, but an opportunity for an enhancement." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "SciPy offers only basic capabilities for working with L-moments. For more advanced features, consider the " }, { "__type": "InlineCode", "__tag": 4051, "value": "lmo" }, { "__type": "Text", "__tag": 4046, "value": " package " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "2" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ], "title": [], "level": 0, "target": null }, "Warns": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Raises": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Yields": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Methods": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Returns": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "out", "annotation": "array", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The L-moment of the random variable of the specified order." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "L-moment or L-moment ratio of positive integer order." } ] } ], "title": [], "level": 0, "target": null }, "Receives": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Warnings": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Attributes": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Parameters": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "order", "annotation": "int", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The positive integer order of the L-moment; i.e. " }, { "__type": "InlineMath", "__tag": 4057, "value": "n" }, { "__type": "Text", "__tag": 4046, "value": " in the formulae above." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "standardize", "annotation": "bool, default: True", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Whether to return L-moment ratios for orders 3 and higher. L-moment ratios are analogous to standardized conventional moments: they are the non-standardized L-moments divided by the L-moment of order 2." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "method", "annotation": "{None, 'formula', 'general', 'order_statistics', 'quadrature_icdf', 'cache'}", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The strategy used to evaluate the L-moment. 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"__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "X_{(1)}, \\dots, X_{(r)}, \\dots, X_{(n)}" }, { "__type": "Text", "__tag": 4046, "value": " are the order statistics of an independent sample of size " }, { "__type": "InlineMath", "__tag": 4057, "value": "n" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The L-moment can also be expressed in terms of the random variable's inverse cumulative distribution function " }, { "__type": "InlineMath", "__tag": 4057, "value": "F^{-1}" }, { "__type": "Text", "__tag": 4046, "value": " and the shifted Legendre polynomial " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\widetilde{P}_{n-1}" }, { "__type": "Text", "__tag": 4046, "value": ":" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\lambda_n(X) = \\int_0^1 F^{-1}(p) \\widetilde{P}_{n-1}(p) dp" }, { "__type": 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