{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "In general, the expectile at level " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\alpha" }, { "__type": "Text", "__tag": 4046, "value": " of a random variable " }, { "__type": "InlineMath", "__tag": 4057, "value": "X" }, { "__type": "Text", "__tag": 4046, "value": " with cumulative distribution function (CDF) " }, { "__type": "InlineMath", "__tag": 4057, "value": "F" }, { "__type": "Text", "__tag": 4046, "value": " is given by the unique solution " }, { "__type": "InlineMath", "__tag": 4057, "value": "t" }, { "__type": "Text", "__tag": 4046, "value": " of:" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\alpha E((X - t)_+) = (1 - \\alpha) E((t - X)_+) \\,." }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Here, " }, { "__type": "InlineMath", "__tag": 4057, "value": "(x)_+ = \\max(0, x)" }, { "__type": "Text", "__tag": 4046, "value": " is the positive part of " }, { "__type": "InlineMath", "__tag": 4057, "value": "x" }, { "__type": "Text", "__tag": 4046, "value": ". 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[1] W. K. Newey and J. L. Powell (1987), \"Asymmetric Least Squares", " Estimation and Testing,\" Econometrica, 55, 819-847.", ".. [2] T. Gneiting (2009). \"Making and Evaluating Point Forecasts,\"", " Journal of the American Statistical Association, 106, 746 - 762.", " :doi:`10.48550/arXiv.0912.0902`" ], "qa": "scipy.stats._stats_py:expectile", "arbitrary": [], "local_refs": [ "a", "alpha", "expectile", "weights" ] }