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  "body": "\n## Definition\n\nLet $(\\mathcal{X}, \\mathcal{A})$ be a sample space and let $\\set{P_\\theta : \\theta \\in \\Theta}$ be a [[statistical model|K3AMMSY8DD]] on it.\nLet $g \\colon \\Theta \\to D$ be a function, the *estimand*, with values in a measurable space $D$.\nA *point estimator* of $g(\\theta)$ is a [[statistic|036756NFD6]] $T \\colon \\mathcal{X} \\to D$, that is, a measurable map that does not depend on $\\theta$.\nFor observed data $x \\in \\mathcal{X}$, the value $T(x)$ is the *point estimate* of $g(\\theta)$.\n*Point estimation* is the problem of choosing such a $T$ so that $T(X)$ is close to $g(\\theta)$ when $X \\sim P_\\theta$, for every $\\theta \\in \\Theta$.\n\n## Remarks\n\nThe general notion of [[estimator|0TTNRP5PFQ]] covers the same object; the word \"point\" stresses that the output is a single element of $D$, not a set.\n\nExample: in the model of $n$ independent Bernoulli($p$) trials with $\\Theta = [0, 1]$ and $g(p) = p$, the map $T(x) = \\frac{1}{n}\\sum_i x_i$ is a point estimator; it is the [[maximum likelihood estimator|0FREPFKNS7]].\n\nNon-example: the map $T(x) = \\frac{1}{n}\\sum_i x_i - p$ is not a point estimator of $p$, because it depends on the unknown parameter $p$.\nAlso, a confidence interval $[T_1(x), T_2(x)]$ is an interval estimator, not a point estimator, since its value is a set, not an element of $D$.\n\nCloseness is measured by a loss function, most often the squared error $(T - g(\\theta))^2$, whose expectation is the mean squared error.\n",
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