{"id":429,"date":"2015-01-05T20:12:47","date_gmt":"2015-01-05T20:12:47","guid":{"rendered":"http:\/\/frees.pajarel.net\/?page_id=429"},"modified":"2015-02-20T19:11:52","modified_gmt":"2015-02-21T01:11:52","slug":"inverse-transform-justification","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/interest-rate-risks-and-simulation\/4-simulation\/4-2-inverse-transform-image-inversedf-needed\/inverse-transform-justification\/","title":{"rendered":"Inverse Transform Justification"},"content":{"rendered":"

Pareto Distribution Example.<\/strong> Suppose that we would like to generate observations from a Pareto distribution with parameters \\(\\alpha\\) and \\(\\theta\\) so that \\(F(x) = 1 – \\left(\\frac{\\theta}{x+\\theta} \\right)^{\\alpha}\\). To compute the inverse transform, we can use the following steps:
\n\\begin{eqnarray*}
\ny = F(x) &\\Leftrightarrow& 1-y = \\left(\\frac{\\theta}{x+\\theta} \\right)^{\\alpha} \\\\
\n&\\Leftrightarrow& \\left(1-y\\right)^{-1\/\\alpha} = \\frac{x+\\theta}{\\theta} = \\frac{x}{\\theta} +1 \\\\
\n&\\Leftrightarrow& \\theta \\left((1-y)^{-1\/\\alpha} – 1\\right) = x = F^{-1}(y) .
\n\\end{eqnarray*}
\nThus, \\(X = \\theta \\left((1-U)^{-1\/\\alpha} – 1\\right)\\) has a Pareto distribution with parameters \\(\\alpha\\) and \\(\\theta\\).<\/p>\n

Inverse Transform Justification<\/strong><\/p>\n

Why does the random variable \\(X = F^{-1}(U)\\) have a distribution function “\\(F\\)”? This is easy to establish in the continuous case.<\/p>\n

Because \\(U\\) is a Uniform random variable on (0,1), we know that \\(Pr(U \\le y) = y\\), for \\(0 \\le y \\le 1\\). Thus,
\n\\begin{eqnarray*}
\n\\Pr(X \\le x) &=& \\Pr(F^{-1}(U) \\le x) \\\\
\n&=& \\Pr(F(F^{-1}(U)) \\le F(x)) \\\\
\n&=& \\Pr(U \\le F(x)) = F(x)
\n\\end{eqnarray*}
\nas required.<\/p>\n

The key step is that \\( F(F^{-1}(u)) = u\\) for each \\(u\\), which is clearly true when \\(F\\) is strictly increasing.
\n

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Pareto Distribution Example. Suppose that we would like to generate observations from a Pareto distribution with parameters \\(\\alpha\\) and \\(\\theta\\) so that \\(F(x) = 1 – \\left(\\frac{\\theta}{x+\\theta} \\right)^{\\alpha}\\). To compute the inverse transform, we can …<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":420,"menu_order":2,"comment_status":"closed","ping_status":"open","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-6V","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/429"}],"collection":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/comments?post=429"}],"version-history":[{"count":3,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/429\/revisions"}],"predecessor-version":[{"id":1648,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/429\/revisions\/1648"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/420"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=429"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}