<html><head><meta http-equiv="Content-Type" content="text/html charset=utf-8"><base></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><base class=""><div class="Apple-Mail-URLShareUserContentTopClass">Caros,</div><div class="Apple-Mail-URLShareUserContentTopClass">este livro sera importante para nós. Abs. Josemar</div><div class="Apple-Mail-URLShareUserContentTopClass"><br class=""></div><div class="Apple-Mail-URLShareWrapperClass" style="position: relative !important;"><blockquote type="cite" style="border-left-style: none; color: inherit; padding: inherit; margin: inherit;" class=""><script type="text/javascript" async="" src="https://ssl.google-analytics.com/ga.js" class=""></script><script src="https://apis.google.com/_/scs/abc-static/_/js/k=gapi.gapi.en.73CODytQm18.O/m=gapi_iframes,googleapis_client,plusone/rt=j/sv=1/d=1/ed=1/am=AAQ/rs=AHpOoo_oUuymWPUqneZS_mOrlV9L5nz06Q/cb=gapi.loaded_0" async="" class=""></script><script class="">(function(){(function(){function e(a){this.t={};this.tick=function(a,c,b){var d=void 0!=b?b:(new Date).getTime();this.t[a]=[d,c];if(void 0==b)try{window.console.timeStamp("CSI/"+a)}catch(e){}};this.tick("start",null,a)}var a;window.performance&&(a=window.performance.timing);var f=a?new e(a.responseStart):new e;window.jstiming={Timer:e,load:f};if(a){var 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Tukey and Parzen used them in the 60s and 70s in describing populations. Since then, the authors of many papers, both theoretical and practical, have used various aspects of quantiles in their work. Until now, however, no one put...</span><span itemprop="url" class=""><a href="https://books.google.com.br/books/about/Statistical_Modelling_with_Quantile_Func.html?hl=pt-BR&id=7c1LimP_e-AC&utm_source=gb-gplus-share" class="">https://books.google.com.br/books/about/Statistical_Modelling_with_Quantile_Func.html?hl=pt-BR&id=7c1LimP_e-AC&utm_source=gb-gplus-share</a></span><span class="main-title" itemprop="name">Statistical Modelling with Quantile Functions</span><img itemprop="image" alt="Statistical Modelling with Quantile Functions" title="Statistical Modelling with Quantile Functions" src="cid:D94A2829-5CB8-418F-8FB2-C0D30D215D5C" class=""></div><div style="display: none" class=""><ol id="ofe-gear-menu-contents" class="gbmcc"><li class="gbe gbmtc"><a class="goog-menuitem-content gbmt" id="" href="https://www.google.com/accounts/Login?service=print&continue=https://books.google.com.br/books%3Fop%3Dlibrary%26hl%3Dpt-BR&hl=pt-BR">Minha biblioteca</a></li><li class="gbe gbmtc"><a class="goog-menuitem-content gbmt" id="" href="http://books.google.com.br/support/topic/4359341?hl=pt-BR-BR">Ajuda</a></li><li class="gbe gbmtc"><a class="goog-menuitem-content gbmt" id="" href="https://books.google.com.br/advanced_book_search?q=derivative+of+quantile+function&hl=pt-BR">Pesquisa de livros avançada</a></li></ol></div><div class="unified-hovercard-device-and-format unified-hovercard-tooltip" style="position: absolute; display: none; z-index: 10000;"><div class="unified-hovercard unified-hovercard-device-content"><div class="unified-hovercard-meta"><table class="unified-hovercard-table-layout"><tbody class=""><tr class=""><td class=""><ul class="unified-hovercard-list"><h4 class="">Bom para:</h4><li class="unified-hovercard-device-supported"><a class="unified-hovercard-device-type" href="https://books.google.com.br/help/ebooks/webreader.html">Web</a></li><li class="unified-hovercard-device-supported"><a class="unified-hovercard-device-type" href="https://books.google.com.br/help/ebooks/devices.html">Tablet/iPad</a></li><li class="unified-hovercard-device-not-supported"><a class="unified-hovercard-device-type" href="https://books.google.com.br/help/ebooks/ereader.html">e-Reader</a></li><li class="unified-hovercard-device-not-supported"><a class="unified-hovercard-device-type" href="https://books.google.com.br/help/ebooks/devices.html">Smartphone</a></li></ul></td><td class=""><ul class="unified-hovercard-list"><h4 class="">Características:</h4><li class="unified-hovercard-device-not-supported"><a href="https://books.google.com.br/support/bin/answer.py?answer=1062949" class="">Texto fluente</a></li><li class="unified-hovercard-device-supported"><a href="https://books.google.com.br/support/bin/answer.py?answer=1062949" class="">Páginas digitalizadas</a></li></ul><div class="unified-hovercard-link"><a target="_blank" href="https://books.google.com.br/support/bin/answer.py?answer=1062949&hl=pt-BR" class="">Ajuda com dispositivos e formatos</a></div></td></tr></tbody></table></div><div class="unified-hovercard-footer"><a href="https://books.google.com/support/bin/topic.py?topic=28528" target="_blank" class="">Saiba mais sobre os livros no Google Play</a></div></div></div><div style="position: absolute; display: none; background-color: rgb(255, 255, 255); color: rgb(51, 51, 51); border: 1px solid rgb(107, 144, 218); padding: 6px; font-size: 83%; width: 350px; cursor: pointer;" class=""><div style="font-weight: bold;" class="">Página 11</div><div class="">Thus the PDF, f(x) = dF/dx, is the <b class="">derivative</b> of the CDF. Figure 1.5 illustrates the <br class=""> form of a PDF. Note that this population <b class="">function</b> corresponds to the sample plot <br class=""> of Dp/Dx against x. Example 1.4: For the uniform distribution F(x) = x, so f(x) = 1.</div></div><div style="position: absolute; display: none; background-color: rgb(255, 255, 255); color: rgb(51, 51, 51); border: 1px solid rgb(107, 144, 218); padding: 6px; font-size: 83%; width: 350px; cursor: pointer;" class=""><div style="font-weight: bold;" class="">Página 12</div><div class="">0 -2 0 0.2 2 4 6 x 0.4 0.6 As mathematics has developed it has ensured that all <br class=""> well-tabulated and used <b class="">functions</b> have <b class="">derivatives</b> that are well tabulated and <br class=""> used. Hence if a distribution has a CDF that can be explicitly expressed in terms <br class=""> of ...</div></div><div style="position: absolute; display: none; background-color: rgb(255, 255, 255); color: rgb(51, 51, 51); border: 1px solid rgb(107, 144, 218); padding: 6px; font-size: 83%; width: 350px; cursor: pointer;" class=""><div style="font-weight: bold;" class="">Página 14</div><div class="">For now it is sufficient to note that we cannot invert this <b class="">quantile function</b> to get p <br class=""> in terms of x, so that no explicit CDF, or consequently PDF, exists. The PDF ... <br class=""> This <b class="">derivative</b> is the Quantile Density Function, QDF, defined by q(p) = dQ(p)/dp.</div></div><div style="position: absolute; display: none; background-color: rgb(255, 255, 255); color: rgb(51, 51, 51); border: 1px solid rgb(107, 144, 218); padding: 6px; font-size: 83%; width: 350px; cursor: pointer;" class=""><div style="font-weight: bold;" class="">Página 19</div><div class="">This model is called the logistic distribution. Figure 1.9 shows the addition <br class=""> operation for the <b class="">quantile</b> density <b class="">functions</b> of the exponential and reflected <br class=""> exponential. As the <b class="">derivative</b> of a sum is the sum of the <b class="">derivatives</b>, the addition <br class=""> of <b class="">quantile</b> ...</div></div><div style="position: absolute; display: none; background-color: rgb(255, 255, 255); color: rgb(51, 51, 51); border: 1px solid rgb(107, 144, 218); padding: 6px; font-size: 83%; width: 350px; cursor: pointer;" class=""><div style="font-weight: bold;" class="">Página 22</div><div class="">Unlike the exponential and the symmetric logistic, this <b class="">quantile function</b> cannot be <br class=""> inverted to give x in terms of p. Thus there is no ... It has a <b class="">quantile function</b> in <br class=""> basic form: S(p) = 1/(1–p)β, β > 0. ... Figure 1.12 shows its <b class="">derivation</b> and form.</div></div><div style="position: absolute; display: none; background-color: rgb(255, 255, 255); color: rgb(51, 51, 51); border: 1px solid rgb(107, 144, 218); padding: 6px; font-size: 83%; width: 350px; cursor: pointer;" class=""><div style="font-weight: bold;" class="">Página 27</div><div class="">The <b class="">quantile</b> density <b class="">function</b> and the p-PDF are closely related. As x = Q(p) and <br class=""> p = F(x) for any pair of values (x, p), it follows from the definition of <b class="">differentiation</b> <br class=""> that (dx/dp)(dp/dx) = 1 so (dQ(p)/dp)(dF(x)/dx)=1, hence q(p)f(x) = 1 and, <br class=""> therefore ...</div></div><div style="position: absolute; display: none; background-color: rgb(255, 255, 255); color: rgb(51, 51, 51); border: 1px solid rgb(107, 144, 218); padding: 6px; font-size: 83%; width: 350px; cursor: pointer;" class=""><div style="font-weight: bold;" class="">Página 28</div><div class="">and its <b class="">derivative</b>, without having to be able to invert Q(p) to get F(x). The plots of <br class=""> PDF in this book are normally obtained in this fashion. The 99 points are best <br class=""> supplemented by additional tail probabilities. For example, p = 0.0005, 0.001, <br class=""> 0.002 ...</div></div><div style="position: absolute; display: none; background-color: rgb(255, 255, 255); color: rgb(51, 51, 51); border: 1px solid rgb(107, 144, 218); padding: 6px; font-size: 83%; width: 350px; cursor: pointer;" class=""><div style="font-weight: bold;" class="">Página 97</div><div class="">... model, then a model-free approach can be used by using the empirical <br class=""> <b class="">quantile function</b> given by , defined in Section 2.2. ... 4.5 Approximation Suppose <br class=""> h(x) is some function of x and h′(x), h′′(x), etc. are the first, second, etc. <br class=""> <b class="">derivatives</b> ...</div></div><div style="position: absolute; display: none; background-color: rgb(255, 255, 255); color: rgb(51, 51, 51); border: 1px solid rgb(107, 144, 218); padding: 6px; font-size: 83%; width: 350px; cursor: pointer;" class=""><div style="font-weight: bold;" class="">Página 103</div><div class="">One simple one, suitable for right-tailed or symmetric distributions, compares <br class=""> <b class="">quantile functions</b> as p approaches one. (See, for example ... This requires the <br class=""> <b class="">derivative</b> of the ratio to be greater than or equal to zero. Differentiating and <br class=""> sorting ...</div></div><div style="position: absolute; display: none; background-color: rgb(255, 255, 255); color: rgb(51, 51, 51); border: 1px solid rgb(107, 144, 218); padding: 6px; font-size: 83%; width: 350px; cursor: pointer;" class=""><div style="font-weight: bold;" class="">Página 128</div><div class="">... that Q(p) + Q(1 – p) = Q(1), for all p. Find a general expression for the IQR of <br class=""> this distribution as a function of k and β. 5. Show that for a standard normal <br class=""> distribution the <b class="">derivatives</b> of the <b class="">quantile function</b> satisfy the relations N′(p) = 1/<br class=""> φp(p), ...</div></div><div style="position: absolute; display: none; background-color: rgb(255, 255, 255); color: rgb(51, 51, 51); border: 1px solid rgb(107, 144, 218); padding: 6px; font-size: 83%; width: 350px; cursor: pointer;" class=""><div style="font-weight: bold;" class="">Página 199</div><div class="">As shown in Section 4.5 the use of the next term in the approximation gives the <br class=""> approximate rankit as μ(r) = Q(pr) + prqr Q′′(pr)/[2(n+ 2)], where pr = r/(n + 1), <br class=""> qr = 1–pr, and Q′′(p) is the second <b class="">derivative</b> of Q(p). Example 9.7: For any ...</div></div><div style="position: absolute; display: none; background-color: rgb(255, 255, 255); color: rgb(51, 51, 51); border: 1px solid rgb(107, 144, 218); padding: 6px; font-size: 83%; width: 350px; cursor: pointer;" class=""><div style="font-weight: bold;" class="">Página 209</div><div class="">The calculation of l(θ) requires the <b class="">derivation</b> of p(r) from x(r) = Q(p(r):θ). This can <br class=""> be carried out directly from F(x(r)) if F(.) is explicit. If not, the iterative method of <br class=""> Section 4.5(c) has to be used. It is seen that in the <b class="">quantile</b>-based likelihood the ...</div></div><div style="position: absolute; display: none; background-color: rgb(255, 255, 255); color: rgb(51, 51, 51); border: 1px solid rgb(107, 144, 218); padding: 6px; font-size: 83%; width: 350px; cursor: pointer;" class=""><div style="font-weight: bold;" class="">Página 210</div><div class="">1. Define distribution Q(p;) 2. Derive <b class="">quantile</b> density q(p;) 3. Set initial or revised <br class=""> parameter values ... Columns (4, 5, 6), (7, 8, 9), ..., (13, 14, 15) give the iterative <br class=""> <b class="">derivation</b> of the column p(r). It will be noted that, as required, column 16, (p(r)), ...</div></div><div style="position: absolute; display: none; background-color: rgb(255, 255, 255); color: rgb(51, 51, 51); border: 1px solid rgb(107, 144, 218); padding: 6px; font-size: 83%; width: 350px; cursor: pointer;" class=""><div style="font-weight: bold;" class="">Página 215</div><div class="">We define two <b class="">functions</b>: The likelihood ratio R(θ) = L(θ)/L() and the log-likelihood <br class=""> ratio r(θ) = ln(R(θ)). ... 0, and calculate the criterion, C( 0) from θ θ Formulae for <br class=""> <b class="">derivatives</b> of l(θ) Sc(θ), I(θ) Initial values for INTERVALS AND REGIONS 215.</div></div><div style="position: absolute; display: none; background-color: rgb(255, 255, 255); color: rgb(51, 51, 51); border: 1px solid rgb(107, 144, 218); padding: 6px; font-size: 83%; width: 350px; cursor: pointer;" class=""><div style="font-weight: bold;" class="">Página 216</div><div class="">Formulae for <b class="">derivatives</b> of l(θ) Sc(θ), I(θ) Initial values for the end points (θ1, θ2) <br class=""> ± √[–2ln(p)/I()] Iterative evaluation for the two end points θnew = θ – [r(θ) – ln(p)]/<br class=""> Sc(θ) θ = (θ1, θ2) 100p% likelihood interval (θ1, θ2) θˆ θˆ Table 9.7. Obtaining ...</div></div><div style="position: absolute; display: none; background-color: rgb(255, 255, 255); color: rgb(51, 51, 51); border: 1px solid rgb(107, 144, 218); padding: 6px; font-size: 83%; width: 350px; cursor: pointer;" class=""><div style="font-weight: bold;" class="">Página 217</div><div class="">These require the setting up of tabular layouts based on initial estimates. We <br class=""> therefore need to consider briefly the <b class="">derivation</b> of these initial values. A number <br class=""> of methods can be used: (a) The method of percentiles often provides a simple <br class=""> and.</div></div><div style="position: absolute; display: none; background-color: rgb(255, 255, 255); color: rgb(51, 51, 51); border: 1px solid rgb(107, 144, 218); padding: 6px; font-size: 83%; width: 350px; cursor: pointer;" class=""><div style="font-weight: bold;" class="">Página 296</div><div class="">Notice that (a) If S(p) is an explicit <b class="">function</b> of p, then fs(p) is also an explicit <br class=""> <b class="">function</b> even if f(x) does not explicitly exist. (b) po) is itself a ... In differentiating / <br class=""> we therefore need the <b class="">derivatives</b> of po with respect to the parameters. These are<br class=""> ...</div></div><div style="position: absolute; display: none; background-color: rgb(255, 255, 255); color: rgb(51, 51, 51); border: 1px solid rgb(107, 144, 218); padding: 6px; font-size: 83%; width: 350px; cursor: pointer;" class=""><div style="font-weight: bold;" class="">Página 317</div><div class="">QDF, see <b class="">Quantile</b> Density <b class="">Function</b> Q-Q plot, 31, 33, 179, 183 Q-transformations, <br class=""> 66, 70, 91, 141 317 Quality ... Rank, 256 Rankit(s), 88 approximate, 98, 199, 202 <br class=""> average of, 200 <b class="">derivation</b> of exact, 258 exact, 199 median, 89, 198, 210, 259, ...</div></div><div id="volume-main" class=""><div id="volume-left" class=""><div id="menu_container" class=""><div id="menu_scroll" role="navigation" style="overflow: auto; border-width: 2px 0px 0px; border-top-style: solid; border-top-color: rgb(235, 239, 249); height: 464px;" class=""><div id="gb-get-book-container" class=""><div role="button" class="jfk-button jfk-button-primary goog-inline-block gb-get-book" tabindex="0" style="-webkit-user-select: none;"><a href="https://play.google.com/store/books/details?id=7c1LimP_e-AC&rdid=book-7c1LimP_e-AC&rdot=1&source=gbs_vpt_read&pcampaignid=books_booksearch_viewport" id="gb-get-book-content" class="">Comprar e-Livro - R$340,40</a></div></div><p id="gb-buy-options-trigger" class="gb-buy-options-link">Obter este livro em versão impressa<span 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Likelihood","Pid":"PA295","PgNum":"295","Order":315},{"Title":"APPENDIX 3 Bivariate Transformations","Pid":"PA299","PgNum":"299","Order":319},{"Title":"References","Pid":"PA301","PgNum":"301","Order":321},{"Title":"Back cover","Pid":"PA325","PgNum":"325","Order":345}]}},"max_resolution_image_width":1280,"max_resolution_image_height":1972,"quality_info":"Sabemos que este é um livro de boa qualidade.","volume_id":"7c1LimP_e-AC","permission_info":"Páginas exibidas com permissão de \u003ca class=link_aux href=\"<a href="https://books.google.com.br/url?id=7c1LimP_e-AC\u0026pg=PA14\u0026q=http://www.crcpress.com\u0026linkid=1\u0026usg=AFQjCNG9sb7YhmU3fNfZ2en-vuWw5jeLKQ\u0026source=gbs_pub_info_r\" class="">https://books.google.com.br/url?id=7c1LimP_e-AC\u0026pg=PA14\u0026q=http://www.crcpress.com\u0026linkid=1\u0026usg=AFQjCNG9sb7YhmU3fNfZ2en-vuWw5jeLKQ\u0026source=gbs_pub_info_r\</a>"\u003eCRC 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Tukey and Parzen used them in the 60s and 70s in describing populations. Since then, the authors of many papers, both theoretical and practical, have used various aspects of quantiles in their work. Until now, however, no one put all the ideas together to form what turns out to be a general approach to statistics.Statistical Modelling with Quantile Functions does just that. It systematically examines the entire process of statistical modelling, starting with using the quantile function to define continuous distributions. The author shows that by using this approach, it becomes possible to develop complex distributional models from simple components. A modelling kit can be developed that applies to the whole model - deterministic and stochastic components - and this kit operates by adding, multiplying, and transforming distributions rather than data.Statistical Modelling with Quantile Functions adds a new dimension to the practice of statistical modelling that will be of value to anyone faced with analyzing data. 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Figure 1.5 illustrates the \u003cbr\u003e\nform of a PDF. Note that this population \u003cb\u003efunction\u003c/b\u003e corresponds to the sample plot \u003cbr\u003e\nof Dp/Dx against x. Example 1.4: For the uniform distribution F(x) = x, so f(x) = 1.","page_url":"<a href="https://books.google.com.br/books?id=7c1LimP_e-AC\u0026pg=PA11\u0026dq=derivative+of+quantile+function\u0026hl=pt-BR" class="">https://books.google.com.br/books?id=7c1LimP_e-AC\u0026pg=PA11\u0026dq=derivative+of+quantile+function\u0026hl=pt-BR</a>"},{"page_id":"PA12","page_number":"12","snippet_text":"0 -2 0 0.2 2 4 6 x 0.4 0.6 As mathematics has developed it has ensured that all \u003cbr\u003e\nwell-tabulated and used \u003cb\u003efunctions\u003c/b\u003e have \u003cb\u003ederivatives\u003c/b\u003e that are well tabulated and \u003cbr\u003e\nused. Hence if a distribution has a CDF that can be explicitly expressed in terms \u003cbr\u003e\nof\u0026nbsp;...","page_url":"<a href="https://books.google.com.br/books?id=7c1LimP_e-AC\u0026pg=PA12\u0026dq=derivative+of+quantile+function\u0026hl=pt-BR" class="">https://books.google.com.br/books?id=7c1LimP_e-AC\u0026pg=PA12\u0026dq=derivative+of+quantile+function\u0026hl=pt-BR</a>"},{"page_id":"PA14","page_number":"14","snippet_text":"For now it is sufficient to note that we cannot invert this \u003cb\u003equantile function\u003c/b\u003e to get p \u003cbr\u003e\nin terms of x, so that no explicit CDF, or consequently PDF, exists. The PDF ... \u003cbr\u003e\nThis \u003cb\u003ederivative\u003c/b\u003e is the Quantile Density Function, QDF, defined by q(p) = dQ(p)/dp.","page_url":"<a href="https://books.google.com.br/books?id=7c1LimP_e-AC\u0026pg=PA14\u0026dq=derivative+of+quantile+function\u0026hl=pt-BR" class="">https://books.google.com.br/books?id=7c1LimP_e-AC\u0026pg=PA14\u0026dq=derivative+of+quantile+function\u0026hl=pt-BR</a>"},{"page_id":"PA19","page_number":"19","snippet_text":"This model is called the logistic distribution. Figure 1.9 shows the addition \u003cbr\u003e\noperation for the \u003cb\u003equantile\u003c/b\u003e density \u003cb\u003efunctions\u003c/b\u003e of the exponential and reflected \u003cbr\u003e\nexponential. As the \u003cb\u003ederivative\u003c/b\u003e of a sum is the sum of the \u003cb\u003ederivatives\u003c/b\u003e, the addition \u003cbr\u003e\nof \u003cb\u003equantile\u003c/b\u003e\u0026nbsp;...","page_url":"<a href="https://books.google.com.br/books?id=7c1LimP_e-AC\u0026pg=PA19\u0026dq=derivative+of+quantile+function\u0026hl=pt-BR" class="">https://books.google.com.br/books?id=7c1LimP_e-AC\u0026pg=PA19\u0026dq=derivative+of+quantile+function\u0026hl=pt-BR</a>"},{"page_id":"PA22","page_number":"22","snippet_text":"Unlike the exponential and the symmetric logistic, this \u003cb\u003equantile function\u003c/b\u003e cannot be \u003cbr\u003e\ninverted to give x in terms of p. Thus there is no ... It has a \u003cb\u003equantile function\u003c/b\u003e in \u003cbr\u003e\nbasic form: S(p) = 1/(1–p)β, β \u0026gt; 0. ... Figure 1.12 shows its \u003cb\u003ederivation\u003c/b\u003e and form.","page_url":"<a href="https://books.google.com.br/books?id=7c1LimP_e-AC\u0026pg=PA22\u0026dq=derivative+of+quantile+function\u0026hl=pt-BR" class="">https://books.google.com.br/books?id=7c1LimP_e-AC\u0026pg=PA22\u0026dq=derivative+of+quantile+function\u0026hl=pt-BR</a>"},{"page_id":"PA27","page_number":"27","snippet_text":"The \u003cb\u003equantile\u003c/b\u003e density \u003cb\u003efunction\u003c/b\u003e and the p-PDF are closely related. As x = Q(p) and \u003cbr\u003e\np = F(x) for any pair of values (x, p), it follows from the definition of \u003cb\u003edifferentiation\u003c/b\u003e \u003cbr\u003e\nthat (dx/dp)(dp/dx) = 1 so (dQ(p)/dp)(dF(x)/dx)=1, hence q(p)f(x) = 1 and, \u003cbr\u003e\ntherefore\u0026nbsp;..."},{"page_id":"PA28","page_number":"28","snippet_text":"and its \u003cb\u003ederivative\u003c/b\u003e, without having to be able to invert Q(p) to get F(x). The plots of \u003cbr\u003e\nPDF in this book are normally obtained in this fashion. The 99 points are best \u003cbr\u003e\nsupplemented by additional tail probabilities. For example, p = 0.0005, 0.001, \u003cbr\u003e\n0.002\u0026nbsp;..."},{"page_id":"PA97","page_number":"97","snippet_text":"... model, then a model-free approach can be used by using the empirical \u003cbr\u003e\n\u003cb\u003equantile function\u003c/b\u003e given by , defined in Section 2.2. ... 4.5 Approximation Suppose \u003cbr\u003e\nh(x) is some function of x and h′(x), h′′(x), etc. are the first, second, etc. \u003cbr\u003e\n\u003cb\u003ederivatives\u003c/b\u003e\u0026nbsp;..."},{"page_id":"PA103","page_number":"103","snippet_text":"One simple one, suitable for right-tailed or symmetric distributions, compares \u003cbr\u003e\n\u003cb\u003equantile functions\u003c/b\u003e as p approaches one. (See, for example ... This requires the \u003cbr\u003e\n\u003cb\u003ederivative\u003c/b\u003e of the ratio to be greater than or equal to zero. Differentiating and \u003cbr\u003e\nsorting\u0026nbsp;..."},{"page_id":"PA128","page_number":"128","snippet_text":"... that Q(p) + Q(1 – p) = Q(1), for all p. Find a general expression for the IQR of \u003cbr\u003e\nthis distribution as a function of k and β. 5. Show that for a standard normal \u003cbr\u003e\ndistribution the \u003cb\u003ederivatives\u003c/b\u003e of the \u003cb\u003equantile function\u003c/b\u003e satisfy the relations N′(p) = 1/\u003cbr\u003e\nφp(p),\u0026nbsp;..."},{"page_id":"PA199","page_number":"199","snippet_text":"As shown in Section 4.5 the use of the next term in the approximation gives the \u003cbr\u003e\napproximate rankit as μ(r) = Q(pr) + prqr Q′′(pr)/[2(n+ 2)], where pr = r/(n + 1), \u003cbr\u003e\nqr = 1–pr, and Q′′(p) is the second \u003cb\u003ederivative\u003c/b\u003e of Q(p). Example 9.7: For any\u0026nbsp;..."},{"page_id":"PA209","page_number":"209","snippet_text":"The calculation of l(θ) requires the \u003cb\u003ederivation\u003c/b\u003e of p(r) from x(r) = Q(p(r):θ). This can \u003cbr\u003e\nbe carried out directly from F(x(r)) if F(.) is explicit. If not, the iterative method of \u003cbr\u003e\nSection 4.5(c) has to be used. It is seen that in the \u003cb\u003equantile\u003c/b\u003e-based likelihood the\u0026nbsp;..."},{"page_id":"PA210","page_number":"210","snippet_text":"1. Define distribution Q(p;) 2. Derive \u003cb\u003equantile\u003c/b\u003e density q(p;) 3. Set initial or revised \u003cbr\u003e\nparameter values ... Columns (4, 5, 6), (7, 8, 9), ..., (13, 14, 15) give the iterative \u003cbr\u003e\n\u003cb\u003ederivation\u003c/b\u003e of the column p(r). It will be noted that, as required, column 16, (p(r)),\u0026nbsp;..."},{"page_id":"PA215","page_number":"215","snippet_text":"We define two \u003cb\u003efunctions\u003c/b\u003e: The likelihood ratio R(θ) = L(θ)/L() and the log-likelihood \u003cbr\u003e\nratio r(θ) = ln(R(θ)). ... 0, and calculate the criterion, C( 0) from θ θ Formulae for \u003cbr\u003e\n\u003cb\u003ederivatives\u003c/b\u003e of l(θ) Sc(θ), I(θ) Initial values for INTERVALS AND REGIONS 215."},{"page_id":"PA216","page_number":"216","snippet_text":"Formulae for \u003cb\u003ederivatives\u003c/b\u003e of l(θ) Sc(θ), I(θ) Initial values for the end points (θ1, θ2) \u003cbr\u003e\n± √[–2ln(p)/I()] Iterative evaluation for the two end points θnew = θ – [r(θ) – ln(p)]/\u003cbr\u003e\nSc(θ) θ = (θ1, θ2) 100p% likelihood interval (θ1, θ2) θˆ θˆ Table 9.7. Obtaining\u0026nbsp;..."},{"page_id":"PA217","page_number":"217","snippet_text":"These require the setting up of tabular layouts based on initial estimates. We \u003cbr\u003e\ntherefore need to consider briefly the \u003cb\u003ederivation\u003c/b\u003e of these initial values. A number \u003cbr\u003e\nof methods can be used: (a) The method of percentiles often provides a simple \u003cbr\u003e\nand."},{"page_id":"PA296","page_number":"296","snippet_text":"Notice that (a) If S(p) is an explicit \u003cb\u003efunction\u003c/b\u003e of p, then fs(p) is also an explicit \u003cbr\u003e\n\u003cb\u003efunction\u003c/b\u003e even if f(x) does not explicitly exist. (b) po) is itself a ... In differentiating / \u003cbr\u003e\nwe therefore need the \u003cb\u003ederivatives\u003c/b\u003e of po with respect to the parameters. These are\u003cbr\u003e\n\u0026nbsp;..."},{"page_id":"PA317","page_number":"317","snippet_text":"QDF, see \u003cb\u003eQuantile\u003c/b\u003e Density \u003cb\u003eFunction\u003c/b\u003e Q-Q plot, 31, 33, 179, 183 Q-transformations, \u003cbr\u003e\n66, 70, 91, 141 317 Quality ... Rank, 256 Rankit(s), 88 approximate, 98, 199, 202 \u003cbr\u003e\naverage of, 200 \u003cb\u003ederivation\u003c/b\u003e of exact, 258 exact, 199 median, 89, 198, 210, 259,\u0026nbsp;..."}],"search_query_escaped":"derivative of quantile function"},{});</script><input type="text" name="history_state0" id="history_state0" style="display:none" class=""><script type="text/javascript" class="">var _gaq = _gaq || [];_gaq.push(['_setAccount', 'UA-18010-1']);_gaq.push(['_trackPageview']);(function() {var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true;ga.src = ('https:' == document.location.protocol ? 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