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<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D">Aos interessados, segue o link do pdf:
<a href="http://bayanbox.ir/view/4479541576394522895/Statistical-Modelling-with-Quantile-function-by-Gilchrist.pdf">
http://bayanbox.ir/view/4479541576394522895/Statistical-Modelling-with-Quantile-function-by-Gilchrist.pdf</a><o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D"><o:p> </o:p></span></p>
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<p class="MsoNormal"><b><span style="font-size:10.0pt;font-family:"Tahoma","sans-serif"">De:</span></b><span style="font-size:10.0pt;font-family:"Tahoma","sans-serif""> abe [mailto:abe-bounces@lists.ime.usp.br]
<b>Em nome de </b>Josemar Rodrigues<br>
<b>Enviada em:</b> segunda-feira, 29 de fevereiro de 2016 17:55<br>
<b>Para:</b> Jorge Bazan; abe-l ABE<br>
<b>Assunto:</b> [ABE-L] Statistical Modelling with Quantile Functions - Warren Gilchrist - Google Livros<o:p></o:p></span></p>
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<p class="MsoNormal"><o:p> </o:p></p>
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<p class="MsoNormal">Caros,<o:p></o:p></p>
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<p class="MsoNormal">este livro sera importante para nós. Abs. Josemar<o:p></o:p></p>
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<p class="MsoNormal">derivative of quantile function<o:p></o:p></p>
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<p class="MsoNormal"><a href="https://books.google.com.br/books?id=7c1LimP_e-AC&pg=PA14&lpg=PA14&dq=derivative+of+quantile+function&source=bl&ots=nakuEstOFg&sig=Ls2_lQYRYJzLf-R-7sDBcN0sGfg&hl=pt-BR&sa=X&output=html_text" title="Usuários de leitores de tela: para usar o modo de acessibilidade, é preciso clicar neste link. O modo de acessibilidade tem os mesmos recursos básicos, mas funciona melhor com seu leitor de tela."><span style="text-decoration:none"><img border="0" width="1" height="1" id="_x0033_0793BCE-079E-4029-80C0-B608DF75FC40" src="cid:image001.gif@01D17EA7.66B7BA10" alt="Usuários de leitores de tela: para usar o modo de acessibilidade, é preciso clicar neste link. O modo de acessibilidade tem os mesmos recursos básicos, mas funciona melhor com seu leitor de tela."></span></a><o:p></o:p></p>
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<h2><a href="https://books.google.com.br/books?hl=pt-BR">Livros</a><o:p></o:p></h2>
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<p class="MsoNormal"><span class="jfk-button-label">Adicionar à minha biblioteca</span><img border="0" width="21" height="21" id="_x0037_6045D0D-F4B6-414B-B911-2FB502567358" src="cid:image008.png@01D17EA7.66B7BA10"><o:p></o:p></p>
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<p class="MsoNormal"><a href="http://books.google.com.br">books.google.com.br</a> - Galton used quantiles more than a hundred years ago in describing data. 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...<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">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 class="main-title">Statistical
Modelling with Quantile Functions</span><img border="0" width="128" height="200" id="_x0000_i1037" src="cid:image012.jpg@01D17EA7.66B7BA10" alt="Statistical Modelling with Quantile Functions"><o:p></o:p></p>
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<p class="MsoNormal" style="background:white"><span style="font-size:10.0pt;color:#333333">Thus the PDF, f(x) = dF/dx, is the
<b>derivative</b> of the CDF. Figure 1.5 illustrates the <br>
form of a PDF. Note that this population <b>function</b> corresponds to the sample plot
<br>
of Dp/Dx against x. Example 1.4: For the uniform distribution F(x) = x, so f(x) = 1.<o:p></o:p></span></p>
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<p class="MsoNormal" style="background:white"><b><span style="font-size:10.0pt;color:#333333">Página 12<o:p></o:p></span></b></p>
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<div>
<p class="MsoNormal" style="background:white"><span style="font-size:10.0pt;color:#333333">0 -2 0 0.2 2 4 6 x 0.4 0.6 As mathematics has developed it has ensured that all
<br>
well-tabulated and used <b>functions</b> have <b>derivatives</b> that are well tabulated and
<br>
used. Hence if a distribution has a CDF that can be explicitly expressed in terms
<br>
of ...<o:p></o:p></span></p>
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<p class="MsoNormal" style="background:white"><b><span style="font-size:10.0pt;color:#333333">Página 14<o:p></o:p></span></b></p>
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<div>
<p class="MsoNormal" style="background:white"><span style="font-size:10.0pt;color:#333333">For now it is sufficient to note that we cannot invert this
<b>quantile function</b> to get p <br>
in terms of x, so that no explicit CDF, or consequently PDF, exists. The PDF ... <br>
This <b>derivative</b> is the Quantile Density Function, QDF, defined by q(p) = dQ(p)/dp.<o:p></o:p></span></p>
</div>
</div>
<div style="border:solid #6B90DA 1.0pt;padding:5.0pt 5.0pt 5.0pt 5.0pt;cursor:pointer">
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<p class="MsoNormal" style="background:white"><b><span style="font-size:10.0pt;color:#333333">Página 19<o:p></o:p></span></b></p>
</div>
<div>
<p class="MsoNormal" style="background:white"><span style="font-size:10.0pt;color:#333333">This model is called the logistic distribution. Figure 1.9 shows the addition
<br>
operation for the <b>quantile</b> density <b>functions</b> of the exponential and reflected
<br>
exponential. As the <b>derivative</b> of a sum is the sum of the <b>derivatives</b>, the addition
<br>
of <b>quantile</b> ...<o:p></o:p></span></p>
</div>
</div>
<div style="border:solid #6B90DA 1.0pt;padding:5.0pt 5.0pt 5.0pt 5.0pt;cursor:pointer">
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<p class="MsoNormal" style="background:white"><b><span style="font-size:10.0pt;color:#333333">Página 22<o:p></o:p></span></b></p>
</div>
<div>
<p class="MsoNormal" style="background:white"><span style="font-size:10.0pt;color:#333333">Unlike the exponential and the symmetric logistic, this
<b>quantile function</b> cannot be <br>
inverted to give x in terms of p. Thus there is no ... It has a <b>quantile function</b> in
<br>
basic form: S(p) = 1/(1–p)β, β > 0. ... Figure 1.12 shows its <b>derivation</b> and form.<o:p></o:p></span></p>
</div>
</div>
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<p class="MsoNormal" style="background:white"><b><span style="font-size:10.0pt;color:#333333">Página 27<o:p></o:p></span></b></p>
</div>
<div>
<p class="MsoNormal" style="background:white"><span style="font-size:10.0pt;color:#333333">The
<b>quantile</b> density <b>function</b> and the p-PDF are closely related. As x = Q(p) and
<br>
p = F(x) for any pair of values (x, p), it follows from the definition of <b>differentiation</b>
<br>
that (dx/dp)(dp/dx) = 1 so (dQ(p)/dp)(dF(x)/dx)=1, hence q(p)f(x) = 1 and, <br>
therefore ...<o:p></o:p></span></p>
</div>
</div>
<div style="border:solid #6B90DA 1.0pt;padding:5.0pt 5.0pt 5.0pt 5.0pt;cursor:pointer">
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<p class="MsoNormal" style="background:white"><b><span style="font-size:10.0pt;color:#333333">Página 28<o:p></o:p></span></b></p>
</div>
<div>
<p class="MsoNormal" style="background:white"><span style="font-size:10.0pt;color:#333333">and its
<b>derivative</b>, without having to be able to invert Q(p) to get F(x). The plots of
<br>
PDF in this book are normally obtained in this fashion. The 99 points are best <br>
supplemented by additional tail probabilities. For example, p = 0.0005, 0.001, <br>
0.002 ...<o:p></o:p></span></p>
</div>
</div>
<div style="border:solid #6B90DA 1.0pt;padding:5.0pt 5.0pt 5.0pt 5.0pt;cursor:pointer">
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<p class="MsoNormal" style="background:white"><b><span style="font-size:10.0pt;color:#333333">Página 97<o:p></o:p></span></b></p>
</div>
<div>
<p class="MsoNormal" style="background:white"><span style="font-size:10.0pt;color:#333333">... model, then a model-free approach can be used by using the empirical
<br>
<b>quantile function</b> given by , defined in Section 2.2. ... 4.5 Approximation Suppose
<br>
h(x) is some function of x and h′(x), h′′(x), etc. are the first, second, etc. <br>
<b>derivatives</b> ...<o:p></o:p></span></p>
</div>
</div>
<div style="border:solid #6B90DA 1.0pt;padding:5.0pt 5.0pt 5.0pt 5.0pt;cursor:pointer">
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<p class="MsoNormal" style="background:white"><b><span style="font-size:10.0pt;color:#333333">Página 103<o:p></o:p></span></b></p>
</div>
<div>
<p class="MsoNormal" style="background:white"><span style="font-size:10.0pt;color:#333333">One simple one, suitable for right-tailed or symmetric distributions, compares
<br>
<b>quantile functions</b> as p approaches one. (See, for example ... This requires the
<br>
<b>derivative</b> of the ratio to be greater than or equal to zero. Differentiating and
<br>
sorting ...<o:p></o:p></span></p>
</div>
</div>
<div style="border:solid #6B90DA 1.0pt;padding:5.0pt 5.0pt 5.0pt 5.0pt;cursor:pointer">
<div>
<p class="MsoNormal" style="background:white"><b><span style="font-size:10.0pt;color:#333333">Página 128<o:p></o:p></span></b></p>
</div>
<div>
<p class="MsoNormal" style="background:white"><span style="font-size:10.0pt;color:#333333">... that Q(p) + Q(1 – p) = Q(1), for all p. Find a general expression for the IQR of
<br>
this distribution as a function of k and β. 5. Show that for a standard normal <br>
distribution the <b>derivatives</b> of the <b>quantile function</b> satisfy the relations N′(p) = 1/<br>
φp(p), ...<o:p></o:p></span></p>
</div>
</div>
<div style="border:solid #6B90DA 1.0pt;padding:5.0pt 5.0pt 5.0pt 5.0pt;cursor:pointer">
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<p class="MsoNormal" style="background:white"><b><span style="font-size:10.0pt;color:#333333">Página 199<o:p></o:p></span></b></p>
</div>
<div>
<p class="MsoNormal" style="background:white"><span style="font-size:10.0pt;color:#333333">As shown in Section 4.5 the use of the next term in the approximation gives the
<br>
approximate rankit as μ(r) = Q(pr) + prqr Q′′(pr)/[2(n+ 2)], where pr = r/(n + 1),
<br>
qr = 1–pr, and Q′′(p) is the second <b>derivative</b> of Q(p). Example 9.7: For any ...<o:p></o:p></span></p>
</div>
</div>
<div style="border:solid #6B90DA 1.0pt;padding:5.0pt 5.0pt 5.0pt 5.0pt;cursor:pointer">
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<p class="MsoNormal" style="background:white"><b><span style="font-size:10.0pt;color:#333333">Página 209<o:p></o:p></span></b></p>
</div>
<div>
<p class="MsoNormal" style="background:white"><span style="font-size:10.0pt;color:#333333">The calculation of l(θ) requires the
<b>derivation</b> of p(r) from x(r) = Q(p(r):θ). This can <br>
be carried out directly from F(x(r)) if F(.) is explicit. If not, the iterative method of
<br>
Section 4.5(c) has to be used. It is seen that in the <b>quantile</b>-based likelihood the ...<o:p></o:p></span></p>
</div>
</div>
<div style="border:solid #6B90DA 1.0pt;padding:5.0pt 5.0pt 5.0pt 5.0pt;cursor:pointer">
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<p class="MsoNormal" style="background:white"><b><span style="font-size:10.0pt;color:#333333">Página 210<o:p></o:p></span></b></p>
</div>
<div>
<p class="MsoNormal" style="background:white"><span style="font-size:10.0pt;color:#333333">1. Define distribution Q(p;) 2. Derive
<b>quantile</b> density q(p;) 3. Set initial or revised <br>
parameter values ... Columns (4, 5, 6), (7, 8, 9), ..., (13, 14, 15) give the iterative
<br>
<b>derivation</b> of the column p(r). It will be noted that, as required, column 16, (p(r)), ...<o:p></o:p></span></p>
</div>
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<p class="MsoNormal" style="background:white"><b><span style="font-size:10.0pt;color:#333333">Página 215<o:p></o:p></span></b></p>
</div>
<div>
<p class="MsoNormal" style="background:white"><span style="font-size:10.0pt;color:#333333">We define two
<b>functions</b>: The likelihood ratio R(θ) = L(θ)/L() and the log-likelihood <br>
ratio r(θ) = ln(R(θ)). ... 0, and calculate the criterion, C( 0) from θ θ Formulae for
<br>
<b>derivatives</b> of l(θ) Sc(θ), I(θ) Initial values for INTERVALS AND REGIONS 215.<o:p></o:p></span></p>
</div>
</div>
<div style="border:solid #6B90DA 1.0pt;padding:5.0pt 5.0pt 5.0pt 5.0pt;cursor:pointer">
<div>
<p class="MsoNormal" style="background:white"><b><span style="font-size:10.0pt;color:#333333">Página 216<o:p></o:p></span></b></p>
</div>
<div>
<p class="MsoNormal" style="background:white"><span style="font-size:10.0pt;color:#333333">Formulae for
<b>derivatives</b> of l(θ) Sc(θ), I(θ) Initial values for the end points (θ1, θ2)
<br>
± √[–2ln(p)/I()] Iterative evaluation for the two end points θnew = θ – [r(θ) – ln(p)]/<br>
Sc(θ) θ = (θ1, θ2) 100p% likelihood interval (θ1, θ2) θˆ θˆ Table 9.7. Obtaining ...<o:p></o:p></span></p>
</div>
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<p class="MsoNormal" style="background:white"><b><span style="font-size:10.0pt;color:#333333">Página 217<o:p></o:p></span></b></p>
</div>
<div>
<p class="MsoNormal" style="background:white"><span style="font-size:10.0pt;color:#333333">These require the setting up of tabular layouts based on initial estimates. We
<br>
therefore need to consider briefly the <b>derivation</b> of these initial values. A number
<br>
of methods can be used: (a) The method of percentiles often provides a simple <br>
and.<o:p></o:p></span></p>
</div>
</div>
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<div>
<p class="MsoNormal" style="background:white"><b><span style="font-size:10.0pt;color:#333333">Página 296<o:p></o:p></span></b></p>
</div>
<div>
<p class="MsoNormal" style="background:white"><span style="font-size:10.0pt;color:#333333">Notice that (a) If S(p) is an explicit
<b>function</b> of p, then fs(p) is also an explicit <br>
<b>function</b> even if f(x) does not explicitly exist. (b) po) is itself a ... In differentiating /
<br>
we therefore need the <b>derivatives</b> of po with respect to the parameters. These are<br>
...<o:p></o:p></span></p>
</div>
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<p class="MsoNormal" style="background:white"><b><span style="font-size:10.0pt;color:#333333">Página 317<o:p></o:p></span></b></p>
</div>
<div>
<p class="MsoNormal" style="background:white"><span style="font-size:10.0pt;color:#333333">QDF, see
<b>Quantile</b> Density <b>Function</b> Q-Q plot, 31, 33, 179, 183 Q-transformations,
<br>
66, 70, 91, 141 317 Quality ... Rank, 256 Rankit(s), 88 approximate, 98, 199, 202
<br>
average of, 200 <b>derivation</b> of exact, 258 exact, 199 median, 89, 198, 210, 259, ...<o:p></o:p></span></p>
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<h1>Statistical Modelling with Quantile Functions<o:p></o:p></h1>
<p class="MsoNormal"><span class="addmd">Por Warren Gilchrist</span><o:p></o:p></p>
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