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About 663 papers for “incomplete beta function”

Derivatives of the Incomplete Beta Function

Robert J. Boik, James F. Robison‐Cox · Journal of Statistical Software · 1998

The incomplete beta function is defined as where Beta(p, q) is the beta function. Dutka (1981) gave a history of the development and numerical evaluation of this function. In this article, an algorithm for computing first and second derivatives of Ix,p,q with respect to p and q is described. The algorithm is useful, for example, when fitting parameters to a censored beta, truncated beta, or a truncated beta-binomial model.

Cited by 31Open access

The Incomplete Beta Function and its Ratio to the Complete Beta Function

David L. Osborn, R. Madey · Mathematics of Computation · 1968

The incomplete beta function, Bx(p, q) = /* yp~l(l -y)q~ldy, and its ratio to the complete beta function Bi(p, q) has been calculated on an IBM-7044 computer to five significant figures and tabulated for the arguments p and q each in the interval from 0.5 to 2.0 with increments of 0.05 and for the parameter x in the interval 0.1 to 1.0 with increments of 0.01.The function was evaluated by first expanding a factor of the integrand in a binomial series and then integrating each term.In order to facilitate computation, one expansion is taken for x ^ 1/2 and another for 1/2 < x ^ 1.The expansions are truncated when the relative error is less than or equal to an arbitrarily predetermined fraction.Typical running time for achieving five significant figures on the IBM-7044 is about one minute for 104 values.Madey and Stephenson [1] and in the authors' present research on the quality factor (QF) for degraded heavy particle spectra observed in solar flare activity, the QF is expressed in terms of complete and incomplete beta functions.The incomplete beta function is defined by the in- Introduction. In a paper published by

Cited by 4Open access

Percentage Points of the Incomplete Beta Function

Robert E. Clark · Journal of the American Statistical Association · 1953

for N=10(1)50, X=1(1)N, and a=.005, .010, .025, and .050, where Ip(X, Y) is Karl Pearson's incomplete beta function ratio.' Values of p for which Ip(X, Y) =.005, 010, .025, .05, .10, .25, .50 have been given by Thompson2 in terms of the arguments vP =2 Y and V2 =2X. The entries in the present table were obtained by inverse linear interpolation of the logarithms of the accumulated frequencies found in Pearson's Tables of the Incomplete Beta Function and by interpolation with Lagrangian coefficients of Thompson's percentage points of the incomplete beta function. For values of p> .2000 these two methods of interpolation gave results which agreed within two units in the fourth significant figure, in spite of the fact that for v2>30 in Thompson's tables double interpolation was employed. Thompson's tables for p .2000 the data were smoothed by taking fourth differences, staying within the limits set by these two methods of interpolation. The data are therefore felt to be accurate within ? 1.5 in the last significant figure. Since binomial sums and the incomplete Beta function appear frequently in statistics the table may be used in a number of problems:

Cited by 10

A Note on the Incomplete Beta Function.

Khursheed Alam · Defense Technical Information Center (DTIC) · 1979

The incomplete beta function arises in various statistical problems. It is known, for example, that the tail probability of the binomial distribution can be expressed as an incomplete beta function. This paper gives some results on a monotonicity property of the incomplete beta function. The given results are shown to have application in a problem of ranking and selection. (Author)

Cited by 0

Computation of the regularized incomplete Beta function

Egorova, V., Gil, A., Segura, J. et al. · BIRD (Basque Center for Applied Mathematics) · 2023

An algorithm for the computation of the regularized incomplete Beta function is described. This function has important applications in areas such as Statistics, Physics and Information Theory. The computation of the function can be carried out through a continued function evaluation supplemented with series and asymptotic expansions when both parameters are large. Numerical tests demonstrate the accuracy of the algorithm and show that our algorithm is more accurate than Matlab’s built-in function betainc for a wide range of parameters.

Cited by 1Open access

Computation of the regularized incomplete Beta function

V. (Vera) Egorova, A. (Amparo) Gil, J. (Javier) Segura et al. · Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the Netherlands · 2023

An algorithm for the computation of the regularized incomplete Beta function is described. This function has important applications in areas such as Statistics, Physics and Information Theory. The computation of the function can be carried out through a continued function evaluation supplemented with series and asymptotic expansions when both parameters are large. Numerical tests demonstrate the accuracy of the algorithm and show that our algorithm is more accurate than Matlab’s built-in function betainc for a wide range of parameters.

Cited by 0Open access

An approximation to the symmetrical incomplete beta function

Joel Cadwell · Biometrika · 1952

An approximation to the symmetrical incomplete beta function Get access J. H. CADWELL J. H. CADWELL Search for other works by this author on: Oxford Academic Google Scholar Biometrika, Volume 39, Issue 1-2, May 1952, Pages 204–207, https://doi.org/10.1093/biomet/39.1-2.204 Published: 01 May 1952

Cited by 5

An asymptotic expansion for the incomplete beta function

B G S Doman · Mathematics of Computation · 1996

A new asymptotic expansion is derived for the incomplete beta function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I left-parenthesis a comma b comma x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">I(a,b,x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which is suitable for large <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, small <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b"> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding="application/x-tex">b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x greater-than 0.5"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0.5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">x &gt; 0.5</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This expansion is of the form <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I left-parenthesis a comma b comma x right-parenthesis tilde upper Q left-parenthesis b comma minus gamma log x right-parenthesis plus StartFraction normal upper Gamma left-parenthesis a plus b right-parenthesis Over normal upper Gamma left-parenthesis a right-parenthesis normal upper Gamma left-parenthesis b right-parenthesis EndFraction x Superscript gamma Baseline sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts upper T Subscript n Baseline left-parenthesis b comma x right-parenthesis slash gamma Superscript n plus 1 Baseline comma"> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="1em"/> <mml:mo>∼</mml:mo> <mml:mspace width="1em"/> <mml:mi>Q</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mo>−</mml:mo> <mml:mi>γ</mml:mi> <mml:mi>log</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mfrac> <mml:mrow> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mfrac> </mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>γ</mml:mi> </mml:mrow> </mml:msup> <mml:munderover> <mml:mo>∑</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:munderover> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msup> <mml:mi>γ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*}I(a,b,x) \quad \sim \quad Q(b, -\gamma \log x) + {\frac {\Gamma (a + b)}{\Gamma (a) \Gamma (b)}} x^{\gamma } \sum ^{\infty }_{n=0}T_{n}(b,x)/ \gamma ^{n+1} , \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q"> <mml:semantics> <mml:mi>Q</mml:mi> <mml:annotation encoding="application/x-tex">Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the incomplete Gamma function ratio and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma equals a plus left-parenthesis b minus 1 right-parenthesis slash 2"> <mml:semantics> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>b</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\gamma = a + (b - 1)/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This form has some advantages over previous asymptotic expansions in this region in which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript n"> <mml:semantics> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">T_{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> depends on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as well as on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b"> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding="application/x-tex">b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Cited by 8Open access

Uniform Asymptotic Expansion for the Incomplete Beta Function

Gergő Nemes, Adri B. Olde Daalhuis · Symmetry Integrability and Geometry Methods and Applications · 2016

1] a uniform asymptotic expansion for the incomplete beta function was derived. It was not obvious from those results that the expansion is actually an asymptotic expansion. We derive a remainder estimate that clearly shows that the result indeed has an asymptotic property, and we also give a recurrence relation for the coefficients.

Cited by 5Open access