H.M. Moustafa

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Bayes Inference in Constant Partially Accelerated Life Tests for the Generalized Exponential Distribution with Progressive Censoring

2014-06-13

Zeinhum F. Jaheen , H.M. Moustafa , G. H. Abd El-Monem

AbstractIn this paper, the problem of constant partially accelerated life tests when the lifetime follows the generalized exponential distribution is considered. Based on progressive type-II censoring scheme, the maximum likelihood … AbstractIn this paper, the problem of constant partially accelerated life tests when the lifetime follows the generalized exponential distribution is considered. Based on progressive type-II censoring scheme, the maximum likelihood and Bayes methods of estimation are used for estimating the distribution parameters and acceleration factor. A Monte Carlo simulation study is carried out to examine the performance of the obtained estimates.Keywords: Bayesian estimationConstant partially accelerated life testsGeneralized exponential distributionMaximum likelihood estimationMonte Carlo simulationProgressive Type-II censoringMathematics Subject Clssification: 62F1062F1562N0162N05

Information geometry and statistical manifold

2002-10-08

Nassar H. Abdel-All , H.N. Abd-Ellah , H.M. Moustafa

Approximate bayes estimation for mixtures of two weibull distributions under type-2 censoring

1997-06-01

K.E. Ahmad , H.M. Moustafa , Ayman M. Abd–Elrahman

Abstract The main object of this paper is the approximate Bayes estimation of the five dimensional vector of parameters and the reliability function of a mixture of two Weibull distributions … Abstract The main object of this paper is the approximate Bayes estimation of the five dimensional vector of parameters and the reliability function of a mixture of two Weibull distributions under Type-2 censoring. Under Type-2 censoring, the posterior distribution is complicated, and the integrals involved cannot be obtained in a simple closed form. In this work, Lindley's (1980) approximate form of Bayes estimation is used in the case of a mixture of two Weibull distributions under Type-2 censoring. Through Monte Carlo simulation, the root mean squared errors (RMSE's) of the Bayes estimates are computed and compared with the corresponding estimated RMSE's of the maximum likelihood estimates. Keywords: Approximate Bayes estimationmixture of two WeibullsMonte Carlo simulationmaximum likelihood estimates

Estimation of a discriminant function from a mixture of two gamma distributions when the sample size is small

1993-09-01

M.A.W. Mahmoud , H.M. Moustafa

Errors of misclassification and their probability distributions when the parent populations are Gompertz

2004-05-12

H.M. Moustafa , S. G. Ramadan

Errors of misclassification associated with gamma distribution

1995-08-01

M.A.W. Mahmoud , H.M. Moustafa

On MLE of a nonlinear discriminant function from a mixture of two Gompertz distributions based on small sample size

2003-11-11

H.M. Moustafa , S. G. Ramadan

The property of identifiability is an important consideration on estimating the parameters in a mixture of distributions. Also classification of a random variable based on a mixture can be meaning … The property of identifiability is an important consideration on estimating the parameters in a mixture of distributions. Also classification of a random variable based on a mixture can be meaning fully discussed only if the class of all finite mixtures is identifiable. The problem of identifiability of finite mixture of Gompertz distributions is studied. A procedure is presented for finding maximum likelihood estimates of the parameters of a mixture of two Gompertz distributions, using classified and unclassified observations. Based on small sample size, estimation of a nonlinear discriminant function is considered. Throughout simulation experiments, the performance of the corresponding estimated nonlinear discriminant function is investigated.

Updating and asymptotic relative efficiency of a non-linear discriminant function estimated from a mixture of two Gompertz populations

2003-09-12

H.M. Moustafa , S. G. Ramadan

Nonlinear Discriminant Functions for Mixed Random Walk Models

2010-10-29

H.M. Moustafa , S. G. Ramadan , Sohair Mohammed

A procedure is presented for finding maximum likelihood estimates of the parameters of a mixture of two random walk distributions in two cases, using classified and unclassified observations. Based on … A procedure is presented for finding maximum likelihood estimates of the parameters of a mixture of two random walk distributions in two cases, using classified and unclassified observations. Based on small sample size, estimation of nonlinear discriminant functions is considered. Throughout simulation experiments, the performance of the corresponding estimated nonlinear discriminant functions is investigated. The total probabilities of misclassification and percentage biases are evaluated and discussed.

On Some Complete Monotonicity of Functions Related to Generalized <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>k</mi> <mo>−</mo> </math>Gamma Function

2021-05-22

H.M. Moustafa , Hanan Almuashi , Mansour Mahmoud

In this paper, we presented two completely monotonic functions involving the generalized <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>k</mi> <mo>−</mo> </math> gamma function <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <msub> <mrow> <mi>Γ</mi> </mrow> <mrow> <mi>k</mi> </mrow> … In this paper, we presented two completely monotonic functions involving the generalized <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>k</mi> <mo>−</mo> </math> gamma function <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <msub> <mrow> <mi>Γ</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> <mfenced open="(" close=")" separators="|"> <mrow> <mi>x</mi> </mrow> </mfenced> </math> and its logarithmic derivative <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <msub> <mrow> <mi>ψ</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> <mfenced open="(" close=")" separators="|"> <mrow> <mi>x</mi> </mrow> </mfenced> </math> , and established some upper and lower bounds for <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <msub> <mi mathvariant="normal">Γ</mi> <mrow> <mi>k</mi> </mrow> </msub> <mfenced open="(" close=")" separators="|"> <mrow> <mi>x</mi> </mrow> </mfenced> </math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6"> <msub> <mrow> <mi>ψ</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> <mfenced open="(" close=")" separators="|"> <mrow> <mi>x</mi> </mrow> </mfenced> </math> .

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An Asymptotic Expansion for the Generalized Gamma Function

2022-07-09

Mansour Mahmoud , Hanan Almuashi , H.M. Moustafa

The symmetric patterns that inequalities contain are reflected in researchers’ studies in many mathematical sciences. In this paper, we prove an asymptotic expansion for the generalized gamma function Γμ(v) and … The symmetric patterns that inequalities contain are reflected in researchers’ studies in many mathematical sciences. In this paper, we prove an asymptotic expansion for the generalized gamma function Γμ(v) and study the completely monotonic (CM) property of a function involving Γμ(v) and the generalized digamma function ψμ(v). As a consequence, we establish some bounds for Γμ(v), ψμ(v) and polygamma functions ψμ(r)(v), r≥1.

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Some bounds for the k-generalized digamma function

2023-09-16

H.M. Moustafa , Mansour Mahmoud , Ahmed Talat

We presented some monotonicity properties for the k-generalized digamma function $\psi_{k}(h)$ and we established some new bounds for $\psi_{k}^{(s)}(h),$ $s\in \mathbb{N}\cup\{0\},$ which refine recent results We presented some monotonicity properties for the k-generalized digamma function $\psi_{k}(h)$ and we established some new bounds for $\psi_{k}^{(s)}(h),$ $s\in \mathbb{N}\cup\{0\},$ which refine recent results

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Some New Bounds for Bateman’s G-Function in Terms of the Digamma Function

2025-04-08

H.M. Moustafa , Waad Al Sayed

In this paper, we present some new symmetric bounds for Bateman’s G-function and its derivatives, in terms of the digamma and polygamma functions, which are better than some recent results. In this paper, we present some new symmetric bounds for Bateman’s G-function and its derivatives, in terms of the digamma and polygamma functions, which are better than some recent results.

Some New Inequalities for the Gamma and Polygamma Functions

2025-04-14

Waad Al Sayed , H.M. Moustafa

In this paper, we present some new symmetric bounds for the gamma and polygamma functions. For this goal, we present two functions involving gamma and polygamma functions and we investigate … In this paper, we present some new symmetric bounds for the gamma and polygamma functions. For this goal, we present two functions involving gamma and polygamma functions and we investigate their complete monotonicity. Also, we investigate their completely monotonic degrees. This concept gives more accuracy in measuring the complete monotonicity property. These new bounds are better than some of the recently published results.

Some New Inequalities for the Gamma and Polygamma Functions

2025-04-14

Waad Al Sayed , H.M. Moustafa

Some New Bounds for Bateman’s G-Function in Terms of the Digamma Function

2025-04-08

H.M. Moustafa , Waad Al Sayed

Some bounds for the k-generalized digamma function

2023-09-16

H.M. Moustafa , Mansour Mahmoud , Ahmed Talat

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An Asymptotic Expansion for the Generalized Gamma Function

2022-07-09

Mansour Mahmoud , Hanan Almuashi , H.M. Moustafa

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On Some Complete Monotonicity of Functions Related to Generalized <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>k</mi> <mo>−</mo> </math>Gamma Function

2021-05-22

H.M. Moustafa , Hanan Almuashi , Mansour Mahmoud

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Bayes Inference in Constant Partially Accelerated Life Tests for the Generalized Exponential Distribution with Progressive Censoring

2014-06-13

Zeinhum F. Jaheen , H.M. Moustafa , G. H. Abd El-Monem

Nonlinear Discriminant Functions for Mixed Random Walk Models

2010-10-29

H.M. Moustafa , S. G. Ramadan , Sohair Mohammed

Errors of misclassification and their probability distributions when the parent populations are Gompertz

2004-05-12

H.M. Moustafa , S. G. Ramadan

On MLE of a nonlinear discriminant function from a mixture of two Gompertz distributions based on small sample size

2003-11-11

H.M. Moustafa , S. G. Ramadan

Updating and asymptotic relative efficiency of a non-linear discriminant function estimated from a mixture of two Gompertz populations

2003-09-12

H.M. Moustafa , S. G. Ramadan

Information geometry and statistical manifold

2002-10-08

Nassar H. Abdel-All , H.N. Abd-Ellah , H.M. Moustafa

Approximate bayes estimation for mixtures of two weibull distributions under type-2 censoring

1997-06-01

K.E. Ahmad , H.M. Moustafa , Ayman M. Abd–Elrahman

Errors of misclassification associated with gamma distribution

1995-08-01

M.A.W. Mahmoud , H.M. Moustafa

Estimation of a discriminant function from a mixture of two gamma distributions when the sample size is small

1993-09-01

M.A.W. Mahmoud , H.M. Moustafa

Coauthor	Papers Together
S. G. Ramadan	4
Mansour Mahmoud	3
Waad Al Sayed	2
M.A.W. Mahmoud	2
Hanan Almuashi	2
Sohair Mohammed	1
H.N. Abd-Ellah	1
Nassar H. Abdel-All	1
K.E. Ahmad	1
Ahmed Talat	1
Ayman M. Abd–Elrahman	1
G. H. Abd El-Monem	1
Zeinhum F. Jaheen	1

Normal Discrimination with Unclassified Observations

1978-12-01

Terence J. O’Neill

Abstract Fisher's linear discriminant rule may be estimated by maximum likelihood estimation using unclassified observations. It is shown that the ratio of the relevant information contained in unclassified observations to … Abstract Fisher's linear discriminant rule may be estimated by maximum likelihood estimation using unclassified observations. It is shown that the ratio of the relevant information contained in unclassified observations to that in classified observations varies from approximately one-fifth to two-thirds for the statistically interesting range of separation of the populations. Thus, more information may be obtained from large numbers of inexpensive unclassified observations than from a small classified sample. Also, all available unclassified and classified data should be used for estimating Fisher's linear discriminant rule.

Complete monotonicity of some functions involving polygamma functions

2009-10-03

Feng Qi , Bai‐Ni Guo

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Variance comparisons for unbiased estimators of probability of correct classification (Corresp.)

1976-01-01

David S. Moore , S. Whitsitt , D. A. Landgrebe

Variance relationships among certain count estimators and posterior probability estimators of probability of correct classification are investigated. An estimator using posterior probabilities is presented for use in stratified sampling designs. … Variance relationships among certain count estimators and posterior probability estimators of probability of correct classification are investigated. An estimator using posterior probabilities is presented for use in stratified sampling designs. A test case involving three normal classes is examined.

Normal Discrimination with Unclassified Observations

1978-12-01

Terence J. O’Neill

Abstract Abstract Fisher's linear discriminant rule may be estimated by maximum likelihood estimation using unclassified observations. It is shown that the ratio of the relevant information contained in unclassified observations … Abstract Abstract Fisher's linear discriminant rule may be estimated by maximum likelihood estimation using unclassified observations. It is shown that the ratio of the relevant information contained in unclassified observations to that in classified observations varies from approximately one-fifth to two-thirds for the statistically interesting range of separation of the populations. Thus, more information may be obtained from large numbers of inexpensive unclassified observations than from a small classified sample. Also, all available unclassified and classified data should be used for estimating Fisher's linear discriminant rule. Key Words: Unclassified observationsNormal discriminationError rateEfficiencyEstimation from mixtures

Estimation of a discriminant function from a mixture of two gamma distributions when the sample size is small

1993-09-01

M.A.W. Mahmoud , H.M. Moustafa

Estimation of a discriminant function from a mixture of two inverse gaussian distributions when sample size is small

1985-01-01

R.K. Amoh

Abstract Estimation of a discriminant function on the basis of a small sample size from a mixture of two inverse Gaussian distributions is considered. Its performance is investigated by a … Abstract Estimation of a discriminant function on the basis of a small sample size from a mixture of two inverse Gaussian distributions is considered. Its performance is investigated by a series of simulation experiments. The relative efficiency of the mixture and classified discrimination procedures are evaluated from the simulation results and compared with available asymptotic relative efficiency results. Keywords: Discriminant FunctionEstimation From MixturesMixture of Inverse Gaussian PopulationsRelative Efficiency

Nonparametric Bayes error estimation using unclassified samples

1973-07-01

Keinosuke Fukunaga , David Kessell

A new nonparametric method of estimating the Bayes risk using an unclassified test sample set as well as a classified design sample set is introduced. The classified design set is … A new nonparametric method of estimating the Bayes risk using an unclassified test sample set as well as a classified design sample set is introduced. The classified design set is used to obtain nonparametric estimates of the conditional Bayes risk of classification at each point of the unclassified test set. The average of these risk estimates is the error estimate. For large numbers of design samples the new error estimate has less variance than does an error-count estimate for classified test samples using the optimum Bayes classifier. The first application of the nonparametric method uses <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -nearest neighbor ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -NN) estimates of the posterior probabilities to form the risk estimate. A large-sample analysis is made of this estimate. The expected value of this estimate is shown to be a lower bound on the Bayes error. A simple modification provides unbiased estimates of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -NN classification error, thus providing an upper bound on the Bayes error. The second application of the method uses Parzen approximation of the density functions to obtain estimates of the risk and subsequently the Bayes error. Results of experiments on simulated data illustrate the small-sample behavior.

Errors of misclassification associated with gamma distribution

1995-08-01

M.A.W. Mahmoud , H.M. Moustafa

The efficiency of a nonlinear discriminant function based on unclassified initial samples from a mixture of two Burr type XII populations

1995-07-01

K.E. Ahmad

Maximum Likelihood from Incomplete Data Via the <i>EM</i> Algorithm

1977-09-01

A. P. Dempster , N. M. Laird , Donald B. Rubin

Summary A broadly applicable algorithm for computing maximum likelihood estimates from incomplete data is presented at various levels of generality. Theory showing the monotone behaviour of the likelihood and convergence … Summary A broadly applicable algorithm for computing maximum likelihood estimates from incomplete data is presented at various levels of generality. Theory showing the monotone behaviour of the likelihood and convergence of the algorithm is derived. Many examples are sketched, including missing value situations, applications to grouped, censored or truncated data, finite mixture models, variance component estimation, hyperparameter estimation, iteratively reweighted least squares and factor analysis.

Goodness of fit for the extreme value distribution

1977-01-01

Michael A. Stephens

In this paper we present goodness of fit tests for the extreme value distribution, based on the empirical distribution function statistics W2, U2and A2. Asymptotic percentage points are given for … In this paper we present goodness of fit tests for the extreme value distribution, based on the empirical distribution function statistics W2, U2and A2. Asymptotic percentage points are given for each of the three statistics, for the three cases where one or both of the parameters of the distribution must be estimated from the data. Slight modifications of the calculated statistics are given to enable the points to be used with small samples.

Estimating the Linear Discriminant Function from Initial Samples Containing a Small Number of Unclassified Observations

1977-06-01

Geoffrey J. McLachlan

Abstract Abstract Estimation of the linear discriminant function L is considered in the case where there are n 1 and n 2 observations from the populations II1 and II2 and … Abstract Abstract Estimation of the linear discriminant function L is considered in the case where there are n 1 and n 2 observations from the populations II1 and II2 and M unclassified observations. Estimates of L using all n 1 + n 2 + M observations are proposed and evaluated in terms of the expected error rate under the assumption that M is small relative to n 1 and n 2. By appropriately weighting the sample means of the unclassified observations, an estimate of L is given which dominates the usual estimate based on just the n 1 + n 2 classified observations. Key Words: Sample discriminant functionsInitial samples incompletely classifiedConditional and expected error rates

Maximum-Likelihood Estimation of the Parameters of the Gompertz Survival Function

1970-01-01

M. L. Garg , B. Raja Rao , Carol Redmond

On MLE of a nonlinear discriminant function from a mixture of two Gompertz distributions based on small sample size

2003-11-11

H.M. Moustafa , S. G. Ramadan

Estimating the components of a mixture of normal distributions

1969-01-01

N. E. Day

The problem of estimating the components of a mixture of two normal distributions, multivariate or otherwise, with common but unknown covariance matrices is examined. The maximum likelihood equations are shown … The problem of estimating the components of a mixture of two normal distributions, multivariate or otherwise, with common but unknown covariance matrices is examined. The maximum likelihood equations are shown to be not unduly laborious to solve and the sampling properties of the resulting estimates are investigated, mainly by simulation. Moment estimators, minimum χ2 and Bayes estimators are discussed but they appear greatly inferior to maximum likelihood except in the univariate case, the inferiority lying either in the sampling properties of the estimates or in the complexity of the computation. The wider problems obtained by allowing the components in the mixture to have different covariance matrices, or by having more than two components in the mixture, are briefly discussed, as is the relevance of this problem to cluster analysis.

On a (p,k)-analogue of the Gamma function and some associated Inequalities

2016-06-21

Kwara Nantomah , Edward Prempeh , Stephen Boakye Twum

Abstract In this paper, we introduce a new two-parameter deformation of the classical Gamma function, which we call a (p,k)-analogue of the Gamma function. We also provide some identities generalizing … Abstract In this paper, we introduce a new two-parameter deformation of the classical Gamma function, which we call a (p,k)-analogue of the Gamma function. We also provide some identities generalizing those satisfied by the classical Gamma function. Furthermore, we establish some inequalities involving this new function.

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Errors of misclassification associated with the inverse gaussion distribution

1986-01-01

R.K. Amoh , Kathleen Kocherlakota

Errors of misclassification and their probabilities are studied for classification problems associated with univariate inverse Gaussian distributions. The effects of applying the linear discriminant function (LDF), based on normality, to … Errors of misclassification and their probabilities are studied for classification problems associated with univariate inverse Gaussian distributions. The effects of applying the linear discriminant function (LDF), based on normality, to inverse Gaussian populations are assessed by comparing probabilities (optimum and conditional) based on the LDF with those based on the likelihood ratio rule (LR) for the inverse Gaussian, Both theoretical and empirical results are presented

Robustness of the linear discriminant function to nonnormality: Johnson's system

1979-01-01

Enock F. Ching'anda , S. Kocherlakota

Monotonicity properties of the gamma function

2006-10-28

Horst Alzer , Necdet Batır

The efficiency of a linear discriminant function based on unclassified initial samples

1978-01-01

S. Gànesalingam , Geoffrey J. McLachlan

Estimation of the optimal linear discriminant function is considered on the basis of a sample of observations known only to belong to a mixture of two univariate normal populations with … Estimation of the optimal linear discriminant function is considered on the basis of a sample of observations known only to belong to a mixture of two univariate normal populations with a common variance. The asymptotic efficiency of the procedure so obtained is evaluated relative to that of Anderson's classification statistic.

Monotonicity properties on k-digamma function and its related inequalities

2020-01-01

Emrah Yıldırım

In this work, we give some monotonicity properties of k -analogues of digamma and polygamma functions and then we obtain some inequalities related to these functions.At last, we give harmonic … In this work, we give some monotonicity properties of k -analogues of digamma and polygamma functions and then we obtain some inequalities related to these functions.At last, we give harmonic mean inequality for k -digamma function for all positive real values of k and x .

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<i>Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables</i>

1966-01-01

Milton Abramowitz , I. A. Stegun , Jacques E. Romain

Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables

1965-02-01

D. B. Owen , Milton Abramowitz , Irene A. Stegun

Sharp inequalities for the psi function and harmonic numbers

2014-01-28

Bai‐Ni Guo , Feng Qi

In this paper, two sharp inequalities for bounding the psi function $\psi$ and the harmonic numbers $H_n$ are established respectively, some results in [I. Muqattash and M. Yahdi, \textit{Infinite family … In this paper, two sharp inequalities for bounding the psi function $\psi$ and the harmonic numbers $H_n$ are established respectively, some results in [I. Muqattash and M. Yahdi, \textit{Infinite family of approximations of the Digamma function}, Math. Comput. Modelling \textbf{43} (2006), 1329\nobreakdash--1336.] are improved, and some remarks are given.

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Random walk density function with unknown origin

1993-10-01

K.E. Ahmad

Errors in Discrimination

1961-12-01

S. John

The probabilities of misclassification involved in the use of estimated discriminant functions are subject to chance variations. The author's purpose in this paper is to derive the distribution laws that … The probabilities of misclassification involved in the use of estimated discriminant functions are subject to chance variations. The author's purpose in this paper is to derive the distribution laws that the probabilities of misclassification follow and to obtain their expected values. The parent populations are assumed to be normal. The first part of the paper considers the univariate case and the second part the multivariate case. The discussion of the multivariate case proceeds in three stages of increasing complexity. When the exact results are complicated, asymptotic results or approximations are given. Finally, the problem of estimating the expected probabilities of misclassification is considered. Interval estimates as well as point estimates are given.

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Some properties of the gamma and psi functions, with applications

2004-05-18

S.-L. Qiu , Матти Вуоринен

In this paper, some monotoneity and concavity properties of the gamma, beta and psi functions are obtained, from which several asymptotically sharp inequalities follow. Applying these properties, the authors improve … In this paper, some monotoneity and concavity properties of the gamma, beta and psi functions are obtained, from which several asymptotically sharp inequalities follow. Applying these properties, the authors improve some well-known results for the volume <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Subscript n"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Omega _n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the unit ball <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Superscript n Baseline subset-of double-struck upper R Superscript n"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>B</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">B^n\subset \mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the surface area <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega Subscript n minus 1"> <mml:semantics> <mml:msub> <mml:mi>ω</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:msub> <mml:annotation encoding="application/x-tex">\omega _{n-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the unit sphere <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript n minus 1"> <mml:semantics> <mml:msup> <mml:mi>S</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:annotation encoding="application/x-tex">S^{n-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and some related constants.

Iterative Reclassification Procedure for Constructing an Asymptotically Optimal Rule of Allocation in Discriminant Analysis

1975-06-01

Geoffrey J. McLachlan

Abstract The construction of a suitable rule of allocation in the two-population discrimination problem is considered in the case where there are initially available from the populations II1, II2, n … Abstract The construction of a suitable rule of allocation in the two-population discrimination problem is considered in the case where there are initially available from the populations II1, II2, n 1, n 2 observations and M unclassified observations. An iterative reclassification procedure based on the n 1 + n 3 + M observations is proposed and found asymptotically optimal when M → ∞ and n 1 and n 2 are moderately large. The case of finite M is evaluated by a Monte Carlo experiment which suggests that the proposed procedure, after only one iteration, gives a rule with smaller average risk than the usual rule based on just the n 1 + n 2 classified observations.

A class of methods for solving nonlinear simultaneous equations

1965-01-01

C. G. Broyden

solution. The functions that require zeroing are real functions of real variables and it will be assumed that they are continuous and differentiable with respect to these variables. In many … solution. The functions that require zeroing are real functions of real variables and it will be assumed that they are continuous and differentiable with respect to these variables. In many practical examples they are extremely complicated anld hence laborious to compute, an-d this fact has two important immediate consequences. The first is that it is impracticable to compute any derivative that may be required by the evaluation of the algebraic expression of this derivative. If derivatives are needed they must be obtained by differencing. The second is that during any iterative solution process the bulk of the computing time will be spent in evaluating the functions. Thus, the most efficient process will tenid to be that which requires the smallest number of function evaluations. This paper discusses certain modificatioins to Newton's method designed to reduce the number of function evaluations required. Results of various numerical experiments are given and conditions under which the modified versions are superior to the original are tentatively suggested.

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Introduction to Multivariate Statistical Analysis.

1959-05-01

William G. Madow , T. W. Anderson

Small sample results for a nonlinear discriminant function estimated from a mixture of two burr type XII distributions

1994-09-01

K.E. Ahmad

Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions

2014-06-21

Feng Qi , Shu-Hong Wang

In the paper, the authors verify the complete monotonicity of the difference $e^{1/t}-\psi'(t)$ on $(0,\infty)$, compute the completely monotonic degree and establish integral representations of the remainder of the Laurent … In the paper, the authors verify the complete monotonicity of the difference $e^{1/t}-\psi'(t)$ on $(0,\infty)$, compute the completely monotonic degree and establish integral representations of the remainder of the Laurent series expansion of $e^{1/z}$, and derive an inequality which gives a lower bound for the first order modified Bessel function of the first kind.

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Some monotonicity properties and inequalities for the (p, k)-gamma function

2018-01-01

Kwara Nantomah , Faton Merovci , Suleman Nasiru

In this paper, the authors present some complete monotonicity properties and some inequalities involving the (p, k)-analogue of the Gamma function. The established results provide the (p, k)-generalizations for some … In this paper, the authors present some complete monotonicity properties and some inequalities involving the (p, k)-analogue of the Gamma function. The established results provide the (p, k)-generalizations for some results known in the literature.

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Robustness of the linear and quadratic discriminant function to certain types of non‐normality

1973-01-01

Peter A. Lachenbruch , Cheryl Sneeringer , Lawrence T. Revo

Estimation of the probabilities of misclassification for a linear discriminant function in the univariate normal case

1971-12-01

Nell Sedransk , Masashi Okamoto

An Introduction to Multivariate Statistical Analysis

1986-01-01

Robb J. Muirhead , T. W. Anderson

Preface to the Third Edition.Preface to the Second Edition.Preface to the First Edition.1. Introduction.2. The Multivariate Normal Distribution.3. Estimation of the Mean Vector and the Covariance Matrix.4. The Distributions and … Preface to the Third Edition.Preface to the Second Edition.Preface to the First Edition.1. Introduction.2. The Multivariate Normal Distribution.3. Estimation of the Mean Vector and the Covariance Matrix.4. The Distributions and Uses of Sample Correlation Coefficients.5. The Generalized T2-Statistic.6. Classification of Observations.7. The Distribution of the Sample Covariance Matrix and the Sample Generalized Variance.8. Testing the General Linear Hypothesis: Multivariate Analysis of Variance9. Testing Independence of Sets of Variates.10. Testing Hypotheses of Equality of Covariance Matrices and Equality of Mean Vectors and Covariance Matrices.11. Principal Components.12. Cononical Correlations and Cononical Variables.13. The Distributions of Characteristic Roots and Vectors.14. Factor Analysis.15. Pattern of Dependence Graphical Models.Appendix A: Matrix Theory.Appendix B: Tables.References.Index.

Handbook of Mathematical Functions

1966-02-01

Donald A. McQuarrie

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Updating a nonlinear discriminant function estimated from a mixture of two Weibull distributions

1994-06-01

K.E. Ahmad , Ayman M. Abd–Elrahman

Identifiability of finite mixtures using a new transform

1988-06-01

Khalaf E. Ahmad

Mixture Densities, Maximum Likelihood and the EM Algorithm

1984-04-01

Richard A. Redner , Homer F. Walker

The problem of estimating the parameters which determine a mixture density has been the subject of a large, diverse body of literature spanning nearly ninety years. During the last two … The problem of estimating the parameters which determine a mixture density has been the subject of a large, diverse body of literature spanning nearly ninety years. During the last two decades, the method of maximum likelihood has become the most widely followed approach to this problem, thanks primarily to the advent of high speed electronic computers. Here, we first offer a brief survey of the literature directed toward this problem and review maximum-likelihood estimation for it. We then turn to the subject of ultimate interest, which is a particular iterative procedure for numerically approximating maximum-likelihood estimates for mixture density problems. This procedure, known as the EM algorithm, is a specialization to the mixture density context of a general algorithm of the same name used to approximate maximum-likelihood estimates for incomplete data problems. We discuss the formulation and theoretical and practical properties of the EM algorithm for mixture densities, focussing in particular on mixtures of densities from exponential families.

On some properties of the gamma function

2007-10-05

Necdet Batır

Small sample results for a linear discriminant function estimated from a mixture of normal populations

1979-06-01

S. Gànesalingam , Geoffrey J. McLachlan

An investigation is undertaken of the performance of the linear discriminant function estimated from a mixture of two multivariate normal populations with a common covariance matrix when the total number … An investigation is undertaken of the performance of the linear discriminant function estimated from a mixture of two multivariate normal populations with a common covariance matrix when the total number of observations available is small. It is concluded from a series of simulation experiments that although the individual estimates of the discriminant function coefficients so obtained may not be very reliable the resulting discriminant function still provides adequate separation between the populations.

Some monotonicity properties and inequalities for the generalized digamma and polygamma functions

2018-09-20

Li Yin , Li-Guo Huang , Zhi-Min Song +1 more

Several monotonicity and concavity results related to the generalized digamma and polygamma functions are presented. This extends and generalizes the main results of Qi and Guo and others. Several monotonicity and concavity results related to the generalized digamma and polygamma functions are presented. This extends and generalizes the main results of Qi and Guo and others.

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Necessary and sufficient conditions for functions involving the tri- and tetra-gamma functions to be completely monotonic

2009-05-31

Feng Qi , Bai‐Ni Guo

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SCHRÖDINGER EQUATION FOR QUANTUM FRACTAL SPACE–TIME OF ORDER n VIA THE COMPLEX-VALUED FRACTIONAL BROWNIAN MOTION

2001-12-20

Guy Jumarie

First remark: Feynman's discovery in accordance of which quantum trajectories are of fractal nature (continuous everywhere but nowhere differentiable) suggests describing the dynamics of such systems by explicitly introducing the … First remark: Feynman's discovery in accordance of which quantum trajectories are of fractal nature (continuous everywhere but nowhere differentiable) suggests describing the dynamics of such systems by explicitly introducing the Brownian motion of fractional order in their equations. The second remark is that, apparently, it is only in the complex plane that the Brownian motion of fractional order with independent increments can be generated, by using random walks defined with the complex roots of the unity; in such a manner that, as a result, the use of complex variables would be compulsory to describe quantum systems. Here one proposes a very simple set of axioms in order to expand the consequences of these remarks. Loosely speaking, a one-dimensional system with real-valued coordinate is in fact the average observation of a one-dimensional system with complex-valued coordinate: It is a strip modeling. Assuming that the system is governed by a stochastic differential equation driven by a complex valued fractional Brownian of order n, one can then obtain the explicit expression of the corresponding covariant stochastic derivative with respect to time, whereby we switch to the extension of Lagrangian mechanics. One can then derive a Schrödinger equation of order n in quite a direct way. The extension to relativistic quantum mechanics is outlined, and a generalized Klein–Gordon equation of order n is obtained. As a by-product, one so obtains a new proof of the Schrödinger equation.

Constant-partially accelerated life tests for Burr type-XII distribution with progressive type-II censoring

2009-02-09

Alaa H. Abdel‐Hamid

Tracer dilution curves in cardiology and random walk and lognormal distributions.

1966-01-01

Wise Me

Progressive stress accelerated life tests under finite mixture models

2007-01-13

Alaa H. Abdel‐Hamid , Essam K. AL-Hussaini

Generalized exponential distribution: Bayesian estimations

2007-06-14

Debasis Kundu , Rameshwar D. Gupta

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Summations for certain series containing the digamma function

2006-03-08

Allen R. Miller

Apparently new summations in terms of well-known special functions are deduced for hypergeometric-type series containing a digamma function as a factor. As a by-product of this investigation new reduction formulae … Apparently new summations in terms of well-known special functions are deduced for hypergeometric-type series containing a digamma function as a factor. As a by-product of this investigation new reduction formulae for the Kampé de Fériet function F0:2;12:1;0[x, x] are obtained.