General Formulas for the Central and Non-Central Moments of the Multinomial Distribution

Ouimet, Frédéric

doi:10.3390/stats4010002

Cited by 9 publications

(3 citation statements)

References 37 publications

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“…These moments were obtained in Ouimet (2020c) by differentiating the moment generating function, cf. Ouimet (2021a). This lemma is used to estimate the ≍ N −1 errors in (3.14) of the proof of Lemma 3.1, and also as a preliminary result for the proof of Lemma A.2 below.…”

Section: A Technical Lemmasmentioning

confidence: 99%

A precise local limit theorem for the multinomial distribution and some applications

Ouimet

2021

Journal of Statistical Planning and Inference

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In Siotani & Fujikoshi (1984), a precise local limit theorem for the multinomial distribution is derived by inverting the Fourier transform, where the error terms are explicit up to order N −1 . In this paper, we give an alternative (conceptually simpler) proof based on Stirling's formula and a careful handling of Taylor expansions, and we show how the result can be used to approximate multinomial probabilities on most subsets of R d . Furthermore, we discuss a recent application of the result to obtain asymptotic properties of Bernstein estimators on the simplex, we improve the main result in Carter ( 2002) on the Le Cam distance bound between multinomial and multivariate normal experiments while simultaneously simplifying the proof, and we mention another potential application related to finely tuned continuity corrections.

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Section: A Technical Lemmasmentioning

confidence: 99%

A precise local limit theorem for the multinomial distribution and some applications

Ouimet

2021

Journal of Statistical Planning and Inference

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“…Below are the joint central moments (up to four) of the multinomial distribution, which were derived in Ouimet (2020c). This lemma is used several times throughout the article.…”

Section: B Known Resultsmentioning

confidence: 99%

Asymptotic properties of Bernstein estimators on the simplex

Ouimet

2021

Journal of Multivariate Analysis

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In this paper, we study various asymptotic properties (bias, variance, mean squared error, mean integrated squared error, asymptotic normality, uniform strong consistency) for Bernstein estimators of cumulative distribution functions and density functions on the d-dimensional simplex. Our results generalize the ones in Leblanc (2012a) and Babu et al. (2002), which treated the case d = 1, and significantly extend those found in Tenbusch (1994) for the density estimators when d = 2. The density estimator (or smoothed histogram) is closely related to the Dirichlet kernel estimator from Ouimet (2020a), and can also be used to analyze compositional data.

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“…𝑥 ′ = 𝑚 10 𝑚 00 and 𝑦 ′ = 𝑚 01 𝑚 00 (3) With (𝑥 ′ , 𝑦 , ) being the coordinate center of the object [69][70] [71]. The x and y coordinate positions obtained are made the center point of this circle and from this center point a circle is then drawn to mark the area of detected spermatozoa in each input video frame.…”

Section: ) Finding and Storing The Position Of Detected Spermmentioning

confidence: 99%

Abnormality Determination of Spermatozoa Motility Using Gaussian Mixture Model and Matching-based Algorithm

Mas Diyasa,

Saputra,

Gunawan

et al. 2024

Journal of Robotics and Control

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Sperm analysis is an initial step in the examination conducted to identify infertility cases in humans. One aspect of sperm analysis involves observing the movement of spermatozoa and determining whether it is normal or abnormal. Normal spermatozoa movement is characterized by progressive motion at an average speed of 20 µm/second, while abnormal movement includes slow or non-motile spermatozoa. Traditional methods can be employed to assess the normality or abnormality of sperm movement, but they have drawbacks such as time-consuming procedures and diverse results depending on the expertise of the examiner. On the other hand, utilizing Computer-Assisted Sperm Analysis (CASA) equipment provides consistent results, albeit at a relatively high cost. Therefore, this research proposes an alternative method for determining sperm movement abnormalities using the Gaussian Mixture Model (GMM) for background subtraction and a matching-based algorithm to track and analyze the formed trajectories, distinguishing between normal and abnormal sperm movement. Human spermatozoa in real-time are used, and their movements are recorded in video format using a bright field microscope. The testing results for determining sperm movement abnormalities based on the GMM method and matching-based algorithm were successful, particularly in videos recorded at 50 fps recording speed, 20 minutes of liquefaction time, and 40x microscope lens magnification. This condition exhibited the highest average accuracy, with a tracking accuracy of 77.3% and an average accuracy for determining sperm motility abnormalities of 87.7%. Therefore, the combined tracking of sperm movement based on the GMM method and matching-based algorithm can be utilized to identify abnormalities in the movement of human spermatozoa.

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General Formulas for the Central and Non-Central Moments of the Multinomial Distribution

Cited by 9 publications

References 37 publications

A precise local limit theorem for the multinomial distribution and some applications

A precise local limit theorem for the multinomial distribution and some applications

Asymptotic properties of Bernstein estimators on the simplex

Abnormality Determination of Spermatozoa Motility Using Gaussian Mixture Model and Matching-based Algorithm

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