The lottery has long captivated the imagination of players worldwide, offering the tantalizing possibility of life-changing wins. While winning the lottery is largely a matter of chance, as lottery drawings are typica...The lottery has long captivated the imagination of players worldwide, offering the tantalizing possibility of life-changing wins. While winning the lottery is largely a matter of chance, as lottery drawings are typically random and unpredictable. Some people use the lottery terminal randomly generates numbers for them, some players choose numbers that hold personal significance to them, such as birthdays, anniversaries, or other important dates, some enthusiasts have turned to statistical analysis as a means to analyze past winning numbers identify patterns or frequencies. In this paper, we use order statistics to estimate the probability of specific order of numbers or number combinations being drawn in future drawings.展开更多
For a GMANOVA-MANOVA model with normal error: Y = XB1Z1 T +B2Z2 T+ E, E- Nq×n(0, In (?) ∑), the present paper is devoted to the study of distribution of MLE, ∑, of covariance matrix ∑. The main results obtaine...For a GMANOVA-MANOVA model with normal error: Y = XB1Z1 T +B2Z2 T+ E, E- Nq×n(0, In (?) ∑), the present paper is devoted to the study of distribution of MLE, ∑, of covariance matrix ∑. The main results obtained are stated as follows: (1) When rk(Z) -rk(Z2) ≥ q-rk(X), the exact distribution of ∑ is derived, where z = (Z1,Z2), rk(A) denotes the rank of matrix A. (2) The exact distribution of |∑| is gained. (3) It is proved that ntr{[S-1 - ∑-1XM(MTXT∑-1XM)-1MTXT∑-1]∑}has X2(q_rk(x))(n-rk(z2)) distribution, where M is the matrix whose columns are the standardized orthogonal eigenvectors corresponding to the nonzero eigenvalues of XT∑-1X.展开更多
Objectives We aim to estimate geographic variability in total numbers of infections and infection fatality ratios(IFR;the number of deaths caused by an infection per 1,000 infected people)when the availability and qua...Objectives We aim to estimate geographic variability in total numbers of infections and infection fatality ratios(IFR;the number of deaths caused by an infection per 1,000 infected people)when the availability and quality of data on disease burden are limited during an epidemic.Methods We develop a noncentral hypergeometric framework that accounts for differential probabilities of positive tests and reflects the fact that symptomatic people are more likely to seek testing.We demonstrate the robustness,accuracy,and precision of this framework,and apply it to the United States(U.S.)COVID-19 pandemic to estimate county-level SARS-CoV-2 IFRs.Results The estimators for the numbers of infections and IFRs showed high accuracy and precision;for instance,when applied to simulated validation data sets,across counties,Pearson correlation coefficients between estimator means and true values were 0.996 and 0.928,respectively,and they showed strong robustness to model misspecification.Applying the county-level estimators to the real,unsimulated COVID-19 data spanning April 1,2020 to September 30,2020 from across the U.S.,we found that IFRs varied from 0 to 44.69,with a standard deviation of 3.55 and a median of 2.14.Conclusions The proposed estimation framework can be used to identify geographic variation in IFRs across settings.展开更多
文摘The lottery has long captivated the imagination of players worldwide, offering the tantalizing possibility of life-changing wins. While winning the lottery is largely a matter of chance, as lottery drawings are typically random and unpredictable. Some people use the lottery terminal randomly generates numbers for them, some players choose numbers that hold personal significance to them, such as birthdays, anniversaries, or other important dates, some enthusiasts have turned to statistical analysis as a means to analyze past winning numbers identify patterns or frequencies. In this paper, we use order statistics to estimate the probability of specific order of numbers or number combinations being drawn in future drawings.
基金supported by the National Naural Science Foundation of China(Grant No.1026 1009)Mathematics Tianyuan Youth Foundation of China,
文摘For a GMANOVA-MANOVA model with normal error: Y = XB1Z1 T +B2Z2 T+ E, E- Nq×n(0, In (?) ∑), the present paper is devoted to the study of distribution of MLE, ∑, of covariance matrix ∑. The main results obtained are stated as follows: (1) When rk(Z) -rk(Z2) ≥ q-rk(X), the exact distribution of ∑ is derived, where z = (Z1,Z2), rk(A) denotes the rank of matrix A. (2) The exact distribution of |∑| is gained. (3) It is proved that ntr{[S-1 - ∑-1XM(MTXT∑-1XM)-1MTXT∑-1]∑}has X2(q_rk(x))(n-rk(z2)) distribution, where M is the matrix whose columns are the standardized orthogonal eigenvectors corresponding to the nonzero eigenvalues of XT∑-1X.
基金K.A.and J.L.were supported by a grant from the Benioff Center for Microbiome MedicineThis research used resources of the Oak Ridge Leadership Computing Facility,which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725+5 种基金This manuscript has been coauthored by UT-Battelle,LLC under contract no.DE-AC05-00OR22725 with the U.S.Department of EnergyThe United States Government retains and the publisher,by accepting the article for publication,acknowledges that the United States Government retains a nonexclusive,paid-up,irrevocable,world-wide license to publish or reproduce the published form of this manuscript,or allow others to do so,for United States Government purposesThe Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan(http://energy.gov/downloads/doe-public-access-plan,last accessed September 16,2020)Work at Oak Ridge and Lawrence Berkeley National Laboratories was supported by the DOE Office of Science through the National Virtual Biotechnology Laboratory,a consortium of DOE national laboratories focused on response to COVID-19,with funding provided by the Coronavirus CARES Actwas facilitated by previous breakthroughs obtained through the Laboratory Directed Research and Development Programs of the Lawrence Berkeley and Oak Ridge National Laboratories.M.P.J.was supported by a grant from the Laboratory Directed Research and Development(LDRD)Program of Lawrence Berkeley National Laboratory under U.S.Department of Energy Contract No.DE-AC02-05CH11231Oak Ridge National Laboratory would also like to acknowledge funding from the U.S.National Science Foundation(EF-2133763).
文摘Objectives We aim to estimate geographic variability in total numbers of infections and infection fatality ratios(IFR;the number of deaths caused by an infection per 1,000 infected people)when the availability and quality of data on disease burden are limited during an epidemic.Methods We develop a noncentral hypergeometric framework that accounts for differential probabilities of positive tests and reflects the fact that symptomatic people are more likely to seek testing.We demonstrate the robustness,accuracy,and precision of this framework,and apply it to the United States(U.S.)COVID-19 pandemic to estimate county-level SARS-CoV-2 IFRs.Results The estimators for the numbers of infections and IFRs showed high accuracy and precision;for instance,when applied to simulated validation data sets,across counties,Pearson correlation coefficients between estimator means and true values were 0.996 and 0.928,respectively,and they showed strong robustness to model misspecification.Applying the county-level estimators to the real,unsimulated COVID-19 data spanning April 1,2020 to September 30,2020 from across the U.S.,we found that IFRs varied from 0 to 44.69,with a standard deviation of 3.55 and a median of 2.14.Conclusions The proposed estimation framework can be used to identify geographic variation in IFRs across settings.
