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MSTAT:020117 Mathematical problem title View problem statement Mstat2 Poisson Ump Mlr Karlin Rubin Mstat2 Poisson Composite Boundary Size Power
Neyman-Pearson testing and power Mathematical Statistics II · Mixed review
— Neyman-Pearson testing and power Mathematical Statistics II · Mixed review
— View access options Mstat2 Exponential Clt Studentized Mean Mstat2 Wald Interval Studentized Mean
Consistency, asymptotic normality, and delta methods Mathematical Statistics II · Mixed review
— Consistency, asymptotic normality, and delta methods Mathematical Statistics II · Mixed review
— View access options Mstat2 Hodges Superefficiency Pointwise Mstat2 Hodges Nonuniform Local Sequence
Consistency, asymptotic normality, and delta methods Mathematical Statistics II · Mixed review
— View access options Mstat2 Beta Binomial Marginal Likelihood Mstat2 Bayes Factor Equal Odds Posterior
Bayesian inference and decision rules Mathematical Statistics II · Mixed review
— View access options Mstat2 Normal Normal Precision Update Mstat2 Posterior Versus Posterior Predictive
Bayesian inference and decision rules Mathematical Statistics II · Mixed review
— View access options Mstat2 Weighted Least Squares Normal Equations Mstat2 Wls Gls Covariance Convention
Linear models and projection geometry Mathematical Statistics II · Mixed review
— Linear models and projection geometry Mathematical Statistics II · Mixed review
— Linear models and projection geometry Mathematical Statistics II · Mixed review
— View access options Mstat2 Exact Sign Test Binomial Tail Mstat2 Sign Test Two Sided Ties Convention
Rank and distribution-free methods Mathematical Statistics II · Mixed review
— View access options Mstat2 Spearman Rho From Rank Differences Mstat2 Spearman Exact Permutation Tail
Rank and distribution-free methods Mathematical Statistics II · Mixed review
— Bootstrap, permutation, and simulation-based inference Mathematical Statistics II · Mixed review
— View access options Mstat2 Parametric Bootstrap Exponential Mle Mstat2 Parametric Bootstrap Bias Variance
Bootstrap, permutation, and simulation-based inference Mathematical Statistics II · Mixed review
— View access options Mstat2 Antithetic Variate Identity Mstat2 Antithetic Variance Reduction Covariance Sign
Bootstrap, permutation, and simulation-based inference Mathematical Statistics II · Mixed review
— Showing 15 of 15 matching problems.
Poisson UMP: Xi iid Pois(theta), n=2, S~Pois(2theta); H0 theta<=1 vs >; LR incr in S; reject S>=5; alpha=1-7e^{-2}; power(theta=2)=1-(103/3)e^{-4}
Holm step-down alpha=.05, sorted p=(.004,.018,.020,.20); thresholds=(.0125,.0166667,.025,.05); reject H1 only and STOP; adj p=(.016,.054,.054,.20)
Exp(rate 2): mu=.5, var=.25; CLT sqrt(n)(xbar-.5)->N(0,.25); S_n->.5; T->N01; n=100, xbar=.58, s=.50 => T=1.6; 95% CI(.482,.678) includes .5
Laplace(0,b=2), f(0)=1/4: sqrt(n) median->N(0,4); n=100 SE=.2, med=.30 => z=1.5; 95% CI(-.092,.692); mean AV=8/n, median AV=4/n, ARE=2
Hodges: 0 if |xbar|<=n^{-1/4} else xbar; pointwise N01 for theta!=0, degenerate at 0; n=1e4 thresh=.1; along theta_n=.5 n^{-1/4}, set 0 w.p.->1 but sqrt(n)(tilde-theta_n)-> -inf
Beta-binomial BF10: n=10,x=8,H0 p=1/2,H1 p~Beta(1,1); m0=45/1024,m1=1/11,BF10=1024/495,P(H1|x)=1024/1519; post Beta(9,3) mean 3/4
Normal-normal: theta~N(10,4), Xi~N(theta,9), n=4, xbar=14; post prec 25/36, mean 314/25, 95% (10.208,14.912); pred N(314/25,261/25)
WLS ledger: X rows (1,0),(1,1),(1,2), W=diag(1,2,1), y=(1,2,2); beta=(5/4,1/2), fitted (5/4,7/4,9/4), wSSE=1/4, Cov=sigma^2 (X'WX)^{-1}
OVB: centered y=beta1 x+beta2 z+u, E[u|x,z]=0; plim=beta1+beta2 Cov(x,z)/Var x; (2,3,Varx=4,Cov=2)=>7/2; z=.5x+v same; zero iff beta2 or Cov vanishes
Ridge: ||y-Xb||^2+lambda||b||^2, X'X=diag(4,1), X'y=(8,3), lambda=1; OLS(2,3), ridge(8/5,3/2), shrink(4/5,1/2), df=13/10, bias=(-2/5,-3/2)
Exact sign test n=10, B=8: Bin(10,1/2), one-sided p=56/1024=7/128=.0546875; doubled two-sided 7/64=.109375; no reject at 5%; ties dropped
Spearman x=(1,2,3,4), y=(1,3,2,4): sum d^2=2, rho=4/5; 4 of 24 have rho>=.8 so one-sided 1/6; 8 have |rho|>=.8 so two-sided 1/3
Jackknife (1,2,4): thetaHat=xbar^2=49/9; loo squares (9,25/4,9/4), mean 35/6; bias=7/9; corrected=14/3; pseudovalues (-5/3,23/6,71/6); V=553/36
Parametric bootstrap Exp rate n=4,sum=10: MLE=2/5; S*~Gamma(4,rate 2/5); E*[4/S*]=8/15, bias=2/15, corrected=4/15, Var*=32/225
Antithetic MC for E[U^2]=1/3 with 2 evals: plain var 2/45; A=(U-1/2)^2+1/4 mean 1/3 var 1/180; Cov=-7/90, corr=-7/8, reduction 8