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Karteikarten

SB2.1

Ben Tidswell Part B uni Englisch 59 Karten

SB2.1

Karten in diesem Stapel (59)

  1. 1Suppose X~f(x;θ) and let T(X) be a statistic. State The Factorisation CriterionSuppose X~f(x;θ) and let T(X) be a statistic. State The Factorisation Criterion
  2. 2State The Invariance Property for MLEsState The Invariance Property for MLEs
  3. 3Suppose Regs 2-4 hold and that 0Suppose Regs 2-4 hold and that 0
  4. 4State Regs 1-4,2'-4'State Regs 1-4,2'-4'
  5. 5Suppose Regs 1,2'-4' hold and that I_X(θ)is not singular. State the Cramer-Rao lower bound in m dimensionsSuppose Regs 1,2'-4' hold and that I_X(θ)is not singular. State the Cramer-Rao lower bound in m dimensions
  6. 6Let X~P_θ and let T be a sufficient statistic. State the Rao-Blackwell Theorem.Let X~P_θ and let T be a sufficient statistic. State the Rao-Blackwell Theorem.
  7. 7State the theorem regarding Completeness for exponential familiesState the theorem regarding Completeness for exponential families
  8. 8State the Lehmann-Scheffé TheoremState the Lehmann-Scheffé Theorem
  9. 9Suppose that ℙ={P_θ:θ∈Θ} is a family of distributions on 𝒳. State the NASC for MVUE Theorem(Should be MVUE)Suppose that ℙ={P_θ:θ∈Θ} is a family of distributions on 𝒳. State the NASC for MVUE Theorem
  10. 10State Bayes' TheoremState Bayes' Theorem
  11. 11State De Finetti's TheoremState De Finetti's Theorem
  12. 12State Stein's ParadoxState Stein's Paradox
  13. 13State the Bayes test for simple hypothesesState the Bayes test for simple hypotheses
  14. 14Suppose X~f(x;θ) for some parameter θ. State what it means for a statistic T(X) to be sufficient for θSuppose X~f(x;θ) for some parameter θ. State what it means for a statistic T(X) to be sufficient for θ
  15. 15Suppose X~f(x;θ) for some parameter θ and T(x) is a sufficient statistic for θ. State what it means for T to be minimal sufficientSuppose X~f(x;θ) for some parameter θ and T(x) is a sufficient statistic for θ. State what it means for T to be minimal sufficient
  16. 16Let {P_θ:θ∈Θ} be a statistical model and T(x) a statistic. Define what it means for {P_θ:θ∈Θ} to be complete and T to be completeLet {P_θ:θ∈Θ} be a statistical model and T(x) a statistic. Define what it means for {P_θ:θ∈Θ} to be complete and T to be complete
  17. 17When is a family 𝒫={P_θ:θ∈Θ} of probabilities on some set 𝒳, indexed by θ, called an exponential familyWhen is a family 𝒫={P_θ:θ∈Θ} of probabilities on some set 𝒳, indexed by θ, called an exponential family
  18. 18Suppose X_1,X_2,...,X_n are i.i.d. Exp(λ) random variables. State the distribution of Σ_(i=1)^n(X_i)Suppose X_1,X_2,...,X_n are i.i.d. Exp(λ) random variables. State the distribution of Σ_(i=1)^n(X_i)
  19. 19State the mean and variance of an Exp(λ) distributionState the mean and variance of an Exp(λ) distribution
  20. 20When Regs 1-3 are satisfied, define the Fisher informationWhen Regs 1-3 are satisfied, define the Fisher information
  21. 21When Regs 1-3 are satisfied, define the score functionWhen Regs 1-3 are satisfied, define the score function
  22. 22In the case that g(θ)=θ, give an if and only if statement for a statistic T(x) attaining the Cramer-Rao Lower Bound relating to the score function and Fisher informationIn the case that g(θ)=θ, give an if and only if statement for a statistic T(x) attaining the Cramer-Rao Lower Bound relating to the score...
  23. 23State the derivative w.r.t. x of the integral of a function h(t) w.r.t. t between two functions g(x) and f(x)State the derivative w.r.t. x of the integral of a function h(t) w.r.t. t between two functions g(x) and f(x)
  24. 24For a given rule Δ∈𝒟 and a parameter θ∈Θ, define the (frequentist) riskFor a given rule Δ∈𝒟 and a parameter θ∈Θ, define the (frequentist) risk
  25. 25Define the Bayes (integrated) risk for a decision rule Δ and a prior π(θ)Define the Bayes (integrated) risk for a decision rule Δ and a prior π(θ)
  26. 26Define what it means for a decision rule Δ to be a Bayes rule w.r.t. a prior π(θ)Define what it means for a decision rule Δ to be a Bayes rule w.r.t. a prior π(θ)
  27. 27State the Bayes rule under the zero-one loss with interval radius b ->0, under the absolute loss, and under the quadratic lossState the Bayes rule under the zero-one loss with interval radius b ->0, under the absolute loss, and under the quadratic loss
  28. 28State the p.m.f. of a Poisson(λ) distributionState the p.m.f. of a Poisson(λ) distribution
  29. 29State the p.m.f. of an NB(r,p) distributionState the p.m.f. of an NB(r,p) distribution
  30. 30Describe what is meant by empirical Bayes methodsDescribe what is meant by empirical Bayes methods
  31. 31State the p.d.f. of a Beta(α,β) distributionState the p.d.f. of a Beta(α,β) distribution
  32. 32Suppose τ_1<<τ_2. State the Kullback-Leibler divergence KL(τ_1∥τ_2)Suppose τ_1&lt;&lt;τ_2. State the Kullback-Leibler divergence KL(τ_1∥τ_2)
  33. 33Suppose τ_1<<τ_2. Define the Radon-Nikodym derivativeSuppose τ_1&lt;&lt;τ_2. Define the Radon-Nikodym derivative
  34. 34Consider a model (L(θ,x))_(θ∈Θ,x∈𝒳). Define what it means for a family of prior distributions (π_γ)_(γ∈Γ) to be conjugateConsider a model (L(θ,x))_(θ∈Θ,x∈𝒳). Define what it means for a family of prior distributions (π_γ)_(γ∈Γ) to be conjugate
  35. 35Suppose that X|θ~P_θ and that θ~π. State hypothesis under which we can find a Bayes estimatorSuppose that X|θ~P_θ and that θ~π. State hypothesis under which we can find a Bayes estimator
  36. 36Let Δ^π be a Bayes rule w.r.t. π with finite Bayes risk. Prove that if Δ^π is unique then it is admissibleLet Δ^π be a Bayes rule w.r.t. π with finite Bayes risk. Prove that if Δ^π is unique then it is admissible
  37. 37State the mean and variance of a Poisson(λ) distributionState the mean and variance of a Poisson(λ) distribution
  38. 38State the variance of a random variable X in terms of it's momentsState the variance of a random variable X in terms of it's moments
  39. 39Suppose (X_1,...,X_n) is a sample of i.i.d. P_θ distributed random variables, where θ∈Θ is the parameter. Let m_r be the r^th moment of X and assume there exists a function h s.t γ=h(m_1,...,m_r). Define the moment estimator for γSuppose (X_1,...,X_n) is a sample of i.i.d. P_θ distributed random variables, where θ∈Θ is the parameter. Let m_r be the r^th moment of X...
  40. 40State the p.d.f. of a N(μ,σ^2) distributionState the p.d.f. of a N(μ,σ^2) distribution
  41. 41Give another term to describe an exponential family as regular, and define itGive another term to describe an exponential family as regular, and define it
  42. 42State which Gamma distribution a χ^2_p is and give its meanState which Gamma distribution a χ^2_p is and give its mean
  43. 43Suppose the functions f(x;θ)=exp[Σ_(j=1)^(k)η_j(θ)T_j(x)-B(θ)]h(x) form a strictly k-parameter exponential family. Let X=(X_1,...,X_n), where X1,...,X_n~f(x;θ) are i.i.d. State a minimal sufficient estimator for θSuppose the functions f(x;θ)=exp[Σ_(j=1)^(k)η_j(θ)T_j(x)-B(θ)]h(x) form a strictly k-parameter exponential family. Let X=(X_1,...,X_n), w...
  44. 44Let f(x,θ) be the density of X assuming θ=θ. State the Bayes factor for the simple hypothesis H_0:θ=θ_0 against the composite hypothesis H_1:θ~g_1(θ), θ∈ΘLet f(x,θ) be the density of X assuming θ=θ. State the Bayes factor for the simple hypothesis H_0:θ=θ_0 against the composite hypothesis ...
  45. 45Let X be a non-negative random variable with finite mean μ and finite variance σ^2. State Markov's inequality and Chebyshev's inequalityLet X be a non-negative random variable with finite mean μ and finite variance σ^2. State Markov's inequality and Chebyshev's inequality
  46. 46State the p.m.f. of a multinomial distribution with n trials and event probabilities p_1,...,p_kState the p.m.f. of a multinomial distribution with n trials and event probabilities p_1,...,p_k
  47. 47Let X be a random variable. State Jensen's inequality and state when it is strictLet X be a random variable. State Jensen's inequality and state when it is strict
  48. 48Define a curved exponential familyDefine a curved exponential family
  49. 49State the marginal likelihood of a random variable X given its likelihood L(θ,x) and a prior π(θ)State the marginal likelihood of a random variable X given its likelihood L(θ,x) and a prior π(θ)
  50. 50State the result regarding reparameterization of Fisher informationState the result regarding reparameterization of Fisher information
  51. 51Assuming regs 1-3, state and prove the expectation of the score functionAssuming regs 1-3, state and prove the expectation of the score function
  52. 52Define the entropy of a distribution πDefine the entropy of a distribution π
  53. 53State a theorem regarding the maximum entropy priorState a theorem regarding the maximum entropy prior
  54. 54Define what it means for a family of distributions to be of monotone likelihood ratioDefine what it means for a family of distributions to be of monotone likelihood ratio
  55. 55Define a uniformly most powerful testDefine a uniformly most powerful test
  56. 56State the three most common loss functionsState the three most common loss functions
  57. 57Show that the Bayes rule minimises the expected posterior lossShow that the Bayes rule minimises the expected posterior loss
  58. 58Define the maximum a posteriori (MAP) testDefine the maximum a posteriori (MAP) test
  59. 59Show that the Bayes estimator under the quadratic loss is the posterior meanShow that the Bayes estimator under the quadratic loss is the posterior mean

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uni Part B Karteikarten am SB2.1, erstellt von Ben Tidswell am 07/04/2023.
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