Computational Statistics in Data Science - Группа авторов

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alt="x element-of script í’³"/>. The constant xi dictates the rate of convergence of the Markov chain. Ergodic Markov chains on finite state spaces are polynomially ergodic. On general state spaces, demonstrating at least polynomial ergodicity usually requires a separate study of the sampler, and we provide some references in Section 6.

      3.1 Means

      Recall that normal upper Lamda equals upper V a r Subscript upper F Baseline left-parenthesis h left-parenthesis upper X right-parenthesis right-parenthesis. For MCMC sampling, a key quantity of interest will be

StartLayout 1st Row 1st Column upper Sigma 2nd Column equals sigma-summation Underscript k equals negative infinity Overscript infinity Endscripts Cov Subscript upper F Baseline left-parenthesis h left-parenthesis upper X 1 right-parenthesis comma h left-parenthesis upper X Subscript 1 plus k Baseline right-parenthesis right-parenthesis 2nd Row 1st Column Blank 2nd Column equals normal upper Lamda plus sigma-summation Underscript k equals 1 Overscript infinity Endscripts left-bracket Cov Subscript upper F Baseline left-parenthesis h left-parenthesis upper X 1 right-parenthesis comma h left-parenthesis upper X Subscript 1 plus k Baseline right-parenthesis right-parenthesis plus Cov Subscript upper F Baseline left-parenthesis h left-parenthesis upper X 1 right-parenthesis comma h left-parenthesis upper X Subscript 1 plus k Baseline right-parenthesis right-parenthesis Superscript upper T Baseline right-bracket EndLayout

      1 IID. Let . If , then, as ,

      2 MCMC. Let be polynomially ergodic of order where such that , then if is positive‐definite, as ,

      3.2 Quantiles

      Let

StartLayout 1st Row 1st Column sigma squared left-parenthesis phi Subscript q Baseline right-parenthesis 2nd Column equals sigma-summation Underscript k equals negative infinity Overscript infinity Endscripts Cov left-parenthesis upper I left-parenthesis upper V 1 less-than-or-equal-to phi Subscript q Baseline right-parenthesis comma upper I left-parenthesis upper V Subscript 1 plus k Baseline less-than-or-equal-to phi Subscript q Baseline right-parenthesis right-parenthesis 2nd Row 1st Column Blank 2nd Column equals upper V a r left-parenthesis upper I left-parenthesis upper V 1 less-than-or-equal-to phi Subscript q Baseline right-parenthesis right-parenthesis plus 2 sigma-summation Underscript k equals 1 Overscript infinity Endscripts Cov left-parenthesis upper I left-parenthesis upper V 1 less-than-or-equal-to phi Subscript q Baseline right-parenthesis comma upper I left-parenthesis upper V Subscript 1 plus k Baseline less-than-or-equal-to phi Subscript q Baseline right-parenthesis right-parenthesis EndLayout

      An asymptotic distribution for sample quantiles is available under both IID Monte Carlo and MCMC.

      Theorem 2.

      Let upper F Subscript h be absolutely continuous, twice differentiable with density f Subscript h, and let f prime Subscript h be bounded within some neighborhood of ModifyingAbove phi With Ì‚ Subscript q.

      1 IID. Let , then

      2 MCMC. [11] If the Markov chain is polynomially ergodic of order and , then

      The density value, f Subscript v Baseline left-parenthesis phi Subscript q Baseline right-parenthesis, can be estimated using a Gaussian kernel density estimator. In addition, sigma squared left-parenthesis phi Subscript q Baseline right-parenthesis is replaced with sigma squared left-parenthesis ModifyingAbove phi With Ì‚ Subscript q Baseline right-parenthesis, the univariate version of upper Sigma for h left-parenthesis upper V Subscript t Baseline right-parenthesis equals upper I left-parenthesis upper V Subscript t Baseline less-than-or-equal-to ModifyingAbove phi With Ì‚ Subscript q Baseline right-parenthesis. We present methods of estimating sigma squared left-parenthesis ModifyingAbove phi With Ì‚ Subscript q Baseline right-parenthesis in Section 4 .

      3.3 Other Estimators