Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   consider an acceptance-rejection (AR) sampler based on deterministic driver sequences.  prove that the discrepancy of an N element sample set generated in this way is bounded by \(\mathcal {O} (N^{-2/3}\log N)\), provided that the target density is twice continuously differentiable with non-vanishing curvature and the AR sampler uses the driver sequence \(\mathcal {K}_M= \{( j \alpha , j \beta ) ~~ mod~~1 \mid j = 1,\ldots ,M\},\) where \(\alpha ,\beta \) are real algebraic numbers such that \(1,\alpha ,\beta \) is a basis of a number field over \(\mathbb {Q}\) of degree 3. For the driver sequence \(\mathcal {F}_k= \{ ({j}/{F_k}, \{{jF_{k-1}}/{F_k}\} ) \mid j=1,\ldots , F_k\},\) where \(F_k\) is the k-th Fibonacci number and \(\{x\}=x-\lfloor x \rfloor \) is the fractional part of a non-negative real number x,  can remove the \(\log \) factor to improve the convergence rate to \(\mathcal {O}(N^{-2/3})\), where again N is the number of samples  accepted.  also introduce a criterion for measuring the goodness of driver sequences. The proposed approach is numerically tested by calculating the star-discrepancy of samples generated for some target densities using \(\mathcal {K}_M\) and \(\mathcal {F}_k\) as driver sequences. These results confirm that achieving a convergence rate beyond \(N^{-1/2}\) is possible in practice using \(\mathcal {K}_M\) and \(\mathcal {F}_k\) as driver sequences in the acceptance-rejection sampler.
['Houying Zhu', 'Josef Dick']
I am writing a research paper on a idea A Discrepancy Bound for Deterministic Acceptance-Rejection Samplers Beyond $N^{-1/2}$ in Dimension 1
[{"from": "human", "value": "I am writing a research paper on a idea A Discrepancy Bound for Deterministic Acceptance-Rejection Samplers Beyond $N^{-1/2}$ in Dimension 1"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   consider an acceptance-rejection (AR) sampler based on deterministic driver sequences.  prove that the discrepancy of an N element sample set generated in this way is bounded by \\(\\mathcal {O} (N^{-2/3}\\log N)\\), provided that the target density is twice continuously differentiable with non-vanishing curvature and the AR sampler uses the driver sequence \\(\\mathcal {K}_M= \\{( j \\alpha , j \\beta ) ~~ mod~~1 \\mid j = 1,\\ldots ,M\\},\\) where \\(\\alpha ,\\beta \\) are real algebraic numbers such that \\(1,\\alpha ,\\beta \\) is a basis of a number field over \\(\\mathbb {Q}\\) of degree 3. For the driver sequence \\(\\mathcal {F}_k= \\{ ({j}/{F_k}, \\{{jF_{k-1}}/{F_k}\\} ) \\mid j=1,\\ldots , F_k\\},\\) where \\(F_k\\) is the k-th Fibonacci number and \\(\\{x\\}=x-\\lfloor x \\rfloor \\) is the fractional part of a non-negative real number x,  can remove the \\(\\log \\) factor to improve the convergence rate to \\(\\mathcal {O}(N^{-2/3})\\), where again N is the number of samples  accepted.  also introduce a criterion for measuring the goodness of driver sequences. The proposed approach is numerically tested by calculating the star-discrepancy of samples generated for some target densities using \\(\\mathcal {K}_M\\) and \\(\\mathcal {F}_k\\) as driver sequences. These results confirm that achieving a convergence rate beyond \\(N^{-1/2}\\) is possible in practice using \\(\\mathcal {K}_M\\) and \\(\\mathcal {F}_k\\) as driver sequences in the acceptance-rejection sampler."}]