Download PDF by Ishiguro M., Sakamoto Y.: A Bayesian approach to binary response curve estimation

By Ishiguro M., Sakamoto Y.

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By Ishiguro M., Sakamoto Y.

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T + fo + -ht + 2" t In the original applications of Faber polynomials, the fi are complex numbers, and the series converges in some neighborhood of infinity. , we work in the ring of formal Laurent series C[[t, fo, hit, 12/t 2, .. ]]. Let F(u) be a polynomial in u of degree r such that F(f) = t + negative powers of t. T Lattice Paths and Faber Polynomials 7 We say that F(u) is a Faber polynomial of f. It is easy to prove by induction that there is exactly one Faber polynomial Fr (u) of degree r, which we call the rth Faber polynomial of f.

Let F(u) be a polynomial in u of degree r such that F(f) = t + negative powers of t. T Lattice Paths and Faber Polynomials 7 We say that F(u) is a Faber polynomial of f. It is easy to prove by induction that there is exactly one Faber polynomial Fr (u) of degree r, which we call the rth Faber polynomial of f. I). Schiffer (1948) gave the generating function ) v- r 1og f (v) - u -_ - L~ F r (u -. 6) may be rewritten in terms of formal power series as log{h(w) - uw} = - r L Fr(u)~ . 00 r=l (1.

In sampling plan terminology [DeGroot (1959)], these games describe sequential sampling plans for binomial populations with y = f-tx + a as the boundary line. It is an easy observation that any game can be represented in terms of a lattice path, by starting in (0,0) and proceeding by a horizontal step if tail (T) was tossed and by a vertical step if head (H) was tossed. 2. The condition (3) is reflected by the fact that any such lattice path, except for the final vertical step, stays below the line y = f-tx + a - 1 (being allowed to touch it).

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A Bayesian approach to binary response curve estimation by Ishiguro M., Sakamoto Y.


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