A bias analysis of Weibull models under heaped data by Augustin T., Wolff J. PDF

By Augustin T., Wolff J.

Retrospectively accrued length information are usually mentioned incorrectly. a big kind of such an mistakes is heaping - respondents are inclined to round-off or round-up the knowledge in keeping with a few rule of thumb. for 2 particular situations of the Weibull version we research the behaviour of the 'naive estimators', which easily forget about the dimension errors because of heaping, and derive closed expressions for the asymptotic bias. those effects supply a proper justification of empirical proof and simulation-based findings mentioned within the literature. also, events the place a amazing bias needs to be anticipated will be pointed out, and a precise bias correction may be played.

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Thus we can write c(i) = (p(i) AND c ( i - D ) OR g ( i ) . We evaluate this recurrence using parallel prefix, where the associative operation is a 2-by-2 Boolean matrix multiplication ( c(i) \ _ fp(i) AND c{i - 1) OR g(i) \ [T)-{ _fp« ~\F T T]{ 9(i)\(c{i-l)\ ) T J = c(i) • c(» -1) *... * c(i) * c(o) * (^ ) where c(-l) = F. The Boolean matrix multiplication is associative, because AND and OR satisfy the same associative and distributive laws as multiplication and addition. This algorithm is called carry look-ahead.

For example, consider the division of 90 by 9 viewed as 8 bit numbers. 90 is given by 01011010 and 9 is given by 00001001 in binary representation. Then 011011010 -00001001 1111011111011010 + 00001001 111110000|11010 + 00001001 11111001111010 + 00001001 1111110011010 + 00001001 000000100110 00001001 111110111|0 + 00001001 000000000| 00001001 11110111 + 00001001 00000000 -> n e g a t i v e -> 0 -> n e g a t i v e -> 0 -> negative -> 0 -> n e g a t i v e -> 0 -> p o s i t i v e -> 1 -> negative -> 0 -> p o s i t i v e -> 1 -> negative -> 0 Remainder The least significant bit is computed last.

For n even, the set of bent functions is the set of functions / : {0,1}™ —> {0,1} such that \Wf(w)\ = 2% Vii)e{0,l}" Find all bent functions for n — 2 and n = 4. Solution 21. We identify x with an integer y(x) £ Z according to y(x) = x0 + 2xx + • • • + 2"- 1 :r n _ 1 and similarly for w. , 2™ — 1 for w and x. The function / is represented as an element of {0,1} 2 ". ,2n-l. The following C++ program finds all bent functions for n = 2. The value for const size_t n should be changed to 4 to find all the bent functions for n = 4.

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A bias analysis of Weibull models under heaped data by Augustin T., Wolff J.


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