Abstract
An important type of statistical decision problems is characterized by the property that only two different (terminal) decisions concerning the parameter ϑ are possible, i.e. that two disjoint subsets H1 and H2 of Θ are selected25 and that
where di is interpreted as ”Hi holds true” (ϑ∈ Hi). In this case one speaks of a (statistical) testing problem; the sets Hi are called hypotheses. The examples (1.7)/(1.14)a) and (1.8)/(1.14)b) are concerned with problems of this type.
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Notes
It will always be assumed that Pϑ1 ≠ Pϑ2 for ϑ1 ≠ ϑ2 i.e. no superfluous parameter values are considered.
The condition \(\hat{t}\left( x \right) = 0 \forall x \notin \left( {{{k}_{1}},{{k}_{2}}} \right) \) is not needed for this proposition.
But even for SPRT’s the consequences of such approximations seem to be unknown.
This somewhat artificial notation allows a simple distinction between transient (i ≤ m) and absorbing (i > m) states.
The spectral radius ρ(Q) of Q is < 1.
There is no connection between these qij(k,l) and the k-step transition probabilities which are often denoted in a similar way.
If A ≥ 0 and C are (m × m)-matrices s.t. ∣ C ∣≤ A then ρ(C) ≤ ρ(A).
The existence of \(E\left( {\sum\nolimits_{{i = 1}}^{N} {\left| {W\left( {{{Y}_{i}}} \right)} \right|} } \right) \) (see (3.15)) ensures that the order of summation has no influence.
For x ∈ Cm,p ∈ (0,∞) we denote (as usual) \(x \in {{C}^{m}},p \in \left( {0,\infty } \right) \)
For an actual implementation one should use the stopping criterion \(\left| {1 - P{{F}_{n}} - O{{C}_{n}}} \right| < \delta OR {{P}_{\vartheta }}\left( {N > n} \right) < \delta \) in order to avoid that an underflow influences the termination of the program. For this reason, one then has to compute Pϑ(N < n) (according to (3.31)(iv)).
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© 1993 Springer-Verlag New York, Inc.
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Schmitz, N. (1993). Sequentially planned tests; sequentially planned probability ratio tests. In: Schmitz, N. (eds) Optimal Sequentially Planned Decision Procedures. Lecture Notes in Statistics, vol 79. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2736-6_3
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DOI: https://doi.org/10.1007/978-1-4612-2736-6_3
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-97908-3
Online ISBN: 978-1-4612-2736-6
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