اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدImportance sampling for maxima on trees
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We consider the distributional fixed-point equation: $$R stackrel{mathcal{D}}{=} Q vee left( bigvee_{i=1}^N C_i R_i right),$$ where the ${R_i}$ are i.i.d.~copies of $R$, independent of the vector $(Q, N, {C_i})$, where $N in mathbb{N}$, $Q, {C_i} geq 0$ and $P(Q > 0) > 0$. By setting $W = log R$, $X_i = log C_i$, $Y = log Q$ it is equivalent to the high-order Lindley equation $$W stackrel{mathcal{D}}{=} maxleft{ Y, , max_{1 leq i leq N} (X_i + W_i) right}.$$ It is known that under Kesten assumptions, $$P(W > t) sim H e^{-alpha t}, qquad t to infty,$$ where $alpha>0$ solves the Cramer-Lundberg equation $E left[ sum_{j=1}^N C_i ^alpha right] = Eleft[ sum_{i=1}^N e^{alpha X_i} right] = 1$. The main goal of this paper is to provide an explicit representation for $P(W > t)$, which can be directly connected to the underlying weighted branching process where $W$ is constructed and that can be used to construct unbiased and strongly efficient estimators for all $t$. Furthermore, we show how this new representation can be directly analyzed using Alsmeyers Markov renewal theorem, yielding an alternative representation for the constant $H$. We provide numerical examples illustrating the use of this new algorithm.
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