Uniformly Accurate Quantile Bounds Via The Truncated Moment Generating Function: The Symmetric Case
| Autor: | Krzysztof Nowicki, Michael J. Klass |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2007 |
| Předmět: |
Statistics and Probability
60G50 quantile approximation 46B09 Absolute value (algebra) Symmetric case Sum of independent rv's Moment-generating function Upper and lower bounds Combinatorics tail probabilities Mathematics::Probability Hoffmann-Jo rgensen/Klass-Nowicki Inequality 46E30 tail distributions Statistics Probability and Uncertainty 60E15 Random variable Mathematics Quantile |
| Zdroj: | Electron. J. Probab. 12 (2007), 1276-1298 |
| Popis: | Let $X_1, X_2, \dots$ be independent and symmetric random variables such that $S_n = X_1 + \cdots + X_n$ converges to a finite valued random variable $S$ a.s. and let $S^* = \sup_{1 \leq n \leq \infty} S_n$ (which is finite a.s.). We construct upper and lower bounds for $s_y$ and $s_y^*$, the upper $1/y$-th quantile of $S_y$ and $S^*$, respectively. Our approximations rely on an explicitly computable quantity $\underline q_y$ for which we prove that $$\frac 1 2 \underline q_{y/2} < s_y^* < 2 \underline q_{2y} \quad \text{ and } \quad \frac 1 2 \underline q_{ (y/4) ( 1 + \sqrt{ 1 - 8/y})} < s_y < 2 \underline q_{2y}. $$ The RHS's hold for $y \geq 2$ and the LHS's for $y \geq 94$ and $y \geq 97$, respectively. Although our results are derived primarily for symmetric random variables, they apply to non-negative variates and extend to an absolute value of a sum of independent but otherwise arbitrary random variables. |
| Databáze: | OpenAIRE |
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