Say $X_1,X_2,\ldots$ are independent identically distributed Bernoulli randomvariables with mean $p$. This paper builds a new estimate $\hat p$ of $p$ thathas the property that the relative error, $\hat p /p - 1$, of the estimate doesnot depend in any way on the value of $p$. This allows the construction ofexact confidence intervals for $p$ of any desired level without needing anysort of limit or approximation. In addition, $\hat p$ is unbiased. For$\epsilon$ and $\delta$ in $(0,1)$, to obtain an estimate where$\mathbb{P}(|\hat p/p - 1| > \epsilon) \leq \delta$, the new algorithm takes onaverage at most $2\epsilon^{-2} p^{-1}\ln(2\delta^{-1})(1 - (14/3)\epsilon)^{-1}$ samples. It is also shown that any such algorithm that applieswhenever $p \leq 1/2$ requires at least $0.2\epsilon^{-2}p^{-1}\ln((2-\delta)\delta^{-1})(1 + 2 \epsilon)$ samples. The same algorithmcan also be applied to estimate the mean of any random variable that falls in$[0,1]$.
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机译：假设$ X_1，X_2，\ ldots $是独立的平均分布的Bernoulli随机变量，均值$ p $。本文建立了$ p $的新估计$ \ hat p $，其具有以下属性：该估计的相对误差$ \ hat p / p-1 $完全不取决于$ p $的值。这允许为任何所需水平的$ p $构造精确的置信区间，而无需任何种类的限制或近似值。另外，$ \ hat p $是无偏的。对于$（0,1）$中的$ \ epsilon $和$ \ delta $，以获得估算值，其中$ \ mathbb {P}（| \ hat p / p-1 |> \ epsilon）\ leq \ delta $，新算法的平均费用最高为$ 2 \ epsilon ^ {-2} p ^ {-1} \ ln（2 \ delta ^ {-1}）（1-（14/3）\ epsilon）^ {-1} $样品。还显示了任何这样的算法，只要$ p \ leq 1/2 $都需要至少$ 0.2 \ epsilon ^ {-2} p ^ {-1} \ ln（（2- \ delta）\ delta ^ {-1 }）（1 + 2 \ epsilon）$个样本。同样的算法也可以用于估计落入$ [0,1] $的任何随机变量的平均值。
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