## The Annals of Statistics

- Ann. Statist.
- Volume 11, Number 3 (1983), 827-836.

### Second Order Approximation to the Risk of a Sequential Procedure

#### Abstract

Given $X_1, X_2, \cdots$, i.i.d. with mean $\mu$ and variance $\sigma^2$, suppose that at stage $n$ one wishes to estimate $\mu$ by the sample mean $\bar{X}_n$, subject to the loss function $L_n = A(\bar{X}_n - \mu)^2 + n, A > 0$. If $\sigma$ is known, the optimal fixed sample size $n_0 \approx A^{1/2}\sigma$ can be used, with corresponding risk $R_{n_0}$, but if $\sigma$ is unknown there is no fixed sample size procedure that will achieve the risk $R_{n_0}$. For the sequential estimation procedure with stopping rule $T = \inf\{n \geq n_A: n^{-1} \sum^n_1 (X_i - \bar{X}_n)^2 \leq A^{-1} n^2\}$, the second order approximation of Woodroofe (1977) to the risk $R_T$ for normal $X_i$ is extended to the distribution-free case. Specifically, if the $X_i$ have finite moments of order greater than eight and are non-lattice, under certain conditions on the delay $n_A$ it is shown that the regret $R_T - R_{n_0} = c + o(1)$ as $A \rightarrow \infty$, where $c$ depends on the first four moments of the distribution of the $X_i$. For the lattice case, bounds of the form $c_1 + o(1) \leq R_T - R_{n_0} \leq c_2 + o(1)$ are obtained, where the $c_j$ are $c \pm 3$. It follows from these approximations that the regret can take arbitrarily large negative values as the distribution of the $X_i$ varies, in contrast to previous results for normal and gamma cases.

#### Article information

**Source**

Ann. Statist., Volume 11, Number 3 (1983), 827-836.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176346249

**Digital Object Identifier**

doi:10.1214/aos/1176346249

**Mathematical Reviews number (MathSciNet)**

MR707933

**Zentralblatt MATH identifier**

0521.62067

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62L12: Sequential estimation

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

**Keywords**

Sequential estimation regret nonlinear renewal theory uniform integrability

#### Citation

Martinsek, Adam T. Second Order Approximation to the Risk of a Sequential Procedure. Ann. Statist. 11 (1983), no. 3, 827--836. doi:10.1214/aos/1176346249. https://projecteuclid.org/euclid.aos/1176346249

#### Corrections

- See Correction: Adam T. Martinsek. Correction: Second Order Approximation to the Risk of a Sequential Procedure. Ann. Statist., Volume 19, Number 4 (1991), 2283--2283.Project Euclid: euclid.aos/1176348404
- See Correction: Adam T. Martinsek. Correction: Second Order Approximation to the Risk of a Sequential Procedure. Ann. Statist., Volume 14, Number 1 (1986), 359--359.Project Euclid: euclid.aos/1176349864