Parametric versus nonparametric tolerance regions in detection problems

Abstract

A major problem in statistical quality control is to detect a change in the distribution of independent sequentially observed random vectors. The case of a Gaussian pre-change distribution has been extensively analyzed. Here we are concerned with the non-normal multivariate case. In this setup it is natural to use tolerance regions as detection tools. These regions are defined in terms of density level sets, which can be estimated in a plug-in fashion. Under a normal mixture model we compare, through a simulation study, the performance of such a detection scheme for two density estimators: a (parametric) normal mixture and a (nonparametric) kernel estimator. The problem of the bandwidth choice for the latter is addressed. We also obtain a result concerning the convergence rates of the error probabilities under a general parametric model. Finally, a real data example is discussed.

Appendix: Proof of the theorem

• (a) Using condition (A2), by the mean value theorem there exists a constant C > 0 such that

  $$\left|f_{\hat{\theta}_{n}}(x)-f_{\theta}(x)\right| \leq C\left\|\hat{\theta}_{n}-\theta\right\|_{1}$$

  (8)

  uniformly in x. If there exists a set A with positive probability on which convergence of c_n to c does not hold, then there exists a δ > 0 and a subsequence $c_{n_{j}}$ such that for all j.

  If $c_{n_{j}}>c+\delta$ then, by (8),$\left|c_{n_{j}}-c\right|>\delta$

  $$\begin{array}{*{20}{c}} {1 - \alpha \leqslant \int\limits_{\left\{ {{f_{\hat \theta n}} \geqslant c + \delta } \right\}} {{f_{\bar \theta }} \leqslant \int\limits_{\left\{ {{f_\theta } \geqslant c + \delta - C{{\left\| {{{\hat \theta }_n} - \theta } \right\|}_1}} \right\}} {{f_{{{\hat \theta }_n}}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} } } \\ { \leqslant \int\limits_{\left\{ {{f_\theta } \geqslant c + \delta - C{{\left\| {{{\hat \theta }_n} - \theta } \right\|}_1}} \right\}} {{f_\theta } + C{{\left\| {{{\hat \theta }_n} - \theta } \right\|}_1}\text{Leb}\left\{ {{f_\theta } \geqslant c + \delta - C{{\left\| {{{\hat \theta }_n} - \theta } \right\|}_1}} \right\}.} } \end{array}$$

  We can take n_j sufficiently large so that $C\left\|\hat{\theta}_{n}-\theta\right\|_{1}<\delta / 2$. Then on A

  $$1-\alpha \leq \int\limits_{\left\{f_{\theta} \geq c+\delta / 2\right\}} f_{\theta}+C\left\|\hat{\theta}_{n}-\theta\right\|_{1} \operatorname{Leb}\left\{f_{\theta} \geq c+\delta / 2\right\},$$

  (9)

  which contradicts the definition of c. If we had assumed that $c_{n_{j}}<c-\delta$ we would have reached a similar contradiction.

• (b) Observe that by (8)

  $$\left|P_{n}-\alpha\right| \leq \int\limits_{\left\{f_{\hat{\theta}_{n}} \geq c_{n}\right\}}\left|f_{\hat{\theta}_{n}}-f_{\theta}\right| \leq C\left\|\hat{\theta}_{n}-\theta\right\|_{1} \operatorname{Leb}\left\{f_{\hat{\theta}_{n}} \geq c_{n}\right\}.$$

  Then the conclusion in (b) follows since, for n large, $\operatorname{Leb}\left\{f_{\hat{\theta}_{n}} \geq c_{n}\right\}$ is bounded almost surely. To see this notice that, with probability one, eventually

  $$\operatorname{Leb}\left\{f_{\hat{\theta}_{n}} \geq c_{n}\right\} \leq \frac{2}{c} c_{n} \operatorname{Leb}\left\{f_{\hat{\theta}_{n}} \geq c_{n}\right\} \leq \frac{2}{c} \int\limits_{\left\{f_{\hat{\theta}_{n}} \geq c_{n}\right\}} f_{\hat{\theta}_{n}}=\frac{2}{c}(1-\alpha).$$

• (c) Let M > 0 be a sufficiently large constant. By (8) and (A1) if γ > 1 then, with probability one and for n large enough,

  $$\begin{array}{*{20}{c}} {\int\limits_{\left\{ {{f_{{{\hat \theta }_n}}} \geqslant c + M\left\| {{{\hat \theta }_n} - \theta } \right\|_1^{1/\gamma }} \right\}} {{f_{{{\hat \theta }_n}}} \leqslant \int\limits_{\left\{ {{f_\theta } \geqslant c + M\left\| {{{\hat \theta }_n} - \theta } \right\|_1^{1/\gamma } - C{{\left\| {{{\hat \theta }_n} - \theta } \right\|}_1}} \right\}} {{f_\theta }\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} } } \\ {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + C\text{Leb}\left\{ {{f_\theta } \geqslant c + M\left\| {{{\hat \theta }_n} - \theta } \right\|_1^{1/\gamma } - C{{\left\| {{{\hat \theta }_n} - \theta } \right\|}_1}} \right\}{{\left\| {{{\hat \theta }_n} - \theta } \right\|}_1}} \\ \;\;\;\;\;\;\;\;\;\;\;\; { \leqslant 1 - \alpha - \int\limits_{\left\{ {c < {f_\theta } < c + M\left\| {{{\hat \theta }_n} - \theta } \right\|_1^{1/\gamma } - C{{\left\| {{{\hat \theta }_n} - \theta } \right\|}_1}} \right\}} {{f_\theta } + {C_1}{{\left\| {{{\hat \theta }_n} - \theta } \right\|}_1}} } \\\;\;\;\;\;\; { \leqslant 1 - \alpha - {{\left\| {{{\hat \theta }_n} - \theta } \right\|}_1}\left[ {{C_2}{{\left( {M - 1} \right)}^\gamma } - {C_1}} \right] < 1 - \alpha ,} \end{array}$$

  where C₁, C₂ > 0 are constants. This implies that $c_{n}<c+M\left\|\hat{\theta}_{n}-\theta\right\|_{1}^{1 / \gamma}$. In a similar way it is easy to check that, for large $n, c_{n}>c-M\left\|\hat{\theta}_{n}-\theta\right\|_{1}^{1 / \gamma}$. Analogously, if γ ≤ 1, with probability one we have that

  $$\int\limits_{\left\{f_{\hat{\theta}_{n}} \geq c+M\left\|\hat{\theta}_{n}-\theta\right\|_{1}\right\}} f_{\hat{\theta}_{n}} \leq 1-\alpha-C_{3}\left[(M-C)\left\|\hat{\theta}_{n}-\theta\right\|_{1}\right]^{\gamma}+C_{4}\left\|\hat{\theta}_{n}-\theta\right\|_{1}<1-\alpha$$

  and

  $$\int\limits_{\left\{f_{\hat{\theta}_{n}} \geq c-M\left\|\hat{\theta}_{n}-\theta\right\|_{1}\right\}}f_{\hat{\theta}_{n}} \geq 1-\alpha+C_{5}\left[(M-C)\left\|\hat{\theta}_{n}-\theta\right\|_{1}\right]^{\gamma}-C_{6}\left\|\hat{\theta}_{n}-\theta\right\|_{1}>1-\alpha,$$

  where the C_i’s are positive constants. This proves (5). Regarding (6), using (A1) we have that with probability one, for n sufficiently large,

  $$\begin{array}{*{20}{c}} {\int\limits_{\left\{ {{f_{{{\hat \theta }_n}}} \geqslant {c_n}} \right\}\vartriangle \left\{ {{f_\theta } \geqslant c} \right\}} {{f_\theta } \leqslant \int\limits_{\left\{ {{c_n} + C{{\left\| {{{\hat \theta }_n} - \theta } \right\|}_1} \geqslant {f_\theta } \geqslant {c_n} - C{{\left\| {{{\hat \theta }_n} - \theta } \right\|}_1}} \right\}} {{f_\theta }\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} } } \\ { = O{{\left( {\max \left( {\left| {{c_n} - c} \right|,{{\left\| {{{\hat \theta }_n} - \theta } \right\|}_1}} \right)} \right)}^\gamma }} \end{array}$$

  which completes the proof of the statement in (c).