Efficiency evaluations of statistical decision probabilities with multiple alternative hypotheses for quality control

Abstract

Efficiency evaluations of statistical decision probabilities with multiple alternative hypotheses are a prerequisite for data quality control in positioning, navigation, and many other applications. Commonly, one uses a time-consuming simulation technique to obtain the statistical decision probabilities or builds lower and/or upper bounds to control the probability, which may be unconvincing when the bounds are loose. We aim to provide a computationally efficient way to calculate the multivariate statistical decision probabilities when performing data snooping in quality control. However, accurate evaluation of those probabilities is complicated considering the complexity of the critical region where the integration intervals contain a variable corresponding to the one with the largest absolute value. Hence, to improve the calculation of statistical decision probabilities, a simplified algorithm for computing the probabilities under the critical region is proposed based on a series of transformation strategies. We implement the proposed algorithm in a simulated numerical experiment and a GPS single-point positioning experiment. The results show that the probabilities computed with the proposed algorithm approximate the results of the simulation technique, but the proposed algorithm is computationally more efficient.

Appendices

Appendix 1

Standard integration form of F ¹

The general form of the lower triangular matrix ${\varvec{C}}$ extracted via the generalized Cholesky decomposition of $\boldsymbol{\varSigma}_{\varvec{w}} = \varvec{CC}^{{\text{T}}}$ with ${\text{rank}}\left( {\boldsymbol{\varSigma}_{\varvec{w}} } \right) = r \le n$ can be further formulated after grouping the rows whose last nonzero elements have the same column number. This requires a series of row permutations of ${\varvec{C}}$, which result in

$$\left[ {\begin{array}{*{20}c} {c_{11} } & 0 & 0 & \cdots & \cdots & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {c_{{l_{1} 1}} } & 0 & 0 & \cdots & \cdots & 0 & \cdots & 0 \\ * & {c_{{(l_{1} + 1)2}} } & 0 & \cdots & \cdots & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ * & {c_{{(l_{1} + l_{2} )2}} } & 0 & \cdots & \cdots & 0 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \ddots & 0 & \cdots & 0 \\ * & * & \cdots & * & {c_{{(l_{1} + \cdots + l_{r - 1} )r}} } & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ * & * & * & * & {c_{{(l_{1} + \cdots + l_{r} )r}} } & 0 & \cdots & 0 \\ \end{array} } \right]$$

(39)

where the subscripts in (39) denote the row and column numbers and the symbol * denotes a zero or nonzero value. In particular, $c_{sj} = 0$ for $j > r$ with $s,\;j = 1, \cdots ,n$ and $l_{1} + \cdots + l_{r} = n$.

Equation (39) can further be normalized to

$${\varvec{C}}^{1} = \left[ {\begin{array}{*{20}c} {1_{11} } & 0 & 0 & \cdots & \cdots & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {1_{{l_{1} 1}} } & 0 & 0 & \cdots & \cdots & 0 & \cdots & 0 \\ * & {1_{{(l_{1} + 1)2}} } & 0 & \cdots & \cdots & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ * & {1_{{(l_{1} + l_{2} )2}} } & 0 & \cdots & \cdots & 0 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \ddots & 0 & \cdots & 0 \\ * & * & \cdots & * & {1_{{(l_{1} + \cdots + l_{r - 1} )r}} } & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ * & * & * & * & {1_{{(l_{1} + \cdots + l_{r} )r}} } & 0 & \cdots & 0 \\ \end{array} } \right]$$

(40)

The components of the lower and upper limits of the inequality $- \varvec T - \varvec{\mu} < \varvec{C\theta} < \varvec T - \varvec{\mu}$ in (18) should also be adjusted accordingly, accompanied by the above permutation and normalization operators. In particular, interchanges of the elements corresponding to the lower and upper limits are necessary when dividing a negative value during normalizations (Bang et al. 2020). Hence, one can then define the transformed vectors ${\varvec{T}}^{1} = \left[ {t_{1}^{1} ,\; \ldots ,\;t_{n}^{1} } \right]^{{\text{T}}}$ and ${\varvec{\mu}}^{1} = [\mu_{1}^{1} ,\; \ldots ,\;\mu_{n}^{1} ]^{{\text{T}}}$, which are the counterparts of $\varvec T$ and $\varvec{\mu}$ after permutations and normalizations.

Combining (18) with (40), one can find that only the first $r$ variables of ${\varvec{\theta}}$ are constrained and one must take the additional $n - r$ constraints on $\varvec{\theta}^{\prime} = \left[ {\theta_{1} ,\; \ldots ,\;\theta_{r} } \right]^{{\text{T}}}$ into account (Genz and Kwong 2000; Bang et al. 2020). Based on the structure of (40), the $n - r$ constraints are distributed into $r$ groups, each containing $l_{k}$ constraints, where $1 \le l_{k} \le n - r$ and $k = 1,\; \ldots ,\;r \le n$. Then, each set of $l_{k}$ constraints is merged to produce a single constraint on $\theta_{k}$, which is given in the following scalar forms (Genz and Kwong 2000):

$$\underbrace {{\mathop {\max }\limits_{{h_{k - 1} \le s \le h_{k} }} \left( { - \mu_{s}^{1} - t_{s}^{1} - \sum\limits_{j = 1}^{k - 1} {c_{sj}^{1} \theta_{j} } } \right)}}_{{L_{k}^{1} (\theta_{1} ,\theta_{2} , \cdots ,\theta_{k - 1} )}} < \theta_{k} < \underbrace {{\max \left( {L_{k}^{1} ,\;\mathop {\min }\limits_{{h_{k - 1} \le s \le h_{k} }} \left( { - \mu_{s}^{1} + t_{s}^{1} - \sum\limits_{j = 1}^{k - 1} {c_{sj}^{1} \theta_{j} } } \right)} \right)}}_{{U_{k}^{1} (\theta_{1} ,\theta_{2} , \cdots ,\theta_{k - 1} )}}$$

(41)

where $c_{sj}^{1}$ denotes the element in the sth row and jth column of ${\varvec{C}}^{1}$ and $h_{k} = \sum\nolimits_{i = 1}^{k} {l_{i} }$.With (40) and (41), equation (18) is equivalent to

$$F^{1} = \frac{1}{{\sqrt {(2\pi )^{r} } }}\int\limits_{{L_{1}^{1} }}^{{U_{1}^{1} }} {e^{{ - \frac{1}{2}\theta_{1}^{2} }} } \int\limits_{{L_{2}^{1} (\theta_{1} )}}^{{U_{2}^{1} (\theta_{1} )}} {e^{{ - \frac{1}{2}\theta_{2}^{2} }} } \cdots \int\limits_{{L_{r}^{1} (\theta_{1} ,\theta_{2} , \cdots ,\theta_{r - 1} )}}^{{U_{r}^{1} (\theta_{1} ,\theta_{2} , \cdots ,\theta_{r - 1} )}} {e^{{ - \frac{1}{2}\theta_{r}^{2} }} {\text{d}}\varvec{\theta}^{\prime}}$$

(42)

Considering the Gaussian cumulative distribution function (CDF) $\Phi \left( x \right) = \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^{x} {e^{{ - \frac{1}{2}y^{2} }} {\text{d}}y}$, each $\theta_{k}$ can be transformed as $\theta_{k} = \Phi^{ - 1} \left( {z_{k} } \right)$. Then, one has

$$F^{1} = \int\limits_{{\overline{L}_{1}^{1} }}^{{\overline{U}_{1}^{1} }} {\int\limits_{{\overline{L}_{2}^{1} (z_{1} )}}^{{\overline{U}_{2}^{1} (z_{1} )}} { \cdots \int\limits_{{\overline{L}_{r}^{1} (z_{1} ,z_{2} , \cdots ,z_{r - 1} )}}^{{\overline{U}_{r}^{1} (z_{1} ,z_{2} , \cdots ,z_{r - 1} )}} {{\text{d}}{\varvec{z}}} } }$$

(43)

where $\mathop {\varvec{z}}\limits_{r \times 1} = \left[ {z_{1} ,\; \ldots ,\;z_{r} } \right]^{{\text{T}}}$, $\overline{L}_{k}^{1} (z_{1} ,z_{2} , \cdots ,z_{k - 1} ) = \Phi \left( {\mathop {\max }\limits_{{h_{k - 1} \le s \le h_{k} }} \left( { - \mu_{s}^{1} - t_{s}^{1} - \sum\nolimits_{j = 1}^{k - 1} {c_{sj}^{1} \Phi^{ - 1} \left( {z_{j} } \right)} } \right)} \right)$, and $\overline{U}_{k}^{1} (z_{1} ,z_{2} , \cdots ,z_{k - 1} ) = \Phi \left( {\max \left( {\overline{L}_{k}^{1} ,\;\mathop {\min }\limits_{{h_{k - 1} \le s \le h_{k} }} \left( { - \mu_{s}^{1} + t_{s}^{1} - \sum\nolimits_{j = 1}^{k - 1} {c_{sj}^{1} \Phi^{ - 1} \left( {z_{j} } \right)} } \right)} \right)} \right)$.

Substitutions are made by setting $z_{k} = \overline{L}_{k}^{1} + \left( {\overline{U}_{k}^{1} - \overline{L}_{k}^{1} } \right)u_{k}$; thus, (43) can be transformed into

$$F^{1} = \left( {\overline{U}_{1}^{1} - \overline{L}_{1}^{1} } \right)\int\limits_{0}^{1} {\left( {\overline{U}_{2}^{1} (u_{1} ) - \overline{L}_{2}^{1} (u_{1} )} \right)} \; \cdots \int\limits_{0}^{1} {\left( {\overline{U}_{r}^{1} (u_{1} ,u_{2} , \cdots ,u_{r - 1} ) - \overline{L}_{r}^{1} (u_{1} ,u_{2} , \cdots ,u_{r - 1} )} \right)} \int\limits_{0}^{1} {{\text{d}}\varvec u}$$

(44)

where $\mathop u\limits_{r \times 1} = \left[ {u_{1} ,\; \ldots ,\;u_{r} } \right]^{{\text{T}}}$, $\overline{L}_{k}^{1} (u_{1} ,u_{2} , \cdots ,u_{k - 1} ) = \Phi \left( {\mathop {\max }\limits_{{h_{k - 1} \le s \le h_{k} }} \left( { - \mu_{s}^{1} - t_{s}^{1} - \sum\nolimits_{j = 1}^{k - 1} {c_{sj}^{1} \Phi^{ - 1} \left( {\overline{L}_{j}^{1} + \left( {\overline{U}_{j}^{1} - \overline{L}_{j}^{1} } \right)u_{j} } \right)} } \right)} \right)$, and $\overline{U}_{k}^{1} (u_{1} ,u_{2} , \cdots ,u_{k - 1} ) = \Phi \left( {\max \left( {\overline{L}_{k}^{1} ,\;\mathop {\min }\limits_{{h_{k - 1} \le s \le h_{k} }} \left( { - \mu_{s}^{1} + t_{s}^{1} - \sum\nolimits_{j = 1}^{k - 1} {c_{sj}^{1} \Phi^{ - 1} \left( {\overline{L}_{j}^{1} + \left( {\overline{U}_{j}^{1} - \overline{L}_{j}^{1} } \right)u_{j} } \right)} } \right)} \right)} \right)$.

Appendix 2

Results by the transformation algorithm of the first experiment

See tables 1 and 2

Table 1 Values of the statistical decision probabilities $\alpha_{00}$ and $\sum \alpha_{0i}$, the critical value $c_{\alpha }$, and the noncentrality parameter $\delta$ as calculated by the transformation algorithm with varying $\rho$ under different numbers of alternative hypotheses

Table 2 Values of the statistical decision probabilities $\beta_{ii}$, $\beta_{i0}$, and $\sum \gamma_{ij}$ as calculated by the transformation algorithm with varying $\rho$ under different numbers of alternative hypotheses