Sequential probability ratio test

As in classical hypothesis testing, SPRT starts with a pair of hypotheses, say ${\displaystyle H_{0}}$ and ${\displaystyle H_{1}}$ for the null hypothesis and alternative hypothesis respectively. They must be specified as follows:

${\displaystyle H_{0}:p=p_{0}}$

${\displaystyle H_{1}:p=p_{1}}$

The next step is to calculate the cumulative sum of the log-likelihood ratio, ${\displaystyle \log \Lambda _{i}}$, as new data arrive: with ${\displaystyle S_{0}=0}$, then, for ${\displaystyle i}$=1,2,...,

${\displaystyle S_{i}=S_{i-1}+\log \Lambda _{i}}$

The stopping rule is a simple thresholding scheme:

• ${\displaystyle a<S_{i}<b}$: continue monitoring (critical inequality)
• ${\displaystyle S_{i}\geq b}$: Accept ${\displaystyle H_{1}}$
• ${\displaystyle S_{i}\leq a}$: Accept ${\displaystyle H_{0}}$

where ${\displaystyle a}$ and ${\displaystyle b}$ (${\displaystyle a<0<b<\infty }$) depend on the desired type I and type II errors, ${\displaystyle \alpha }$ and ${\displaystyle \beta }$. They may be chosen as follows:

${\displaystyle a\approx \log {\frac {\beta }{1-\alpha }}}$ and ${\displaystyle b\approx \log {\frac {1-\beta }{\alpha }}}$

In other words, ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ must be decided beforehand in order to set the thresholds appropriately. The numerical value will depend on the application. The reason for being only an approximation is that, in the discrete case, the signal may cross the threshold between samples. Thus, depending on the penalty of making an error and the sampling frequency, one might set the thresholds more aggressively. The exact bounds are correct in the continuous case.

A textbook example is parameter estimation of a probability distribution function. Consider the exponential distribution:

${\displaystyle f_{\theta }(x)=\theta ^{-1}e^{-{\frac {x}{\theta }}},\qquad x,\theta >0}$

The hypotheses are

${\displaystyle {\begin{cases}H_{0}:\theta =\theta _{0}\\H_{1}:\theta =\theta _{1}\end{cases}}\qquad \theta _{1}>\theta _{0}.}$

Then the log-likelihood function (LLF) for one sample is

${\displaystyle {\begin{aligned}\log \Lambda (x)&=\log \left({\frac {\theta _{1}^{-1}e^{-{\frac {x}{\theta _{1}}}}}{\theta _{0}^{-1}e^{-{\frac {x}{\theta _{0}}}}}}\right)\\&=\log \left({\frac {\theta _{0}}{\theta _{1}}}e^{{\frac {x}{\theta _{0}}}-{\frac {x}{\theta _{1}}}}\right)\\&=\log \left({\frac {\theta _{0}}{\theta _{1}}}\right)+\log \left(e^{{\frac {x}{\theta _{0}}}-{\frac {x}{\theta _{1}}}}\right)\\&=-\log \left({\frac {\theta _{1}}{\theta _{0}}}\right)+\left({\frac {x}{\theta _{0}}}-{\frac {x}{\theta _{1}}}\right)\\&=-\log \left({\frac {\theta _{1}}{\theta _{0}}}\right)+\left({\frac {\theta _{1}-\theta _{0}}{\theta _{0}\theta _{1}}}\right)x\end{aligned}}}$

The cumulative sum of the LLFs for all x is

${\displaystyle S_{n}=\sum _{i=1}^{n}\log \Lambda (x_{i})=-n\log \left({\frac {\theta _{1}}{\theta _{0}}}\right)+\left({\frac {\theta _{1}-\theta _{0}}{\theta _{0}\theta _{1}}}\right)\sum _{i=1}^{n}x_{i}}$

Accordingly, the stopping rule is:

${\displaystyle a<-n\log \left({\frac {\theta _{1}}{\theta _{0}}}\right)+\left({\frac {\theta _{1}-\theta _{0}}{\theta _{0}\theta _{1}}}\right)\sum _{i=1}^{n}x_{i}<b}$

After re-arranging we finally find

${\displaystyle a+n\log \left({\frac {\theta _{1}}{\theta _{0}}}\right)<\left({\frac {\theta _{1}-\theta _{0}}{\theta _{0}\theta _{1}}}\right)\sum _{i=1}^{n}x_{i}<b+n\log \left({\frac {\theta _{1}}{\theta _{0}}}\right)}$

The thresholds are simply two parallel lines with slope ${\displaystyle \log(\theta _{1}/\theta _{0})}$. Sampling should stop when the sum of the samples makes an excursion outside the continue-sampling region.

• CUSUM
• Computerized classification test
• Wald test
• Likelihood-ratio test

• Ghosh, Bhaskar Kumar (1970). Sequential Tests of Statistical Hypotheses. Reading: Addison-Wesley.
• Holger Wilker: Sequential-Statistik in der Praxis, BoD, Norderstedt 2012, ISBN 978-3848232529.

• Wald's Sequential Probability Ratio Test for R by Stéphane Bottine
• Wald's Sequential Probability Ratio Test for Python by Zhenning Yu