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Some binary start-up demonstration tests and associated inferential methods

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Abstract

During the past few decades, substantial research has been carried out on start-up demonstration tests. In this paper, we study the class of binary start-up demonstration tests under a general framework. Assuming that the outcomes of the start-up tests are described by a sequence of exchangeable random variables, we develop a general form for the exact waiting time distribution associated with the length of the test (i.e., number of start-ups required to decide on the acceptance or rejection of the equipment/unit under inspection). Approximations for the tail probabilities of this distribution are also proposed. Moreover, assuming that the probability of a successful start-up follows a beta distribution, we discuss several estimation methods for the parameters of the beta distribution, when several types of observed data have been collected from a series of start-up tests. Finally, the performance of these estimation methods and the accuracy of the suggested approximations for the tail probabilities are illustrated through numerical experimentation.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada

    N. Balakrishnan

  2. Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli & Dimitriou St., 185 34 , Piraeus, Greece

    M. V. Koutras

  3. Department of Mathematics and Statistics, University of Cyprus, P.O. Box: 20537, 1678 , Nicosia, Cyprus

    F. S. Milienos

Authors
  1. N. Balakrishnan
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  2. M. V. Koutras
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  3. F. S. Milienos
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Correspondence to F. S. Milienos.

Appendices

Appendix A

Before proceeding to the partial derivatives of the log-likelihood function from (11), it is necessary to mention that

$$\begin{aligned} \frac{\partial B(x,y)}{\partial x}=B(x,y)(\Psi (x)\!-\!\Psi (x+y)) \text { and } \frac{\partial B(x,y)}{\partial y}\!=\!B(x,y)(\Psi (y)-\Psi (x+y)), \end{aligned}$$

where \(\Psi (x)=\Gamma '(x)/\Gamma (x)\) is the digamma function. The following recurrence relation is also very useful in the sequel:

$$\begin{aligned} \Psi (x + 1) = \Psi (x) + \frac{1}{x}, \text { for every } x. \end{aligned}$$

Thus, the partial derivatives of the log-likelihood function for Case C (see (11)) are as follows:

$$\begin{aligned}&\frac{\partial l(\alpha ,\beta )}{\partial \alpha }=-m (\Psi (\alpha )-\Psi (\alpha +\beta ))\\&\qquad +\sum \limits _{i=1}^{m} \frac{\sum _{k=0}^{n_i}v_{n_ik}B(\alpha +k,\beta +n_i-k)(\Psi (\alpha +k)-\Psi (\alpha +\beta +n_i))}{\sum _{t=0}^{n_i}v_{n_it} B(\alpha +t,\beta +n_i-t)}\\&\quad =-m (\Psi (\alpha )-\Psi (\alpha +\beta ))-\sum _{i=1}^{m} \Psi (\alpha +\beta +n_i)\nonumber \\&\qquad +\sum \limits _{i=1}^{m} \frac{\sum _{k=0}^{n_i}v_{n_ik}B(\alpha +k,\beta +n_i-k)\Psi (\alpha +k)}{\sum _{t=0}^{n_i}v_{n_it} B(\alpha +t,\beta +n_i-t)}\\&\quad =-m (\Psi (\alpha )-\Psi (\alpha +\beta ))-\sum \limits _{i=1}^{m}\left( \Psi (\alpha +\beta )+\sum \limits _{r=1}^{n_i}\frac{1}{\alpha +\beta +n_i-r}\right) \\&\qquad +\sum \limits _{i=1}^{m} \frac{\sum _{k=0}^{n_i}v_{n_ik}B(\alpha +k,\beta +n_i-k)(\Psi (\alpha )+\sum _{r=0}^{k-1}\frac{1}{\alpha +r})}{\sum _{t=0}^{n_i}v_{n_it} B(\alpha +t,\beta +n_i-t)}\\&\quad =-m (\Psi (\alpha )-\Psi (\alpha +\beta ))-m\Psi (\alpha +\beta )-\sum \limits _{i=1}^{m}\sum \limits _{r=1}^{n_i}\frac{1}{\alpha +\beta +n_i-r}\\&\qquad \!+\!\sum \limits _{i=1}^{m} \frac{\sum _{k=0}^{n_i}v_{n_ik}B(\alpha \!+\!k,\beta \!+\!n_i-k)\Psi (\alpha )\!+\!\sum _{k=0}^{n_i}v_{n_ik}B(\alpha \!+\!k,\beta \!+\!n_i-k)\sum _{r=0}^{k-1}\frac{1}{\alpha \!+\!r}}{\sum _{t=0}^{n_i}v_{n_it} B(\alpha \!+\!t,\beta \!+\!n_i-t)}\\&\quad =-\sum \limits _{i=1}^{m}\sum \limits _{r=1}^{n_i}\frac{1}{\alpha +\beta +n_i-r}+\sum \limits _{i=1}^{m} \frac{\sum _{k=0}^{n_i}v_{n_ik}B(\alpha +k,\beta +n_i-k)\sum _{r=0}^{k-1}\frac{1}{\alpha +r}}{\sum _{t=0}^{n_i}v_{n_it}B(\alpha +t,\beta +n_i-t)}\\&\quad =-\sum \limits _{i=1}^{m}\sum \limits _{r=0}^{n_i-1}\frac{1}{\alpha +\beta +r}+\sum \limits _{i=1}^{m}\sum \limits _{k=0}^{n_i}\sum \limits _{r=0}^{k-1}\frac{\pi _{ik}(\alpha ,\beta )}{\alpha +r} \end{aligned}$$

and

$$\begin{aligned} \frac{\partial l(\alpha ,\beta )}{\partial \beta }&= -m (\Psi (\beta )-\Psi (\alpha +\beta ))\\&\quad \!+\!\sum \limits _{i=1}^{m} \frac{\sum _{k=0}^{n_i}v_{n_ik}B(\alpha +k,\beta \!+\!n_i-k)(\Psi (\beta \!+\!n_i-k)\!-\!\Psi (\alpha \!+\!\beta \!+\!n_i))}{\sum _{t=0}^{n_i}v_{n_it} B(\alpha \!+\!t,\beta \!+\!n_i-t)}\\&= -m (\Psi (\beta )-\Psi (\alpha +\beta ))-\sum _{i=1}^{m} \Psi (\alpha +\beta +n_i)\\&\quad +\sum \limits _{i=1}^{m} \frac{\sum _{k=0}^{n_i}v_{n_ik}B(\alpha +k,\beta +n_i-k)\Psi (\beta +n_i-k)}{\sum _{t=0}^{n_i}v_{n_it} B(\alpha +t,\beta +n_i-t)}\\&= \!-\!\sum \limits _{i=1}^{m}\sum \limits _{r=0}^{n_i-1}\frac{1}{\alpha \!+\!\beta \!+\!r}\!+\!\sum \limits _{i=1}^{m} \frac{\sum _{k=0}^{n_i}v_{n_ik}B(\alpha \!+\!k,\beta \!+\!n_i-k)\sum _{r=0}^{n_i-k-1}\frac{1}{\beta \!+\!r}}{\sum _{t=0}^{n_i}v_{n_it}B(\alpha \!+\!t,\beta \!+\!n_i-t)}\\&= -\sum \limits _{i=1}^{m}\sum \limits _{r=0}^{n_i-1}\frac{1}{\alpha +\beta +r}+\sum \limits _{i=1}^{m}\sum \limits _{k=0}^{n_i}\sum \limits _{r=0}^{n_i-k-1}\frac{\pi _{ik}(\alpha ,\beta )}{\beta +r}. \end{aligned}$$

Appendix B

The partial derivatives of \(\pi _{ik}(\alpha ,\beta )\) are as follows:

$$\begin{aligned}&\frac{\partial \pi _{ik}(\alpha ,\beta )}{\partial \alpha }=\frac{v_{n_ik}B(\alpha +k,\beta +n_i-k)}{\sum _{t=0}^{n_i}v_{n_it}B(\alpha +t,\beta +n_i-t)}\\&\quad =\frac{v_{n_ik}B'(\alpha +k,\beta +n_i-k)\sum _{t=0}^{n_i}v_{n_it}B(\alpha +t,\beta +n_i-t)}{\left( \sum _{t=0}^{n_i}v_{n_it}B(\alpha +t,\beta +n_i-t)\right) ^2}\\&\qquad -\frac{v_{n_ik}B(\alpha +k,\beta +n_i-k)\sum _{t=0}^{n_i}v_{n_it}B'(\alpha +t,\beta +n_i-t)}{\left( \sum _{t=0}^{n_i}v_{n_it}B(\alpha +t,\beta +n_i-t)\right) ^2}\\&\quad =\frac{v_{n_ik}B(\alpha \!+\!k,\beta \!+\!n_i-k)\left( \Psi (\alpha \!+\!k)\!-\!\Psi (\alpha \!+\!\beta \!+\!n_i)\right) \sum _{t=0}^{n_i}v_{n_it}B(\alpha \!+\!t,\beta \!+\!n_i-t)}{\left( \sum _{t=0}^{n_i}v_{n_it}B(\alpha \!+\!t,\beta \!+\!n_i-t)\right) ^2}\\&\qquad - \frac{v_{n_ik}B(\alpha \!+\!k,\beta \!+\!n_i\!-\!k)\sum _{t=0}^{n_i}v_{n_it}B(\alpha \!+\!t,\beta \!+\!n_i-t)\left( \Psi (\alpha \!+\!t)-\Psi (\alpha \!+\!\beta \!+\!n_i)\right) }{\left( \sum _{t=0}^{n_i}v_{n_it}B(\alpha \!+\!t,\beta \!+\!n_i-t)\right) ^2}\\&\quad =\frac{v_{n_ik}B(\alpha +k,\beta +n_i-k)\Psi (\alpha +k)\sum _{t=0}^{n_i}v_{n_it}B(\alpha +t,\beta +n_i-t)}{\left( \sum _{t=0}^{n_i}v_{n_it}B(\alpha +t,\beta +n_i-t)\right) ^2}\\&\qquad - \frac{v_{n_ik}B(\alpha +k,\beta +n_i-k)\sum _{t=0}^{n_i}v_{n_it}B(\alpha +t,\beta +n_i-t)\Psi (\alpha +t)}{\left( \sum _{t=0}^{n_i}v_{n_it}B(\alpha +t,\beta +n_i-t)\right) ^2}\\&\quad =\Psi (\alpha +k)\pi _{ik}(\alpha ,\beta )-\pi _{ik}(\alpha ,\beta )\sum \limits _{t=0}^{n_i}\pi _{it}(\alpha ,\beta )\Psi (\alpha +t)\\&\quad =\Psi (\alpha +k)\pi _{ik}(\alpha ,\beta )-\pi _{ik}(\alpha ,\beta )\sum \limits _{t=0}^{n_i}\pi _{it}(\alpha ,\beta )\left( \Psi (\alpha )+\sum \limits _{j=0}^{t-1}\frac{1}{\alpha +j}\right) \\&\quad =\Psi (\alpha +k)\pi _{ik}(\alpha ,\beta )-\pi _{ik}(\alpha ,\beta )\left( \Psi (\alpha )+\sum \limits _{t=0}^{n_i}\sum \limits _{j=0}^{t-1}\frac{\pi _{it}(\alpha ,\beta )}{\alpha +j}\right) \\&\quad =\Psi (\alpha +k)\pi _{ik}(\alpha ,\beta )-\pi _{ik}(\alpha ,\beta )\Psi (\alpha )-\pi _{ik}(\alpha ,\beta )\sum \limits _{t=0}^{n_i}\sum \limits _{j=0}^{t-1}\frac{\pi _{it}(\alpha ,\beta )}{\alpha +j}\\&\quad =\pi _{ik}(\alpha ,\beta ) \left[ \sum \limits _{j=0}^{k-1}\frac{1}{\alpha +j}-\sum \limits _{t=0}^{n_i}\sum \limits _{j=0}^{t-1}\frac{\pi _{it}(\alpha ,\beta )}{\alpha +j}\right] \end{aligned}$$

and

$$\begin{aligned}&\frac{\partial \pi _{ik}(\alpha ,\beta )}{\partial \beta }=\frac{v_{n_ik}B(\alpha +k,\beta +n_i-k)}{\sum _{t=0}^{n_i}v_{n_it}B(\alpha +t,\beta +n_i-t)}\\&\quad =\frac{v_{n_ik}B'(\alpha +k,\beta +n_i-k)\sum _{t=0}^{n_i}v_{n_it}B(\alpha +t,\beta +n_i-t)}{\left( \sum _{t=0}^{n_i}v_{n_it}B(\alpha +t,\beta +n_i-t)\right) ^2}\\&\qquad -\frac{v_{n_ik}B(\alpha +k,\beta +n_i-k)\sum _{t=0}^{n_i}v_{n_it}B'(\alpha +t,\beta +n_i-t)}{\left( \sum _{t=0}^{n_i}v_{n_it}B(\alpha +t,\beta +n_i-t)\right) ^2}\\&\quad =\frac{v_{n_ik}B(\alpha \!+\!k,\beta \!+\!n_i-k)\left( \Psi (\beta \!+\!n_i-k)\!-\!\Psi (\alpha \!+\!\beta \!+\!n_i)\right) \sum _{t=0}^{n_i}v_{n_it}B(\alpha \!+\!t,\beta \!+\!n_i-t)}{\left( \sum _{t=0}^{n_i}v_{n_it}B(\alpha +t,\beta +n_i-t)\right) ^2}\\&\qquad \!-\frac{v_{n_ik}B(\alpha \!+\!k,\beta \!+\!n_i-k)\sum _{t=0}^{n_i}v_{n_it}B(\alpha \!+\!t,\beta \!+\!n_i-t)\left( \Psi (\beta \!+\!n_i-t)\!-\!\Psi (\alpha \!+\!\beta \!+\!n_i)\right) }{\left( \sum _{t=0}^{n_i}v_{n_it}B(\alpha +t,\beta +n_i-t)\right) ^2}\\&\quad =\frac{v_{n_ik}B(\alpha +k,\beta +n_i-k)\Psi (\beta +n_i-k)\sum _{t=0}^{n_i}v_{n_it}B(\alpha +t,\beta +n_i-t)}{\left( \sum _{t=0}^{n_i}v_{n_it}B(\alpha +t,\beta +n_i-t)\right) ^2}\\&\qquad -\frac{v_{n_ik}B(\alpha +k,\beta +n_i-k)\sum _{t=0}^{n_i}v_{n_it}B(\alpha +t,\beta +n_i-t)\Psi (\beta +n_i-t)}{\left( \sum _{t=0}^{n_i}v_{n_it}B(\alpha +t,\beta +n_i-t)\right) ^2}\\&\quad =\Psi (\beta +n_i-k)\pi _{ik}(\alpha ,\beta )-\pi _{ik}(\alpha ,\beta )\sum \limits _{t=0}^{n_i}\pi _{it}(\alpha ,\beta )\Psi (\beta +n_i-t)\\&\quad =\Psi (\beta +n_i-k)\pi _{ik}(\alpha ,\beta )-\pi _{ik}(\alpha ,\beta )\sum \limits _{t=0}^{n_i}\pi _{it}(\alpha ,\beta )\left( \Psi (\beta )+\sum \limits _{j=0}^{n_i-t-1}\frac{1}{\beta +j}\right) \\&\quad =\Psi (\beta +n_i-k)\pi _{ik}(\alpha ,\beta )-\pi _{ik}(\alpha ,\beta )\left( \Psi (\beta )+\sum \limits _{t=0}^{n_i}\sum \limits _{j=0}^{n_i-t-1}\frac{\pi _{it}(\alpha ,\beta )}{\beta +j}\right) \\&\quad =\Psi (\beta +n_i-k)\pi _{ik}(\alpha ,\beta )-\pi _{ik}(\alpha ,\beta )\Psi (\beta )-\pi _{ik}(\alpha ,\beta )\sum \limits _{t=0}^{n_i}\sum \limits _{j=0}^{n_i-t-1}\frac{\pi _{it}(\alpha ,\beta )}{\beta +j}\\&\quad =\pi _{ik}(\alpha ,\beta ) \left[ \sum \limits _{j=0}^{n_i-k-1}\frac{1}{\beta +j}-\sum \limits _{t=0}^{n_i}\sum \limits _{j=0}^{n_i-t-1}\frac{\pi _{it}(\alpha ,\beta )}{\beta +j}\right] . \end{aligned}$$

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Balakrishnan, N., Koutras, M.V. & Milienos, F.S. Some binary start-up demonstration tests and associated inferential methods. Ann Inst Stat Math 66, 759–787 (2014). https://doi.org/10.1007/s10463-013-0424-y

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