Optimal sample size for estimating the mean concentration of invasive organisms in ballast water via a semiparametric Bayesian analysis

Abstract

We consider the determination of optimal sample sizes to estimate the concentration of organisms in ballast water via a semiparametric Bayesian approach involving a Dirichlet process mixture based on a Poisson model. This semiparametric model provides greater flexibility to model the organism distribution than that allowed by competing parametric models and is robust against misspecification. To obtain the optimal sample size we use a total cost minimization criterion, based on the sum of a Bayes risk and a sampling cost function. Credible intervals obtained via the proposed model may be used to verify compliance of the water with international standards before deballasting.

Appendix A

Algorithm 1: :

Drawing samples from the joint posterior distribution of the $\lambda _i$.

Step 1. :

Simulate initial values for $\lambda _i$, $i=1,\ldots ,n$ from $F_0$;

Step 2. :

Under a Gibbs sampling scheme, update $\lambda _i$, $i=1,\ldots ,n$ using (4);

Step 3. :

Update the values obtained in Step 2 using (5);

Step 4. :

Repeat steps 2-3 a number of times as a burn-in; the values obtained in the last iteration are the required values.

Algorithm 2::

Drawing samples of the random mean $\overline{\lambda }$.

Step 1. :

Set a value for $\epsilon$, set $\overline{\lambda }_1^\ell =0$ and take $\overline{\lambda }_1^u$ as the largest internal bit value of the computer being employed (in our case, $1.79\times 10^{308}$);

Step 2. :

Update the upper and lower quantities using (8) and (9);

Step 3. :

If the absolute difference between the two quantities is smaller than $\epsilon$, the required value $\overline{\lambda }$ may be taken as either $\overline{\lambda }_t^u$ or $\overline{\lambda }_t^\ell$. Otherwise, return to step 2.

Algorithm 3: :

Drawing samples from the distribution of $\overline{\lambda }^{(n)}$

Step 1. :

Simulate $B_*$ from a $\text {Beta}(n, \alpha )$ distribution;

Step 2. :

Simulate $\overline{\lambda }$ using Algorithm 2;

Step 3. :

Simulate $D_i$, $i=1,\ldots ,n$ from a multivariate uniform distribution;

Step 4. :

Simulate $(Z_1,\ldots ,Z_n)$ from $\nu (d{\varvec{\lambda }}_n\vert {\varvec{x}}_n)$ using Algorithm 1;

Step 5. :

Obtain the required value using the quantities generated in steps 1-4 and (10) of the article.