An algorithm to construct Monte Carlo confidence intervals for an arbitrary function of probability distribution parameters

Abstract

We derive a new algorithm for calculating an exact confidence interval for a parameter of location or scale family, based on a two-sided hypothesis test on the parameter of interest, using some pivotal quantities. We use this algorithm to calculate approximate confidence intervals for the parameter or a function of the parameter of one-parameter continuous distributions. After appropriate heuristic modifications of the algorithm we use it to obtain approximate confidence intervals for a parameter or a function of parameters for multi-parameter continuous distributions. The advantage of the algorithm is that it is general and gives a fast approximation of an exact confidence interval. Some asymptotic (analytical) results are shown which validate the use of the method under certain regularity conditions. In addition, numerical results of the method compare well with those obtained by other known methods of the literature on the exponential, the normal, the gamma and the Weibull distribution.

Appendices

Appendix A: Some theoretical results

First we show that the confidence interval for the parameter $\mu$ of a normal distribution $N(\mu ,\sigma ^{2})$ is asymptotically equivalent to a Wald-type interval. For the normal distribution we define $\varvec{\theta } = \left( {\mu ,\sigma } \right),\, {{\varvec{T}}}({{{{\underline{\varvec{x}}}}}}) = (T_{1} ({{{{\underline{\varvec{x}}}}}}),T_{2} ({{{\underline{\varvec{x}}}}}))$, where $T_{1} ({{{{\underline{\varvec{x}}}}}}) =\underline{\mu }, \text{ and} \, T_{2} ({{{{\underline{\varvec{x}}}}}}) = \underline{\sigma }$ are the MLE of $\mu$ and $\sigma$ respectively.

Then, following the notation of the preceding sections we have $\beta :=h\left( {\mu ,\sigma } \right) =\mu ,h\left( {{\varvec{T}}} \right) =T_{1}$ and $P(b({{{{\underline{\varvec{x}}}}}}) <\lambda \left( \varvec{\theta } \right)) =\alpha /{2},P(b({{{{\underline{\varvec{x}}}}}}) >\upsilon \left( \varvec{\theta } \right)) =\alpha /{2}$ which imply that $\lambda =\mu +\varPhi ^{-{1}}\left( {\alpha /{2}} \right)\sigma /\sqrt{n}$ and $\upsilon =\mu +\varPhi ^{-{1}}\left( {{1 }-\alpha /{2}} \right)\sigma /\sqrt{n}$. Now from (18) we obtain

$$\begin{aligned} \frac{d\varvec{\gamma } }{d\varvec{\theta } }=\left[ {{\begin{array}{l} {\frac{d\lambda }{d\varvec{\theta } }} \\ {\frac{d\beta }{d\varvec{\theta } }} \\ {\frac{d\upsilon }{d\varvec{\theta } }} \\ \end{array} }} \right]=\left[ {{\begin{array}{lc} 1&{\varPhi ^{-1}(\alpha /2)/\sqrt{n}} \\ 1&0 \\ 1&{\varPhi ^{-1}(1-\alpha /2)/\sqrt{n}} \\ \end{array} }} \right] \end{aligned}$$

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It is also easy to prove that asymptotically $\left[{{\begin{array}{l} {\underline{\mu } -\mu } \\ {\underline{\sigma } -\sigma } \\ \end{array} }} \right]\sim N\left( {\left[ {{\begin{array}{l} 0 \\ 0 \\ \end{array} }} \right],\frac{\sigma ^{2}}{n}\left[ {{\begin{array}{ll} 1&0 \\ 0&{1/2} \\ \end{array} }} \right]} \right)$, thus $\underline{\mu } \sim N\left( {\mu ,\sigma ^{\mathrm{2}}/n} \right)$ and $\underline{\sigma } \sim N({\sigma , \sigma ^{2}/2n})$. We also have that $\varPhi ^{-{1}}\left( {{1 }-\alpha /{2}} \right) = -\varPhi ^{-{1}}\left( {\alpha /{2}} \right)$.

From (12), (13) we derive $l=\underline{\mu } -\frac{\underline{\sigma } \varPhi ^{-1}(1-\alpha /2)}{\sqrt{n}\cdot \frac{d\upsilon }{d\mu }}$ and $u =\underline{\mu } +\frac{\underline{\sigma } \varPhi ^{-1}(1-\alpha /2)}{\sqrt{n}\cdot \frac{d\lambda }{d\mu }}$. From (22) we have that $\frac{d\lambda }{d\mu }=\frac{d\upsilon }{d\mu }=1-\frac{(\varPhi ^{-1}(1-\alpha /2))^{2}}{4n}$. A $1 - {\alpha }$ confidence interval for $\mu$ is $\left( {\underline{\mu } -t_{n-1}} (1 -\alpha /2)\frac{\sigma }{\sqrt{n}},\mu +t_{n-1} (1 -\alpha /2)\frac{\sigma }{\sqrt{n}} \right)$ (e.g. Papoulis and Pillai 2002, p. 309). Now we have that $\mathop {\lim }\limits _{n\rightarrow \infty } \frac{{\varPhi ^{-1}(1-\alpha /2)}/{\left( {\frac{d\upsilon }{d\mu }} \right)}}{{t}_{n-1} (1-\alpha /2)}=1$, which proves that the confidence interval obtained by (14) is asymptotically exact.

We will also show that the confidence interval obtained by our method is asymptotically equivalent to a Wald-type interval for two-parameter regular distributions. According to Casella and Berger (2002, p. 472) $\sqrt{n}(\underline{\varvec{\theta }} -\varvec{\theta }) \mathop \rightarrow \limits ^\mathrm{d} N(\mathbf{0},{{\varvec{I}}}^{-{1}})$, where $\underline{{\varvec{\theta }}}$ is the MLE of ${\varvec{\theta }}$, and I is the Fisher Information Matrix with elements ${{\varvec{I}}}_{jk} = \text{ E}\left( {-\frac{\partial ^{2}\ln f(x|\varvec{\theta } )}{\partial \theta _j \theta _k }} \right)$. This means that $\sqrt{n}(\theta _1 -\theta _1 ) \mathop \rightarrow \limits ^\mathrm{d} N(0,{{\varvec{I}}}_{11}^{-{1}} )$ and$\sqrt{n}(\theta _2 -\theta _2 ) \mathop \rightarrow \limits ^\mathrm{d} N(0,{{\varvec{I}}}_{22}^{-{1}} )$. We conclude that $\sqrt{n}(\beta -\beta )\mathop \rightarrow \limits ^\mathrm{d} {N}(0,\sigma _\beta ^2 )$, where $\sigma _\beta ^2$ depends only on $\theta _{1}$ and $\theta _{2}$. Suppose that we seek a $1 - {\alpha }$ confidence interval for $\beta$. Then it is easy to show that asymptotically $\lambda \left( \beta \right) =\beta -\varPhi ^{-{1}}\left( {{1 }-\alpha /{2}} \right)\sigma _\beta /\sqrt{n},\upsilon \left( \beta \right) =\beta +\varPhi ^{-{1}}\left( {{1 }-\alpha /{2}} \right)\sigma _\beta /\sqrt{n}$. Now we have $\text{ Var} (\underline{\theta } _{1} ) =I_{11}^{-1} /n,{ \text{ Var}}(\underline{\theta } _\mathrm{2} ) =I_{22}^{-1} /n$ and

$$\begin{aligned}&\frac{d\varvec{\gamma } }{d\varvec{\theta } }=\left[ {{\begin{array}{l} {\frac{d\lambda }{d\varvec{\theta } }} \\ {\frac{d\beta }{d\varvec{\theta } }} \\ {\frac{d\upsilon }{d\varvec{\theta } }} \\ \end{array} }} \right]=\left[ {{\begin{array}{cc} {\frac{\partial \beta }{\partial \theta _1 }\!-\!\varPhi ^{\!-\!1}(1\!-\!\alpha /2)\frac{\partial \sigma \beta }{\partial \theta _1 }/\sqrt{n}}&{\frac{\partial \beta }{\partial \theta _2 }\!-\!\varPhi ^{\!-\!1}(1\!-\!\alpha /2)\frac{\partial \sigma \beta }{\partial \theta _2 }/\sqrt{n}} \\ {\frac{\partial \beta }{\partial \theta _1 }}&{\frac{\partial \beta }{\partial \theta _2 }} \\ {\frac{\partial \beta }{\partial \theta _1 }\!+\!\varPhi ^{\!-\!1}(1\!-\!\alpha /2)\frac{\partial \sigma \beta }{\partial \theta _1 }/\sqrt{n}}&{\frac{\partial \beta }{\partial \theta _2 }+\varPhi ^{\!-\!1}(1-\alpha /2)\frac{\partial \sigma \beta }{\partial \theta _2 }/\sqrt{n}} \\ \end{array} }} \right] \end{aligned}$$

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$$\begin{aligned} \nonumber \\&\frac{q_{31} \!+\!q_{32} }{q_{21} \!+\!q_{22} }\nonumber \\&\qquad \quad =\frac{[\varPhi ^{\!-\!1}(1\!-\!\alpha /2)]^{2}\left[ {\left( {\frac{\partial _{\sigma \beta } }{\partial \theta _1 }} \right)^{2}{{\varvec{I}}}_{11}^{\!-\!1} \!+\!\left( {\frac{\partial _{\sigma \beta } }{\partial \theta _2 }} \right)^{2}{{\varvec{I}}}_{22}^{\!-\!1} } \right]\!-\!\varPhi ^{\!-\!1}(1\!-\!\alpha /2)\sqrt{n}\left[ {\frac{\partial _{\sigma \beta } \partial \beta }{\partial \theta _1 \partial \theta _1 }{{\varvec{I}}}_{11}^{\!-\!1} \!+\!\left( {\frac{\partial _{\sigma \beta } \partial \beta }{\partial \theta _2 \partial \theta _2 }} \right)^{2}{{\varvec{I}}}_{22}^{\!-\!1} } \right]}{\sqrt{n}\left[ {\varPhi ^{\!-\!1}(1\!-\!\alpha /2)\left( {\frac{\partial _{\sigma \beta } \partial \beta }{\partial \theta _1 \partial \theta _1 }\!+\!\frac{\partial _{\sigma \beta } \partial \beta }{\partial \theta _2 \partial \theta _2 }} \right)\!-\!2\sqrt{n}\left( {\left( {\frac{\partial \beta }{\partial \theta _1 }} \right)^{2}{{\varvec{I}}}_{11}^{\!-\!1} \!+\!\left( {\frac{\partial \beta }{\partial \theta _2 }} \right)^{2}{{\varvec{I}}}_{22}^{\!-\!1} } \right)} \right]}\nonumber \\&\qquad -\frac{2n\left( {\left( {\frac{\partial \beta }{\partial \theta _1 }} \right)^{2}{{\varvec{I}}}_{11}^{-1} +\left( {\frac{\partial \beta }{\partial \theta _2 }} \right)^{2}{{\varvec{I}}}_{22}^{-1} } \right)}{\sqrt{n}\left[ {\varPhi ^{-1}(1-\alpha /2)\left( {\frac{\partial _{\sigma \beta } \partial \beta }{\partial \theta _1 \partial \theta _1 }+\frac{\partial _{\sigma \beta } \partial \beta }{\partial \theta _2 \partial \theta _2 }} \right)-2\sqrt{n}\left( {\left( {\frac{\partial \beta }{\partial \theta _1 }} \right)^{2}{{\varvec{I}}}_{11}^{-1} +\left( {\frac{\partial \beta }{\partial \theta _2 }} \right)^{2}{{\varvec{I}}}_{22}^{-1} } \right)} \right]} \end{aligned}$$

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It is obvious that $\mathop {\lim }\limits _{n\rightarrow \infty } \frac{d\upsilon }{d\beta }=\mathop {\lim }\limits _{n\rightarrow \infty } \frac{q_{31} +q_{32} }{q_{21} +q_{22} }=1$. In a similar way we can find that $\mathop {\lim }\limits _{n\rightarrow \infty } \frac{d\lambda }{d\beta }=\mathop {\lim }\limits _{n\rightarrow \infty } \frac{q_{12} +q_{13} }{q_{22} +q_{23} }=1$. Now substituting to (12), (13) we obtain $l=\underline{\beta } -\varPhi ^{-{1}}\left( {{1 }-\alpha /{2}} \right)\sigma _\beta /\sqrt{n},\, u=\underline{\beta } +\varPhi ^{-{1}}\left( {{1 }-\alpha /{2}} \right)\sigma _\beta /\sqrt{n}\,$ which is an asymptotically equivalent to a Wald-type interval according to Casella and Berger (2002, p. 497).

Repeating the same procedure for three-parameter distributions, we obtain the same results.

Appendix B: Application of the algorithm on a historical river flows dataset

In this Appendix we apply the algorithm on a historical river flow data set using the hydrological statistical software Hydrognomon (2009–2012), suitable for the processing and the analysis of hydrological time series, which has already incorporated the proposed method. The case study is performed on an important basin in Greece, which is currently part of the water supply system of Athens and has a history, as regards hydraulic infrastructure and management, that goes back to at least 3,500 years ago. Modelling attempts with good performance have already been done on the hydrosystem (Rozos et al. 2004). A long-term dataset of the catchment runoff, extending from 1906 to 2008, is available. The example presented in Fig. 12 is for the January monthly flow record at the Boeoticos Kephisos river outlet at the Karditsa station measured in $\text{ hm}^{3}$. The gamma distribution is often used to model monthly river flows. Confidence limits of quantiles of distributions are of interest to hydrologists. Here we derived confidence intervals for the scale and the shape parameters of the gamma distribution. Comparison of the results of the different methods used show that the MCCI and “Ripley scale” limits are close to the Bayesian ones. In addition, Fig. 13 gives confidence limits of the distribution percentiles using the same dataset, this time constructed using Hydrognomon (2009–2012).

Fig. 12

Confidence intervals for the scale (upper) and shape (lower) parameter of a gamma distribution, used to model the Boeoticos Kephisos river January monthly flows with $n = 102$, $\hat{{k}}= {3.842}$ and $\hat{{\theta }} = {15.218}$. Here the number of samples $m = 120,000$ for MCCI and $m = 60,000$ for the “Ripley location” and “Ripley scale” cases, $\updelta k = 0.3$ and $\updelta \theta = 0.3$

Fig. 13

A graph (normal probability plot) produced by the Hydrognomon software referring to the monthly flow of Boeoticos Kephisos river for the month of January (1993–2006). The sample (dots plotted using Weibul plotting positions) was modelled by a gamma distribution (central line) with $\hat{{k}}= {3.842}$ and $\hat{{\theta }} = {15.218}$. Dotted lines represent 95 % prediction intervals for these parameter values (denoted as $\lambda$ and $\upsilon$ in the text) and dashed lines represent 95 % confidence intervals (MCCI denoted as $l$ and $u$ in the text) for the distribution percentiles