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I want to calculate the N% confidence interval for some time-series data set. I have the standard errors for this data series and the error variance of the time-series is assumed to be Gaussian. I understand that we can use a factor of ±1.96 multiplied by the standard error of each data point to obtain the 95% confidence interval (because it is the quantile 0.975 of the Gaussian distribution) [the exact specifics of the standard errors is not important].

I know I am able to get this value of 1.96 by calculating the Inverse Cumulative Distribution Function for the Normal distribution (`InvCDF(0.975) ~ 1.96), what I don't understand is how to map the quartile value (0.975) to the desired confidence interval. What I want is to be able to say is: "I want the 87% confidence interval" and then calculate InvCDF(X), but what is X in this case? And what function relates the quantile value to the confidence interval?

Thanks for your time.

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  • $\begingroup$ Are you asking about how one obtains $0.975$ from the 95% specification? $\endgroup$
    – whuber ♦
    Commented Jan 7, 2015 at 17:28
  • $\begingroup$ Yes. That is exactly why I am asking... $\endgroup$
    – MoonKnight
    Commented Jan 7, 2015 at 17:55

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For a two-sided confidence interval, a N% confidence interval requires the quantile 1-(1-N)/2 for the upper bound of the interval and (1-N)/2 for the lower bound. In this way, the (1-N)% of the mass is split symmetrically in both tails of the normal distribution. For a 87% confidence interval you would have 1.87/2=0.935 and (1-0.87)/2=0.065 and the quantile values are ± 1.514.

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