# Confidence interval of amplitude and horizontal shift

I have a linear model with sine and cosine wave as predictor. Y~$$\alpha$$sine(...)+$$\beta$$cosine(...). From the parameter estimation, I know the estimation of the coefficients are: $$\hat\alpha$$ = 0.65(sd = 0.15), $$\hat\beta$$ =0.54 (sd=0.15). The correlation between $$\alpha$$ and $$\beta$$ is 0. Using bivariate normal approximation, I can plot the 0.95 confidence ellipse:

p <- rmvnorm(1000, c(0.65,0.54), matrix(c(0.15^2,0,0,0.15^2),2,2))
plot(p, pch=20, xlab="sine", ylab="cosine")
ellipse(mu=colMeans(p), sigma=cov(p), alpha = .05, npoints = 250, col="red")


Then I wish to calculate the confidence interval for $$\sqrt{\alpha^2+\beta^2}$$. Any suggestions is appreciated.

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• Rewrite your model in the form given in my answer at stats.stackexchange.com/a/77865/919, which incorporates $\sqrt{\alpha^2+\beta^2}$ directly, and compute a confidence interval for it using the usual techniques. – whuber Aug 14 at 16:03
• your answer does not include calculating the confidence interval. @whuber – Huang Rui Aug 15 at 20:22
• As I indicated, that's a standard textbook calculation--there's little more to say about it. Your software will automatically report the confidence interval, for instance. – whuber Aug 15 at 21:13