# Confidence interval in a nonlinear model

Thanks for your suggestions.Actually i have the following model that i explains to you.

Suppose i have observations structure like \begin{align} y_1 &=&v_1+e_1 \\ y_2 &=&v_2+e_2 \\ y_3&=&v_1^2-v_1v_2(\cos(x_1-x_2)+\sin(x_1-x_2))+e_3 \end{align}, and so on......

Every error term assume to follow normal distribution with mean zero but different variance. In model form above observations can be written as

Y=XB+e.(heterscedastic and nonlinear model) Here v1,v2 x1,x2....are unknown values and surely model is nonlinear in unknown parameters. We use Newton type method to estimates the unknown parameters. Guass Newton method work well to find estimates in this problem. I can find Hessian matrix or fisher Information matrix by this method that can be used to find standard errors of the estimates.

Now my question is related to confidence intervals for unknown parameters. i construct the confidence intervals for unknown parameters by using the standards that are calculated from Hessian matrix .I construct individual confidence intervals for each parameter by using t statistics?(Wald Type).When i simulate this model by knowing the actual values of parameters i found that coverage probabilities are very low special for unknown parameters angles.

Appreciate if you could suggest the technique to tackle this problem.

• What model is it? There is a very general class of models that allows you to construct a theoretically sound (albeit valid only asymptotically) confidence interval by inverting a Wald test, or by using the delta method – shadowtalker Jan 13 '15 at 16:57
• Dear Harvey Motulsky:Could you please comments now to the problem – zahid khan Jan 18 '15 at 14:55