Why should there be two solutions for each parameter of likelihood ratio equation for Weibull-distribution?What are the disadvantages of the profile likelihood?Likelihood ratio test for sex ratioLikelihood Ratio of two-sample Uniform DistributionLikelihood Ratio Test statistic for the exponential distributionMysterious results from likelihood ratio confidence bounds on a Weibull reliability estimateHypotheses for the likelihood ratio testDetermining normalizing constant for Weibull distributionMLE: Marginal vs Full Likelihood
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Why should there be two solutions for each parameter of likelihood ratio equation for Weibull-distribution?
What are the disadvantages of the profile likelihood?Likelihood ratio test for sex ratioLikelihood Ratio of two-sample Uniform DistributionLikelihood Ratio Test statistic for the exponential distributionMysterious results from likelihood ratio confidence bounds on a Weibull reliability estimateHypotheses for the likelihood ratio testDetermining normalizing constant for Weibull distributionMLE: Marginal vs Full Likelihood
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I try to calculate the confidence ínterval of a Weibull distribution by means of the Likelihood method as described in ReliaWiki. In order to find the confidence intervals, I have to solve the Likelihood ratio equation:
$$L(theta_1,theta_2)=L(hattheta_1,hattheta_2)cdot e^-chi^2_alpha;1/2$$
In the case of a Weibull distribution, one has to iterate over both parameters (one is held constant, over the other is iterated and vice versa) to find a solution. But in the examples on ReliaWiki for each parameter $theta_1$ and $theta_2$ a minimal and a maximal solution is found. This fact is also mentioned on the webside:
"The task now is to find the values of the parameters $theta_1$ and $theta_2$ so that the equality in the likelihood ratio equation shown above is satisfied. Unfortunately, there is no closed-form solution; therefore, these values must be arrived at numerically. One way to do this is to hold one parameter constant and iterate on the other until an acceptable solution is reached. This can prove to be rather tricky, since there will be two solutions for one parameter if the other is held constant. In situations such as these, it is best to begin the iterative calculations with values close to those of the MLE values, so as to ensure that one is not attempting to perform calculations outside of the region of the contour plot where no solution exists." (citated ReliaWiki)
I do not understand, why should there be two solutions for each parameter. Can somebody explain me, why it behaves that way?
confidence-interval likelihood likelihood-ratio weibull numerical-integration
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I try to calculate the confidence ínterval of a Weibull distribution by means of the Likelihood method as described in ReliaWiki. In order to find the confidence intervals, I have to solve the Likelihood ratio equation:
$$L(theta_1,theta_2)=L(hattheta_1,hattheta_2)cdot e^-chi^2_alpha;1/2$$
In the case of a Weibull distribution, one has to iterate over both parameters (one is held constant, over the other is iterated and vice versa) to find a solution. But in the examples on ReliaWiki for each parameter $theta_1$ and $theta_2$ a minimal and a maximal solution is found. This fact is also mentioned on the webside:
"The task now is to find the values of the parameters $theta_1$ and $theta_2$ so that the equality in the likelihood ratio equation shown above is satisfied. Unfortunately, there is no closed-form solution; therefore, these values must be arrived at numerically. One way to do this is to hold one parameter constant and iterate on the other until an acceptable solution is reached. This can prove to be rather tricky, since there will be two solutions for one parameter if the other is held constant. In situations such as these, it is best to begin the iterative calculations with values close to those of the MLE values, so as to ensure that one is not attempting to perform calculations outside of the region of the contour plot where no solution exists." (citated ReliaWiki)
I do not understand, why should there be two solutions for each parameter. Can somebody explain me, why it behaves that way?
confidence-interval likelihood likelihood-ratio weibull numerical-integration
edited 4 hours ago
StubbornAtom
3,7111 gold badge7 silver badges39 bronze badges
asked 8 hours ago
WaldiWaldi
83 bronze badges
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Waldi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$begingroup$
I try to calculate the confidence ínterval of a Weibull distribution by means of the Likelihood method as described in ReliaWiki. In order to find the confidence intervals, I have to solve the Likelihood ratio equation:
$$L(theta_1,theta_2)=L(hattheta_1,hattheta_2)cdot e^-chi^2_alpha;1/2$$
In the case of a Weibull distribution, one has to iterate over both parameters (one is held constant, over the other is iterated and vice versa) to find a solution. But in the examples on ReliaWiki for each parameter $theta_1$ and $theta_2$ a minimal and a maximal solution is found. This fact is also mentioned on the webside:
"The task now is to find the values of the parameters $theta_1$ and $theta_2$ so that the equality in the likelihood ratio equation shown above is satisfied. Unfortunately, there is no closed-form solution; therefore, these values must be arrived at numerically. One way to do this is to hold one parameter constant and iterate on the other until an acceptable solution is reached. This can prove to be rather tricky, since there will be two solutions for one parameter if the other is held constant. In situations such as these, it is best to begin the iterative calculations with values close to those of the MLE values, so as to ensure that one is not attempting to perform calculations outside of the region of the contour plot where no solution exists." (citated ReliaWiki)
I do not understand, why should there be two solutions for each parameter. Can somebody explain me, why it behaves that way?
confidence-interval likelihood likelihood-ratio weibull numerical-integration
edited 4 hours ago
StubbornAtom
3,7111 gold badge7 silver badges39 bronze badges
asked 8 hours ago
WaldiWaldi
83 bronze badges
New contributor
Waldi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I try to calculate the confidence ínterval of a Weibull distribution by means of the Likelihood method as described in ReliaWiki. In order to find the confidence intervals, I have to solve the Likelihood ratio equation:
$$L(theta_1,theta_2)=L(hattheta_1,hattheta_2)cdot e^-chi^2_alpha;1/2$$
In the case of a Weibull distribution, one has to iterate over both parameters (one is held constant, over the other is iterated and vice versa) to find a solution. But in the examples on ReliaWiki for each parameter $theta_1$ and $theta_2$ a minimal and a maximal solution is found. This fact is also mentioned on the webside:
"The task now is to find the values of the parameters $theta_1$ and $theta_2$ so that the equality in the likelihood ratio equation shown above is satisfied. Unfortunately, there is no closed-form solution; therefore, these values must be arrived at numerically. One way to do this is to hold one parameter constant and iterate on the other until an acceptable solution is reached. This can prove to be rather tricky, since there will be two solutions for one parameter if the other is held constant. In situations such as these, it is best to begin the iterative calculations with values close to those of the MLE values, so as to ensure that one is not attempting to perform calculations outside of the region of the contour plot where no solution exists." (citated ReliaWiki)
I do not understand, why should there be two solutions for each parameter. Can somebody explain me, why it behaves that way?
confidence-interval likelihood likelihood-ratio weibull numerical-integration
confidence-interval likelihood likelihood-ratio weibull numerical-integration
edited 4 hours ago
StubbornAtom
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asked 8 hours ago
WaldiWaldi
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edited 4 hours ago
StubbornAtom
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asked 8 hours ago
WaldiWaldi
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edited 4 hours ago
StubbornAtom
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edited 4 hours ago
StubbornAtom
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edited 4 hours ago
StubbornAtom
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asked 8 hours ago
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asked 8 hours ago
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$colorwhite^textLikelihood contour showing the usual case two boundary points for\^texta given choice of one parameter when the selected value is within the region$
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$colorwhite^textLikelihood contour showing the usual case two boundary points for\^texta given choice of one parameter when the selected value is within the region$
answered 7 hours ago
Glen_b♦Glen_b
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$colorwhite^textLikelihood contour showing the usual case two boundary points for\^texta given choice of one parameter when the selected value is within the region$
answered 7 hours ago
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$colorwhite^textLikelihood contour showing the usual case two boundary points for\^texta given choice of one parameter when the selected value is within the region$
answered 7 hours ago
Glen_b♦Glen_b
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$colorwhite^textLikelihood contour showing the usual case two boundary points for\^texta given choice of one parameter when the selected value is within the region$
answered 7 hours ago
Glen_b♦Glen_b
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answered 7 hours ago
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