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You apply a relatively narrow definition of frequentism and MLE - if we are a bit more generous and define

• Frequentism: goal of consistency, (asymptotic) optimality, unbiasedness, and controlled error rates under repeated sampling, independent of the true parameters

• MLE = point estimate + confidence intervals (CIs)

then it seems pretty clear that MLE satisfies all frequentist ideals. In particular, CIs in MLE, as p-values, control the error rate under repeated sampling, and do not give the 95% probability region for the true parameter value, as many people think - hence they are through and through frequentist.

Not all of these ideas were already present in Fisher's foundational 1922 paper "On the mathematical foundations of theoretical statistics", but the idea of optimality and unbiasedness is, and Neyman latter added the idea of constructing CIs with fixed error rates. Efron, 2013, "A 250-year argument: Belief, behavior, and the bootstrap", summarizes in his very readable history of the Bayesian/Frequentist debate:

The frequentist bandwagon really got rolling in the early 1900s. Ronald Fisher developed the maximum likelihood theory of optimal estimation, showing the best possible behavior for an estimate, and Jerzy Neyman did the same for confidence intervals and tests. Fisher’s and Neyman’s procedures were an almost perfect fit to the scientific needs and the computational limits of twentieth century science, casting Bayesianism into a shadow existence.

Regarding your more narrow definition - I mildly disagree with your premise that minimization of frequentist risk (FR) is the main criterion to decide if a method follows frequentist philosophy. I would say the fact that minimizing FR is a desirable property follows from frequentist philosophy, rather than preceding it. Hence, a decision rule / estimator does not have to minimize FR to be frequentist, and minimizing FR is also does not necessarily say that a method is frequentist, but a frequentist would in doubt prefer minimization of FR.

If we look at MLE specifically: Fisher showed that MLE is asymptotically optimal (broadly equivalent to minimizing FR), and that was certainly one reason for promoting MLE. However, he was aware that optimality did not hold for finite sample size. Still, he was happy with this estimator due to other desirable properties such as consistency, asymptotic normality, invariance under parameter transformations, and let's not forget: ease to calculate. Invariance in particular is stressed abundantly in the 1922 paper - from my reading, I would say maintaining invariance under parameter transformation, and the ability to get rid of the priors in general, were one of his main motivations in choosing MLE. If you want to understand his reasoning better, I really recommend the 1922 paper, it's beautifully written and he explains his reasoning very well.

You apply a relatively narrow definition of frequentism and MLE - if we are a bit more generous and define

• Frequentism: goal of consistency, (asymptotic) optimality, unbiasedness, and controlled error rates under repeated sampling, independent of the true parameters

• MLE = point estimate + confidence intervals (CIs)

then it seems pretty clear that MLE satisfies all frequentist ideals. In particular, CIs in MLE, as p-values, control the error rate under repeated sampling, and do not give the 95% probability region for the true parameter value, as many people think - hence they are through and through frequentist.

Not all of these ideas were already present in Fisher's foundational 1922 paper "On the mathematical foundations of theoretical statistics", but the idea of optimality and unbiasedness is, and Neyman latter added the idea of constructing CIs with fixed error rates. Efron, 2013, "A 250-year argument: Belief, behavior, and the bootstrap", summarizes in his very readable history of the Bayesian/Frequentist debate:

The frequentist bandwagon really got rolling in the early 1900s. Ronald Fisher developed the maximum likelihood theory of optimal estimation, showing the best possible behavior for an estimate, and Jerzy Neyman did the same for confidence intervals and tests. Fisher’s and Neyman’s procedures were an almost perfect fit to the scientific needs and the computational limits of twentieth century science, casting Bayesianism into a shadow existence.

Regarding your more narrow definition - I mildly disagree with your premise that minimization of frequentist risk (FR) is the main criterion to decide if a method follows frequentist philosophy. I would say the fact that minimizing FR is a desirable property follows from frequentist philosophy, rather than preceding it. Hence, a decision rule / estimator does not have to minimize FR to be frequentist, and minimizing FR is also does not necessarily say that a method is frequentist, but a frequentist would in doubt prefer minimization of FR.

If we look at MLE specifically: Fisher showed that MLE is asymptotically optimal (broadly equivalent to minimizing FR), and that was certainly one reason for promoting MLE. However, he was aware that optimality did not hold for finite sample size. Still, he was happy with this estimator due to other desirable properties such as consistency, asymptotic normality, invariance under parameter transformations, and let's not forget: ease to calculate. Invariance in particular is stressed abundantly in the 1922 paper - from my reading, I would say maintaining invariance under parameter transformation, and the ability to get rid of the priors in general, were one of his main motivations in choosing MLE. If you want to understand his reasoning better, I really recommend the 1922 paper, it's beautifully written and he explains his reasoning very well.

You apply a relatively narrow definition of frequentism and MLE - if we are a bit more generous and define

• Frequentism: goal of consistency, (asymptotic) optimality, unbiasedness, and controlled error rates under repeated sampling, independent of the true parameters

• MLE = point estimate + confidence intervals (CIs)

then it seems pretty clear that MLE satisfies all frequentist ideals. In particular, CIs in MLE, as p-values, are defined to control the error rate under repeated sampling, and not to give the 95% probability region of the true parameter value, as many people think - hence they are through and through frequentist.

Not all of these ideas were already present in Fisher's foundational 1922 paper "On the mathematical foundations of theoretical statistics", but the idea of optimality and unbiasedness is, and Neyman latter added the idea of constructing CIs with fixed error rates. Efron, 2013, "A 250-year argument: Belief, behavior, and the bootstrap", summarizes in his very readable history of the Bayesian/Frequentist debate:

The frequentist bandwagon really got rolling in the early 1900s. Ronald Fisher developed the maximum likelihood theory of optimal estimation, showing the best possible behavior for an estimate, and Jerzy Neyman did the same for confidence intervals and tests. Fisher’s and Neyman’s procedures were an almost perfect fit to the scientific needs and the computational limits of twentieth century science, casting Bayesianism into a shadow existence.

Regarding your more narrow definition - I mildly disagree with your premise that minimization of frequentist risk (FR) is the sole criterion to decide if a method follows frequentist philosophy. The fact that minimizing FR is a desirable property follows from frequentist philosophy, rather than preceding it. Hence, a decision rule / estimator does not have to minimize FR to be frequentist, but a frequentist would in doubt prefer these property.

If we look at FR of MLE: Fisher showed that MLE is asymptotically optimal (broadly equivalent to minimizing FR), and that was certainly one reason for promoting MLE. However, he was aware that optimality did not hold for finite sample size. Still, he was happy with this estimator due to other desirable properties such as consistency, asymptotic normality, and invariance under parameter transformations, and let's not forget: ease to calculate. Invariance in particular is stressed abundantly in the 1922 paper - from my reading, I would say maintaining invariance under parameter transformation, and the ability to get rid of the priors in general, were one of his main motivations in choosing MLE. If you want to understand his reasoning better, I really recommend the 1922 paper, it's beautifully written and he explains his reasoning very well.

You apply a relatively narrow definition of frequentism and MLE - if we are a bit more generous and define

• Frequentism: goal of consistency, (asymptotic) optimality, unbiasedness, and controlled error rates under repeated sampling, independent of the true parameters

• MLE = point estimate + confidence intervals (CIs)

then it seems pretty clear that MLE satisfies all frequentist ideals. In particular, CIs in MLE, as p-values, control the error rate under repeated sampling, and do not give the 95% probability region for the true parameter value, as many people think - hence they are through and through frequentist.

Not all of these ideas were already present in Fisher's foundational 1922 paper "On the mathematical foundations of theoretical statistics", but the idea of optimality and unbiasedness is, and Neyman latter added the idea of constructing CIs with fixed error rates. Efron, 2013, "A 250-year argument: Belief, behavior, and the bootstrap", summarizes in his very readable history of the Bayesian/Frequentist debate:

The frequentist bandwagon really got rolling in the early 1900s. Ronald Fisher developed the maximum likelihood theory of optimal estimation, showing the best possible behavior for an estimate, and Jerzy Neyman did the same for confidence intervals and tests. Fisher’s and Neyman’s procedures were an almost perfect fit to the scientific needs and the computational limits of twentieth century science, casting Bayesianism into a shadow existence.

Regarding your more narrow definition - I mildly disagree with your premise that minimization of frequentist risk (FR) is the main criterion to decide if a method follows frequentist philosophy. I would say the fact that minimizing FR is a desirable property follows from frequentist philosophy, rather than preceding it. Hence, a decision rule / estimator does not have to minimize FR to be frequentist, and minimizing FR is also does not necessarily say that a method is frequentist, but a frequentist would in doubt prefer minimization of FR.

If we look at MLE specifically: Fisher showed that MLE is asymptotically optimal (broadly equivalent to minimizing FR), and that was certainly one reason for promoting MLE. However, he was aware that optimality did not hold for finite sample size. Still, he was happy with this estimator due to other desirable properties such as consistency, asymptotic normality, invariance under parameter transformations, and let's not forget: ease to calculate. Invariance in particular is stressed abundantly in the 1922 paper - from my reading, I would say maintaining invariance under parameter transformation, and the ability to get rid of the priors in general, were one of his main motivations in choosing MLE. If you want to understand his reasoning better, I really recommend the 1922 paper, it's beautifully written and he explains his reasoning very well.

You apply a relatively narrow definition of frequentism and MLE - if we are a bit more generous and define

• Frequentism: goal of optimality, unbiasedness, and controlled error rates under repeated sampling, independent of the true parameters

• MLE = point estimate + confidence intervals (CIs)

then it seems pretty clear that MLE satisfies all frequentist ideals (btw., it is often not appreciated that CIs in MLE, as p-values, are defined to control the error rate under repeated sampling, and do not give the 95% probability region of the true parameter value - hence they are through and through frequentist).

Not all of these ideas were already present in Fisher's foundational 1922 paper "On the mathematical foundations of theoretical statistics", but the idea of optimality and unbiasedness is, and Neyman latter added the idea of constructing CIs with fixed error rates. Efron, 2013, "A 250-year argument: Belief, behavior, and the bootstrap", summarizes in his very readable history of the Bayesian/Frequentist debate:

The frequentist bandwagon really got rolling in the early 1900s. Ronald Fisher developed the maximum likelihood theory of optimal estimation, showing the best possible behavior for an estimate, and Jerzy Neyman did the same for confidence intervals and tests. Fisher’s and Neyman’s procedures were an almost perfect fit to the scientific needs and the computational limits of twentieth century science, casting Bayesianism into a shadow existence.

You apply a relatively narrow definition of frequentism and MLE - if we are a bit more generous and define

• Frequentism: goal of consistency, (asymptotic) optimality, unbiasedness, and controlled error rates under repeated sampling, independent of the true parameters

• MLE = point estimate + confidence intervals (CIs)

then it seems pretty clear that MLE satisfies all frequentist ideals. In particular, CIs in MLE, as p-values, are defined to control the error rate under repeated sampling, and not to give the 95% probability region of the true parameter value, as many people think - hence they are through and through frequentist.

Not all of these ideas were already present in Fisher's foundational 1922 paper "On the mathematical foundations of theoretical statistics", but the idea of optimality and unbiasedness is, and Neyman latter added the idea of constructing CIs with fixed error rates. Efron, 2013, "A 250-year argument: Belief, behavior, and the bootstrap", summarizes in his very readable history of the Bayesian/Frequentist debate:

The frequentist bandwagon really got rolling in the early 1900s. Ronald Fisher developed the maximum likelihood theory of optimal estimation, showing the best possible behavior for an estimate, and Jerzy Neyman did the same for confidence intervals and tests. Fisher’s and Neyman’s procedures were an almost perfect fit to the scientific needs and the computational limits of twentieth century science, casting Bayesianism into a shadow existence.

Regarding your more narrow definition - I mildly disagree with your premise that minimization of frequentist risk (FR) is the sole criterion to decide if a method follows frequentist philosophy. The fact that minimizing FR is a desirable property follows from frequentist philosophy, rather than preceding it. Hence, a decision rule / estimator does not have to minimize FR to be frequentist, but a frequentist would in doubt prefer these property.

If we look at FR of MLE: Fisher showed that MLE is asymptotically optimal (broadly equivalent to minimizing FR), and that was certainly one reason for promoting MLE. However, he was aware that optimality did not hold for finite sample size. Still, he was happy with this estimator due to other desirable properties such as consistency, asymptotic normality, and invariance under parameter transformations, and let's not forget: ease to calculate. Invariance in particular is stressed abundantly in the 1922 paper - from my reading, I would say maintaining invariance under parameter transformation, and the ability to get rid of the priors in general, were one of his main motivations in choosing MLE. If you want to understand his reasoning better, I really recommend the 1922 paper, it's beautifully written and he explains his reasoning very well.