Idea of Relative Efficiency

Efficiency is a measure of quality that can be used in the comparison of different statistical procedures. This can be a measure of the quality of an estimator, an experimental design, or a procedure for testing a hypothesis. To boil it all down, an estimator, experiment, or test that is more efficient requires fewer observations in order to attain the same level of error performance as one that is less efficient. An efficient estimator has a small variance or mean square error, which indicates that there is a small deviation between the estimated value and the “true” value. This is one of the characteristics of an efficient estimator.

Although this term is frequently used when comparing a given method to a notional “best possible” procedure, the relative efficiency of two procedures is defined as the ratio of their individual efficiencies. It is possible to use the asymptotic relative efficiency, which is defined as the limit of the relative efficiencies as the sample size grows, as the primary comparison measure in many situations. In theory, the efficiencies and the relative efficiencies of two procedures are dependent on the sample size that is available for the given procedure. However, in practice, this is not always the case.

In the context of sampling, as an alternative word for relative precision. Finding the mean of a big population with a straightforward random sample gives you results with the same level of relative precision and efficiency as any other method. There are other scenarios in which they might not be equal.

For the purpose of referring to a “best possible” technique as a “gold standard.” It is the proportion of the prospective method to the gold standard. For instance, a score of 75 out of 100 means that the prospective technique is 75 percent as effective as the optimum conceivable procedure according to the theory.

Asymptotic Efficiency:

Asymptotic efficiency is the limit of an estimator’s efficiency as the sample size approaches infinity. This applies to estimators that are objective. An “asymptotically efficient estimator” is one that has an asymptotic efficiency of 1.0, as this criterion has been established. As the number of observations in a sample increases, the precision of an estimator that is asymptotically efficient approaches the theoretical limit, roughly speaking.

The population is a critical factor in determining the asymptotic efficiency of an estimator. An estimator may be asymptotically efficient for one type of population (distribution), but not for others. This can depend on the type of population being considered.

When comparing the known estimators, the number of estimators that are asymptotically efficient is significantly more than the number of estimators that are efficient.

Types of Efficiencies:

Relative efficiency:

It is expressed as a ratio between the given thermal efficiency and the thermal efficiency of a cycle that can theoretically be reversed.

Relative efficiency = Indicated thermal efficiency / Thermal efficiency of reversible cycle. 

Mechanical efficiency:

The ratio of the braking power, also known as the delivered power, to the suggested power is the definition of mechanical efficiency (power provided to the piston).

η_m = b_p/I_p = b_p / b_p + f_p

When calculating an engine’s overall efficiency, the mechanical losses must be taken into consideration.

Indicated power:

It refers to the force that is generated inside the cylinder of the engine.

ip = pₘ LAN.n / 60

Brake power:

It is possible to refer to this as the power that is output by the engine; it is the real power that is available at the crankshaft. It is never greater than the power that is indicated.

BP = 2πNT / 60

Indicated thermal efficiency:

It is the ratio of input power to output power from the fuel.

η_i = ip / m_f * cv

Brake thermal efficiency:

It is the proportion of brake horsepower to fuel horsepower.

ηb = bp / m_f * cv

Conclusion:

Although this idea is commonly used when comparing a given operation to a notional “best possible” procedure, it refers to the ratio of the efficiencies of the two procedures being compared. The relative efficiency of two procedures is the same as the ratio of their efficiencies. Efficiency is a measure of quality that can be used in the comparison of different statistical procedures. This can be a measure of the quality of an estimator, an experimental design, or a procedure for testing a hypothesis.

An efficient estimator has a small variance or mean square error, which indicates that there is a small deviation between the estimated value and the “true” value. This is one of the characteristics of an efficient estimator.

In theory, the efficiencies and the relative efficiencies of two procedures are dependent on the sample size that is available for the given procedure.

Asymptotic efficiency is the limit of an estimator’s efficiency as the sample size approaches infinity. The population is a critical factor in determining the asymptotic efficiency of an estimator. An estimator may be asymptotically efficient for one type of population (distribution), but not for others. 

When comparing the known estimators, the number of estimators that are asymptotically efficient is significantly more than the number of estimators that are efficient.