4.4.1 - Properties of 'Good' Estimators | STAT 800 (2024)
Note!
We should stop here and explain why we use the estimated standard error and not the standard error itself when constructing a confidence interval. Remember we are using the known values from our sample to estimate the unknown populationvalues.Therefore we cannot use the actual population values!This is actuallyeasier to see by presenting the formulas.If we used the following as the standard error, we would not have the values for \(p\) (because this is the population parameter):
\(\sqrt{\dfrac{p(1-p)}{n}}\)
Instead we have to use the estimated standard error by using \(\hat{p}\) In this case the estimated standard error is...
For the case for estimating the population mean, the population standard deviation, \(\sigma\), may also be unknown. When it is unknown, we can estimate it with the sample standard deviation, s. Then the estimated standard error of the sample mean is...
\(\dfrac{s}{\sqrt{n}}\)
As an experienced statistician and data analyst with a background in quantitative research, I've delved deeply into the intricacies of statistical inference and hypothesis testing. My expertise extends to the construction of confidence intervals and the nuances involved in choosing the appropriate standard error for such calculations.
Let's address the fundamental concept emphasized in your article – the use of estimated standard error over the standard error itself when constructing a confidence interval. This practice is rooted in the principles of statistical estimation, which involves using information from a sample to make inferences about a population.
In statistical analysis, we encounter situations where the actual population parameters are unknown, and we rely on sample data to estimate them. This leads us to use the estimated standard error, a crucial step in constructing confidence intervals.
The first formula mentioned in the article pertains to estimating the standard error for a population proportion ((p)). The standard error for (p) is given by:
[ \sqrt{\dfrac{p(1-p)}{n}} ]
However, in practice, we often don't have the actual population proportion ((p)) because it's a population parameter. Instead, we use the sample proportion ((\hat{p})) as an estimate. Thus, the estimated standard error becomes:
[ \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}} ]
This adjustment is crucial when working with sample data to infer about a larger population.
Moving on to estimating the population mean, another scenario arises when the population standard deviation ((\sigma)) is unknown. In such cases, we resort to estimating it using the sample standard deviation ((s)). The formula for the estimated standard error of the sample mean ((\bar{x})) is:
[ \dfrac{s}{\sqrt{n}} ]
This formula reflects the incorporation of sample data to estimate the standard error, providing a more realistic reflection of the uncertainty inherent in using sample statistics to make inferences about population parameters.
In conclusion, the utilization of estimated standard errors is a vital aspect of statistical inference, allowing us to bridge the gap between sample data and population parameters. These adjustments are essential for constructing accurate and reliable confidence intervals, showcasing the practical application of statistical principles in real-world scenarios.
In determining what makes a good estimator, there are two key features: The center of the sampling distribution for the estimate is the same as that of the population. When this property is true, the estimate is said to be unbiased. The most often-used measure of the center is the mean.
1. Unbiasedness. An estimator is said to be unbiased if its expected value is identical with the population parameter being estimated. That is if θ is an unbiased estimate of θ, then we must have E (θ) = θ.
A good confidence interval should have two desirable properties: coverage probability and margin of error. Coverage probability means that the confidence interval will contain the true population parameter with a certain level of confidence.
The most efficient point estimator is the one with the smallest variance of all the unbiased and consistent estimators. The variance measures the level of dispersion from the estimate, and the smallest variance should vary the least from one sample to the other.
An almost essential property is that the estimator should be consistent: T is a consistent estimator of θ if T converges to θ in probability as n → ∞. Consistency implies that, as the sample size increases, any bias in T tends to 0 and the variance of T also tends to 0.
The three desirable properties of an estimator are unbiasedness, efficiency, and consistency. An unbiased estimator is one whose expected value (the mean of its sampling distribution) equals the parameter it is intended to estimate.
The one which is unbiased,Consistent(preferably consistent for small sample size rather than being asymptotically consistent),and Sufficent. For being the best one it should be Minimum variance bound unbiased estimator and sufficient. Originally Answered: What are the four qualities of a good estimator?
Efficiency: The most efficient estimator among a group of unbiased estimators is the one with the smallest variance. For example, both the sample mean and the sample median are unbiased estimators of the mean of a normally distributed variable. However, X has the smallest variance.
A 95% confidence interval (CI) of the mean is a range with an upper and lower number calculated from a sample. Because the true population mean is unknown, this range describes possible values that the mean could be.
2) Sufficiency: It is a property that refers to the use of all the information that could be derived from a sample to estimate the corresponding parameter. In other words, there does not exist any other statistic that could provide more information about the parameter value, when calculated from the same sample.
An unbiased estimator is when a statistic does not overestimate or underestimate a population parameter. In other words, a value is unbiased when it is the same as the actual value of a particular parameter.
Ratio Scale: The Ratio Scale is the highest level of measurement scales. This has the properties of an interval scale together with a fixed (absolute) zero point. The absolute zero points allow us to construct a meaningful ratio. Examples of ratio scales include weights, lengths, and times.
Interval data cannot be multiplied or divided, however, it can be added or subtracted. Interval data is measured on an interval scale. A simple example of interval data: The difference between 100 degrees Fahrenheit and 90 degrees Fahrenheit is the same as 60 degrees Fahrenheit and 70 degrees Fahrenheit.
The interval scale of measurement has the properties of identity, magnitude, and equal intervals. A perfect example of an interval scale is the Fahrenheit scale to measure temperature.
Interval variables have the property that differences in the numbers represent real differences in the variable. Another way to say this is that equal equal differences in the numbers on the scale represent equal differences in the underlying variables being measured.
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