Point estimation and confidence intervals



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  • The following topics will be covered in this lecture:

    • General concepts in point estimation
    • Bias of estimators
    • Variance of estimators
    • Standard Error
    • An introduction to confidence intervals

General concepts of point estimation

  • Recall, any function of a random sample, i.e., any statistic, is modeled as a random variable.

  • If \( h \) is a general function used to compute some statistic, we thus define

    \[ \hat{\Theta} = h(X_1, \cdots, X_n) \]

    to be a random variable that will depend on the particular realizations of \( X_1,\cdots, X_n \).

  • We call the probability distribution of a statistic a sampling distribution.

    Sampling Distribution
    The probability distribution of a statistic is called a sampling distribution.
  • The sample mean

    \[ \hat{\Theta} = \overline{X} = h(X_1, \cdots, X_n)= \frac{1}{n}\sum_{i=1}^n X_i \]

    is now one example for which we have a model of the sampling distribution.

  • Specifically, the central limit theorem says that the sampling distribution of the sample mean is \( \overline{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right) \) when \( X \) is normal, or if \( n \) is sufficiently large.

General concepts of point estimation continued

  • Recall, we had a special name for \( \hat{\Theta} \) in relation to the true parameter value \( \theta \):
  • Point estimators
    A point estimate of some population parameter \( \theta \) is a single numerical value \( \hat{\theta} \) of a statistic \( \hat{\Theta} \). This is a particular realization of the random variable \( \hat{\Theta} \), viewed as a random variable; \( \hat{\Theta} \) is called the point estimator.
  • We want an estimator to be “close” in some sense to the true value of the unknown parameter, but we know that it happens to be a random variable.

  • In this way, we need to describe how close this estimator is to the true value in a probabilistic sense.

  • As we have seen before, there are important parameters that describe a probability distribution or a data set:

    1. the “center” of the data / distribution; and
    2. the “spread” of the data / distribution.
  • The central limit theorem actually provided both of these (and the sampling distribution) for the sample mean:

    1. the “center” of the distribution for \( \hat{\Theta}=\overline{X} \) was given by \( \mu \), the true population mean;
    2. the “spread” of the distribution for \( \hat{\Theta}=\overline{X} \) was given by \( \frac{\sigma}{\sqrt{n}} \), the standard deviation of the population, divided by the square-root of the sample size.
  • The two above parameters thus give us a means of describing “how close” the sample mean \( \overline{X} \) tends to be to the population mean \( \mu \) in a probabilistic sense.

Bias of estimators

  • The notion of the “center” of the sampling distribution can be useful as a general criteria for estimators.

  • Formally, we say that \( \hat{\Theta} \) is an unbiased estimator of \( \theta \) if the expected value of \( \hat{\theta} \) is equal to \( \theta \).

  • This is equivalent to saying that the mean of the probability distribution of \( \hat{\Theta} \) (or the mean of the sampling distribution of \( \hat{\Theta} \)) is equal to \( \theta \).

Bias of an Estimator
The point estimator \( \hat{\Theta} \) is an unbiased estimator for the parameter \( \theta \) if \[ \mathbb{E}\left[\hat{\Theta}\right] = \theta \] If the estimator is not unbiased, then the difference \[ \mathbb{E}\left[\hat{\Theta}\right] - \theta \] is called the bias of the estimator \( \hat{\Theta} \). When an estimator is unbiased, the bias is zero; that is, \[ \begin{align} \mathbb{E}\left[\hat{\Theta}\right] - \theta &= \theta - \theta \\ &=0 \end{align} \]
  • If we consider the expected value to represent the average value over infinite replications;

    • the above says that “over infinite replications of a random sample of size \( n \), the average value of the point estimator \( \hat{\Theta} \) will equal the true population parameter \( \theta \)”.
  • A particular realization of \( \hat{\Theta} \) will generally not equal the true value \( \theta \).

  • However, replications of the experiment will give a good approximation of the true value \( \theta \).

Variance of estimators

  • We use the bias as discussed already to measure the center of a sampling distribution
    • An unbiased estimator will have a distribution centered at the true population parameter.
  • Yet suppose we have two estimators of the same parameter \( \theta \), which we will denote \( \hat{\Theta}_1 \) and \( \hat{\Theta}_2 \) respectively.
  • It is possible that they are both unbiased (the sampling distributions have the same center), yet they have different spread.
  • That is to say, one estimator might tend to vary more than the other.
Sampling distributions with same mean and different variance.

Courtesy of Montgomery & Runger, Applied Statistics and Probability for Engineers, 7th edition

  • The spread is a critical measure of how much variation is encountered with respect to resampling.
  • We might describe the two concepts with an estimator as follows:
    • Accuracy of an estimator - this is represented by the estimator being unbiased, so that we expect it to give an accurate result on average.
    • Precision of an estimator - this is represented by the estimator having a small spread, so that the estimates don’t differ wildly from sample to sample.
  • It is possible, in general, for an estimator to be either, both or neither of the above.
  • We are often interested, thus, in unbiased estimators with a minimum variance as a first choice.
  • In some situations biased estimators will actually be preferred, though a general discussion of the tradoffs is beyond our scope.

Variance of estimators continued

  • As a formal definition, we will introduce the following idea:
  • Minimum Variance Unbiased Estimator
    If we consider all unbiased estimators of \( \theta \), the one with the smallest variance is called the minimum variance unbiased estimator (MVUE).
  • The practical interpretation again is demonsrated by the last figure:
Sampling distributions with same mean and different variance.

Courtesy of Montgomery & Runger, Applied Statistics and Probability for Engineers, 7th edition

  • Suppose that \( \hat{\Theta}_1 \) is the MVUE, and \( \hat{\Theta}_2 \) is any other unbiased estimator.
  • Then, \[ \mathrm{var}\left(\hat{\Theta}_1\right) \leq \mathrm{var}\left(\hat{\Theta}_2\right). \]
  • Practically speaking, the MVUE is the most precise unbiased estimator, as its value changes the least with respect to resampling.
  • An important example of a MVUE is actually the sample mean.
  • If \( X_1, X_2 , \cdots , X_n \) is a random sample of size \( n \) from a normal distribution with mean \( \mu \) and variance \( \sigma^2 \), the sample mean \( \overline{X} \) is the MVUE for \( \mu \).
  • Again, other choices exist to estimate \( \mu \), but among all unbiased estimators, the sample mean is the most precise.
  • For non-normal distributions, however, a better choice might be, e.g., a biased estimator.

Standard error of an estimator

  • As noted before, the variance is a “natural” measure of spread mathematically for theoretical reasons, but it is in the units squared of the original units.

  • For this reason, when we talk about the spread of an estimator's sampling distribution, we typically discuss the standard error.

    The standard error Let \( \hat{\Theta} \) be an estimator of \( \theta \). The standard error error of \( \hat{\Theta} \) is its standard deviation given by \[ \sigma_\hat{\Theta} = \sqrt{\mathrm{var}\left(\hat{\Theta}\right)}. \] If the standard error involves unknown parameters that can be estimated, substitution of those values into the equation above produces an estimated standard error denoted \( \hat{\sigma}_\hat{\Theta} \). It is also common to write the standard error as \( \mathrm{SE}\left(\hat{\Theta}\right) \).
  • Q: can anyone recall what the standard error is of the sample mean? That is, what is the standard deviation of the sampling distribution (for a normal sample or \( n \) large)?

    • A: the central limit theorem states that \( \overline{X} \) follows (exactly for a normal sample or \( n \) large, approximately) a sampling distribution

    \[ \overline{X}\sim N\left(\mu, \frac{\sigma^2}{n}\right). \]

    • Therefore, the standard error of the sample mean is precisely,

    \[ \sigma_{\overline{X}} = \frac{\sigma}{\sqrt{n}}. \]

Standard error of an estimator

  • As was discussed before, there are times that we may not know all the parameters that describe the standard error.

  • For example, suppose we draw \( X_1, \cdots, X_n \) from a normal population, for which we know neither the mean nor the variance.

  • Let the unknown and unobservable theoretical parameters be denoted \( \mu \) and \( \sigma \) as usual.

  • The sample mean has the sampling distribution,

    \[ \overline{X} \sim N\left( \mu, \frac{\sigma^2}{n}\right), \]

    and therefore standard error \( \sigma_{\overline{X}} = \frac{\sigma}{\sqrt{n}} \).

  • However, we stated that \( \sigma \) itself is unknown.

  • In this case, we will estimate the standard error as

    \[ \hat{\sigma}_\overline{X} = \frac{s}{\sqrt{n}} \] with the sample standard deviation \( s \).

  • This is what is meant to estimate the standard error.

  • This particular example will be extremely important for confidence intervals.

Standard error of an estimator – example

  • An article in the Journal of Heat Transfer (Trans. ASME, Sec. C, 96, 1974, p. 59) described a new method of measuring the thermal conductivity of Armco iron.

  • Using a temperature of \( 100^\circ \) F and a power input of 550 watts, the following 10 measurements of thermal conductivity (in Btu/hr-ft-∘ F) were obtained:

    \[ 41.60, 41.48, 42.34, 41.95, 41.86, 42.18, 41.72, 42.26, 41.81, 42.04 \]

  • A point estimate of the mean thermal conductivity at \( 100^\circ \) F and 550 watts is the sample mean or

    \[ \overline{x} = 41.924 \]

  • The standard error of the sample mean is \( \sigma_\overline{X}=\frac{\sigma}{\sqrt{n}} \);

    • however, \( \sigma \) is unknown so that we estimate it by the sample standard deviation \( s = 0.284 \) to obtain

    \[ \hat{\sigma}_\overline{X} = \frac{s}{\sqrt{n}}= \frac{0.284}{\sqrt{10}} \approx 0.0898 \]

  • Notice that the standard error is about 0.2 percent of the sample mean, implying that we have obtained a relatively precise point estimate of thermal conductivity.

Standard error of an estimator – example

  • Assume that thermal conductivity is normally distributed, then two times the standard error is

    \[ 2\hat{\sigma}_\overline{X} = 2(0.0898) = 0.1796. \]

  • The empirical rule says that about 95% of realizations of the sample mean lie within two standard deviations of the true mean \( \mu \).

  • Therefore, we are highly confident that the true mean thermal conductivity is within the interval 41.924 ± 0.1796 or between \( [41.744 , 42.104] \).

  • We will now formalize this logic into confidence intervals.

Introduction to confidence intervals

  • Engineers are often involved in estimating parameters.

  • For example, there is an ASTM Standard E23 that defines a technique called the Charpy V-notch method for notched bar impact testing of metallic materials.

  • The impact energy is often used to determine whether the material experiences a ductile-to-brittle transition as the temperature decreases.

  • Suppose that we have tested a sample of \( 10 \) specimens of a particular material with this procedure. We know that we can use the sample average \( \overline{X} \) to estimate the true mean impact energy \( \mu \).

  • However, we also know that the true mean impact energy is unlikely to be exactly equal to your estimate due to sampling error.

  • Reporting the results of your test as a single number is unappealing because \( \overline{X} \) may be an accurate estimator, but it doesn't tell us how precise the estimate is.

    • Rather, measures of the spread of the sampling distribution, e.g., the standard error tell us how precise the estimate is.
  • A way to quantify our uncertainty is to report the estimate in terms of a range of plausible values called a confidence interval.

Introduction to confidence intervals – continued

  • Suppose \( \hat{\Theta} \) is an unbiased estimator for the speed of light \( \theta \).
  • We are assuming that \( \theta \) is a deterministic but unknown constant.

    • For sake of example, also suppose that \( \hat{\Theta} \) has standard deviation \( \sigma_\hat{\Theta} \) = 100 km/sec.
    • Recall Chebyshev’s inequality,

    \[ \begin{align} P\left(\vert \hat{\Theta} - \theta \vert \geq k \sigma_\hat{\Theta}\right) \leq \frac{1}{k^2} \end{align} \]

    • We find

    \[ \begin{align} P\left(\vert \hat{\Theta} - \theta \vert < 2 \sigma_\hat{\Theta}\right) > \frac{3}{4} \end{align} \]

    • This tells us that there is a probability of at least 75% that \( \hat{\Theta} \) is within 200 km/sec of the speed of light \( \theta \).
    • Equivalently, \( \theta \in \left(\hat{\Theta}-200, \hat{\Theta}+200\right) \) with probability 75%.
    • Notice that \( \left(\hat{\Theta}-200, \hat{\Theta}+200\right) \) is a random interval but we again assume that \( \theta \) is a fixed constant.

Introduction to confidence intervals – continued

  • Now we will suppose the estimate \( \hat{\Theta} \) gives us based on some data is \( \hat{\theta}=299852.4 \)

  • We can say that \( \theta \in (299652.4, 300 052.4) \) with confidence 75%.

    • Note that \( \theta \) is an an unknown constant – it is either in the interval or not and there is nothing random about the above statement.
    • Therefore, we cannot say that the probability of \( \theta \in (299652.4, 300 052.4) \) is 75%, but we used information to guarantee that our procedure for estimation will work 75% of the time.
  • Confidence intervals give us a systematic procedure as above to guarantee with a level of confidence that our plausible values for the parameter \( \theta \) include the true value.

  • In the remaining course, we will be concerned with dual questions:

    1. With what confidence can we estimate a parameter \( \theta \) as a range of plausible values? And
    2. how unlikely would it be for \( \theta \) to be outside of some range based on our observations?
  • These two ideas are known as confidence intervals and hypothesis testing respectively.

Introduction to confidence intervals – continued

  • Suppose that \( X_1 , X_2, \cdots , X_n \) is a random sample from a normal distribution with unknown mean \( \mu \) and known variance \( \sigma^2 \).

  • By the central limit theorem, we know that the sample mean \( \overline{X} \) is distributed

    \[ \overline{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right) \]

  • We may standardize \( \overline{X} \) by subtracting its mean and dividing by its standard deviation,

    \[ \begin{align} Z = \frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}}. \end{align} \]

  • The random variable \( Z \) has a standard normal distribution.

  • A confidence interval estimate for \( \mu \) is an interval of the form

    \[ l ≤ \mu ≤ u, \] where the end-points \( l \) and \( u \) are computed from the sample data.

  • Because different samples will produce different values of \( l \) and \( u \), these end-points are values of random variables \( L \) and \( U \), respectively.

Introduction to confidence intervals – continued

  • Suppose that we can determine values of \( L \) and \( U \) such that the following probability statement is true:

    \[ P\{L \leq \mu \leq U\} = 1 − \alpha \] where \( 0 \leq \alpha \leq 1 \).

  • There is a probability of \( 1 − \alpha \) of selecting a sample for which the CI will contain the true value of \( \mu \).

  • Once we have selected the sample, so that \( X_1 = x_1 , X_2 = x_2 , \cdots , X_n = x_n \), and computed \( l \) and \( u \), the resulting confidence interval for \( \mu \) is

    \[ l \leq \mu \leq u. \]

  • The end-points or bounds \( l \) and \( u \) are called the lower- and upper-confidence limits (bounds), respectively, and \( 1 − \alpha \) is called the confidence coefficient (or level).

  • Again,

    $$l \leq \mu \leq u.$$ is a fixed numerical statement with nothing random about it, this is true or untrue.

  • The goal then is to find the procedure that defines the random variables \( L,U \) for which the procedure will succeed \( (1-\alpha)\times 100\% \) of the time.

Introduction to confidence intervals – continued

Common confidence levels.

Courtesy of Mario Triola, Essentials of Statistics, 6th edition

  • Let’s take an example confidence level of \( 95\% \) – this corresponds to a rate of failure of \( 5\% \) over infinitely many replications.
  • Generally, we will write the confidence level as, \[ (1 - \alpha) \times 100\% \] so that we can associate this confidence level with its rate of failure \( \alpha \).
  • In our problem, because \( Z = \frac{\overline{X} - \mu }{\frac{\sigma}{\sqrt{n}}} \) has a standard normal distribution, we may write \[ \begin{align} P\left(-z_\frac{\alpha}{2} \leq \frac{\overline{X} - \mu }{\frac{\sigma}{\sqrt{n}}} \leq z_\frac{\alpha}{2}\right) = 1 - \alpha \end{align} \] where
  • we want to find the critical value \( z_\frac{\alpha}{2} \) for which:
    • \( (1-\frac{\alpha}{2})\times 100\% \) of the area under the normal density lies to the left of \( z_\frac{\alpha}{2} \); and
    • \( (1-\frac{\alpha}{2})\times 100\% \) of the area under the normal density lies to the right of \( -z_\frac{\alpha}{2} \).
  • Put together, \( (1-\alpha)\times 100\% \) of values lie within \( [-z_\frac{\alpha}{2},z_\frac{\alpha}{2}] \) in the standard normal.
Area between alpha over two critical value.

Introduction to confidence intervals – continued