Estimates and Estimation

Post about Estimates and Estimation that include point estimation, interval estimation, level of significance, Properties of estimators, and real-life examples about estimate and estimation.

Point Estimation of Parameters

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Introduction to Point Estimation of Parameters

The objective of point estimation of parameters is to obtain a single number from the sample which will represent the unknown value of the parameter.

Practically we did not know about the population mean and standard deviation i.e. population parameters such as mean, standard deviation, etc. However, our goal is to measure (estimate) the mean and standard deviation of the population we are interested in from sample information to save time, cost, etc.  This can be done by estimating the sample mean and standard deviation as the best guess for the true population mean and standard deviation.  We can call this estimate a “best guess” and termed a “point estimate” as it is a single number summarized one.

Point Estimate

A Point Estimate is a statistic (a statistical measure from the sample) that gives a plausible estimate (or possibly a best guess) for the value in question.

$\overline{x}$ is a point estimate for $\mu$ and s is a point estimate for $\sigma$.

Or we can say that

A statistic used to estimate a parameter is called a point estimator or simply an estimator. The actual numerical value which we obtain for an estimator in a given problem is called an estimate.

Generally symbol $\theta$ (unknown constant) is used to denote a population parameter which may be a proportion, mean, or some measure of variability. The available information is in the form of a random sample $X_1, X_2, \cdots, X_n$ of size n drawn from the population. We wish to formulate a function of the sample observations $X_1, X_2, \cdots, X_n$; that is, we look for a statistic such that its value computed from the sample data would reflect the value of the population parameter as closely as possible. The estimator of $\theta$ is commonly denoted by $\hat{\theta}$. Different random samples usually provide different values of the statistic $\hat{\theta}$ having its sampling distribution.

Note that Unbiasedness, Efficiency, Consistency, and Sufficiency are the criteria (statistical properties of the estimator) to identify whether a statistic is a “good” estimator.

Application of Point Estimator Confidence Intervals

We can build intervals with confidence as we are not only interested in finding the point estimate for the mean but also in determining how accurate the point estimate is. Here the Central Limit Theorem plays a very important role in building confidence interval.  We assume that the sample standard deviation is close to the population standard deviation (which will almost always be true for large samples). The standard deviation of the sampling distribution of the estimator (here for mean) is

\[\sigma_x \approx \frac{\sigma}{\sqrt{n}}\]

Our interest is to find an interval around $\overline{x}$ such that there is a large probability that the actual (true) mean falls inside the computed interval.  This interval is called a confidence interval and the large probability is called the confidence level.

Example of Point Estimation of Parameters

Question: Suppose that we check for clarity in 50 locations in Lake and discover that the average depth of clarity of the lake is 14 feet with a standard deviation of 2 feet.  What can we conclude about the average clarity of the lake with a 95% confidence level?

Solution: Variable $x$ (depth of lack at 50 locations) can be used to provide a point estimate for $\mu$ and s to provide a point estimate for $s$. To answer how accurate is $x$ as a point estimate, we can construct a 95% confidence interval for $\mu$ as follows.

normal curve: Point Estimation of Parameters

Draw the picture given below and use the standard normal table to find the z-score associated with the probability of .025 (there is .025 to the left and .025 to the right i.e. two-tailed case).

The Z-score for a 95% confidence level is about $\pm 1.96$.

\begin{align*}
Z&=\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\\
\pm 1.96&=\frac{\overline{x}-\mu}{\frac{2}{\sqrt{n}}}\\
\overline{x}-14&=\pm 0.5488
\end{align*}

Note that $Z\frac{\sigma}{\sqrt{n}}$ is called the margin of error.

The 95% confidence interval for the mean clarity will be (13.45, 14.55)

In other words, there is a 95% chance that the mean clarity is between 13.45 and 14.55.

In general, if $z$ is the standard normal table value associated with a given level of confidence then a $\alpha$% confidence interval for the mean is

\[\overline{x} \pm Z_{\alpha}\frac{\sigma}{\sqrt{n}}\]

See more at Wikipedia about Point Estimation of Parameters

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Unbiasedness of the Estimator (2013)

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The unbiasedness of the estimator is probably the most important property that a good estimator should possess. In statistics, the bias (or bias function) of an estimator is the difference between this estimator’s expected value and the true value of the parameter being estimated. An estimator is said to be unbiased if its expected value equals the corresponding population parameter; otherwise, it is said to be biased. Let us discuss in detail the unbiasedness of the estimator.

Unbiasedness of the Estimator

Suppose in the realization of a random variable $X$ taking values in probability space i.e. ($\chi, \mathfrak{F}, P_\theta$), such that $\theta \varepsilon \Theta$, a function $f:\Theta \rightarrow \Omega $ has to be estimated, mapping the parameter set $\Theta$ into a certain set $\Omega$, and that as an estimator of $f(\theta)$ a statistic $T=T(X)$ is chosen. if $T$ is such that
\[E_\theta[T]=\int_\chi T(x) dP_\theta(x)=f(\theta)\]
holds for $\theta\varepsilon \Theta$ then $T$ is called an unbiased estimator of $f(\theta)$. The unbiased estimator is frequently called free of systematic errors.

Unbiased Estimator

Suppose $\hat{\theta}$ be an estimator of a parameter $\theta$, then $\hat{\theta}$ is said to be unbiased estimator if $E(\hat{\theta})=0$.

  • If $E(\hat{\theta})=\theta$ then $\hat{\theta}$ is an unbiased estimator of a parameter $\theta$.
  • If $E(\hat{\theta})<\theta$ then $\hat{\theta}$ is a negatively biased estimator of a parameter $\theta$.
  • If $E(\hat{\theta})>\theta$ then $\hat{\theta}$ is a positively biased estimator of a parameter $\theta$.

Bias of an estimator $\theta$ can be found by $$[E(\hat{\theta})-\theta]$$

  • $\overline{X}$ is an unbiased estimator of the mean of a population (whose mean exists).
  • $\overline{X}$ is an unbiased estimator of $\mu$ in a Normal distribution i.e. $N(\mu, \sigma^2)$.
  • $\overline{X}$ is an unbiased estimator of the parameter $p$ of the Bernoulli distribution.
  • $\overline{X}$ is an unbiased estimator of the parameter $\lambda$ of the Poisson distribution.

In each of these cases, the parameter $\mu, p$ or $\lambda$ is the mean of the respective population being sampled.

However, sample variance $S^2$ is not an unbiased estimator of population variance $\sigma^2$, but consistent.

It is possible to have more than one unbiased estimator for an unknown parameter. The sample mean and the sample median are unbiased estimators of the population mean $\mu$ if the population distribution is symmetrical.

Unbiasedness of the Estimator

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What is Standard Error of Sampling? (2012)

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The standard error (SE) of a statistic is the standard deviation of the sampling distribution of that statistic. The standard error of sampling reflects how much sampling fluctuation a statistic will show. The inferential (deductive) statistics involved in constructing confidence intervals and significance testing are based on standard errors. Increasing the sample size decreases the standard error.

In practical applications, the true value of the standard deviation of the error is unknown. As a result, the term standard error is often used to refer to an estimate of this unknown quantity.

The size of the SE is affected by two values.

  1. The Standard Deviation of the population affects the standard errors. The larger the population’s standard deviation ($\sigma$), the larger is SE i.e. $\frac {\sigma}{\sqrt{n}}$. If the population is homogeneous (which results in a small population standard deviation), the SE will also be small.
  2. The standard errors are affected by the number of observations in a sample. A large sample will result in a small SE of estimate (indicates less variability in the sample means)

Application of Standard Error of Sampling

The SEs are used in different statistical tests such as

  • to measure the distribution of the sample means
  • to build confidence intervals for means, proportions, differences between means, etc., for cases when population standard deviation is known or unknown.
  • to determine the sample size
  • in control charts for control limits for means
  • in comparison tests such as z-test, t-test, Analysis of Variance,
  • in relationship tests such as Correlation and Regression Analysis (standard error of regression), etc.

(1) Standard Error Formula Means

The SE for the mean or standard deviation of the sampling distribution of the mean measures the deviation/ variation in the sampling distribution of the sample mean, denoted by $\sigma_{\bar{x}}$ and calculated as the function of the standard deviation of the population and respective size of the sample i.e

$\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$                      (used when population is finite)

If the population size is infinite then ${\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}} \times \sqrt{\frac{N-n}{N}}}$ because $\sqrt{\frac{N-n}{N}}$ tends towards 1 as N tends to infinity.

When the population’s standard deviation ($\sigma$) is unknown, we estimate it from the sample standard deviation. In this case SE formula is $\sigma_{\bar{x}}=\frac{S}{\sqrt{n}}$

Standard Error of sampling

(2) Standard Error Formula for Proportion

The SE for a proportion can also be calculated in the same manner as we calculated the standard error of the mean, denoted by $\sigma_p$ and calculated as $\sigma_p=\frac{\sigma}{\sqrt{n}}\sqrt{\frac{N-n}{N}}$.

In case of finite population $\sigma_p=\frac{\sigma}{\sqrt{n}}$
in case of infinite population $\sigma=\sqrt{p(1-p)}=\sqrt{pq}$, where $p$ is the probability that an element possesses the studied trait and $q=1-p$ is the probability that it does not.

(3) Standard Error Formula for Difference Between Means

The SE for the difference between two independent quantities is the square root of the sum of the squared standard errors of both quantities i.e $\sigma_{\bar{x}_1+\bar{x}_2}=\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}$, where $\sigma_1^2$ and $\sigma_2^2$ are the respective variances of the two independent population to be compared and $n_1+n_2$ are the respective sizes of the two samples drawn from their respective populations.

Unknown Population Variances
Suppose the variances of the two populations are unknown. In that case, we estimate them from the two samples i.e. $\sigma_{\bar{x}_1+\bar{x}_2}=\sqrt{\frac{S_1^2}{n_1}+\frac{S_2^2}{n_2}}$, where $S_1^2$ and $S_2^2$ are the respective variances of the two samples drawn from their respective population.

Equal Variances are assumed
In case when it is assumed that the variance of the two populations are equal, we can estimate the value of these variances with a pooled variance $S_p^2$ calculated as a function of $S_1^2$ and $S_2^2$ i.e

\[S_p^2=\frac{(n_1-1)S_1^2+(n_2-1)S_2^2}{n_1+n_2-2}\]
\[\sigma_{\bar{x}_1}+{\bar{x}_2}=S_p \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\]

(4) Standard Error for Difference between Proportions

The SE of the difference between two proportions is calculated in the same way as the SE of the difference between means is calculated i.e.
\begin{eqnarray*}
\sigma_{p_1-p_2}&=&\sqrt{\sigma_{p_1}^2+\sigma_{p_2}^2}\\
&=& \sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}
\end{eqnarray*}
where $p_1$ and $p_2$ are the proportion for infinite population calculated for the two samples of sizes $n_1$ and $n_2$.

FAQs about Standard Error

  1. Define the Standard Error of Mean.
  2. Standard Error is affected by which two values?
  3. Write the formula of the standard error of mean, proportion, and difference between means.
  4. What is the application of standard error of mean in Sampling?
  5. Discuss the importance of standard error?
https://itfeature.com Standard Error

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