SMHS Estimation - Revision history

Glenbrau: /* Problems */

2015-04-27T15:43:18Z

‎Problems

Glenbrau: /* Confidence Intervals (CIs) */

2015-04-27T15:39:11Z

‎Confidence Intervals (CIs)

Glenbrau: /* Theory */

2015-04-27T15:37:33Z

‎Theory

Glenbrau: /* Confidence Intervals (CIs) */

2015-04-07T17:58:36Z

‎Confidence Intervals (CIs)

Glenbrau: /* Confidence Intervals (CIs) */

2015-04-07T16:06:28Z

‎Confidence Intervals (CIs)

Glenbrau: /* Theory */

2015-04-07T14:06:55Z

‎Theory

Glenbrau: /* Motivation */

2015-04-07T14:06:27Z

‎Motivation

Glenbrau: /* Overview */

2015-04-07T13:55:07Z

‎Overview

Dinov: /* Characteristics of Estimators */

2014-09-11T20:33:59Z

‎Characteristics of Estimators

Dinov: /* Characteristics of Estimators */

2014-09-11T20:32:28Z

‎Characteristics of Estimators

@@ Line 95: / Line 95: @@
 ===Problems===
-* Which of the following statements is true.
+* Which of the following statements is true?
 : a.  When the margin of error is small, the confidence level is high.
 : b. When the margin of error is small, the confidence level is low.
@@ Line 102: / Line 102: @@
 : e. None of the above.
-* Which of the following statements is true.
+* Which of the following statements is true?
 : a. The standard error is computed solely from sample attributes.
 : b. The standard deviation is computed solely from sample attributes.

@@ Line 43: / Line 43: @@
 *Find the critical probability $(p^*): p^* = 1 -\frac {\alpha} {2}$
 *To express the critical value as a $z$ score, find the $z$ score having a cumulative probability equal to the critical probability $(p^*)$.
-*To express the critical value as a t score, follow these steps. Find the degree of freedom (DF): when estimating a mean score or a proportion from a single sample, DF is equal to the sample size minus one. For other applications, the degrees of freedom may be calculated differently. We will describe those computations as they come up.
+*To express the critical value as a $t$ score, follow these steps. Find the degree of freedom (DF): when estimating a mean score or a proportion from a single sample, DF is equal to the sample size minus one. For other applications, the degrees of freedom may be calculated differently. We will describe those computations as they come up.
 : The critical $t$ score $(t^*)$ is the $t$ score having degrees of freedom equal to DF and a cumulative probability equal to the critical probability $(p^*)$.

@@ Line 21: / Line 21: @@
 ===Theory===
 An estimate of a population parameter may be expressed in two ways:
-*''Point estimate'': a single value of estimate. For example, sample mean is a point estimate of the population mean.
+*''Point estimate'': A single value of estimate. For example, sample mean is a point estimate of the population mean.
-*''Interval estimate'': an interval estimate is defined by two numbers, between which a population parameter is said to lie.
+*''Interval estimate'': An interval estimate is defined by two numbers, between which a population parameter is said to lie.
 ====Confidence Intervals (CIs)====

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 : The critical $t$ score $(t^*)$ is the $t$ score having degrees of freedom equal to DF and a cumulative probability equal to the critical probability $(p^*)$.
-: Should you express the critical value as a t score or as a $z$ score? As a practical matter, when the sample size is large (greater than 40), it doesn't make much difference. Both approaches yield similar results. Strictly speaking, when the population standard deviation is unknown or when the sample size is small, the $t$ score is preferred. Nevertheless, many introductory statistics texts use the $z$ score exclusively.
+: Should you express the critical value as a $t$ score or as a $z$ score? As a practical matter, when the sample size is large (greater than 40), it doesn't make much difference. Both approaches yield similar results. Strictly speaking, when the population standard deviation is unknown or when the sample size is small, the $t$ score is preferred. Nevertheless, many introductory statistics texts use the $z$ score exclusively.
 * ''Standard error'': an estimate of the standard deviation of a statistic. When the values of population parameters are unknown, it is valuable to compute the standard error as an unbiased estimate of the standard deviation of a statistic. It is computed form known sample statistic. The table below shows how to compute the standard error for simple random samples assuming that the population size is at least 10 times larger than the sample size.

@@ Line 25: / Line 25: @@
 ====Confidence Intervals (CIs)====
-CIs describe the uncertainty of a sampling method and contains a confidence level, a statistic and a margin of error. The statistic and the margin of error define an interval estimate, which represent the precision of the method. Confidence Interval is expressed as sample statistic ± margin of error.
+CIs describe the uncertainty of a sampling method and contain a ''confidence level'', a ''statistic'' and a ''margin of error''. The statistic and the margin of error define an interval estimate, which represents the precision of the method. Confidence Interval is expressed as sample statistic plus the margin of error.
-* Confidence level: the probability part of a confidence interval. It describes the likelihood that a particular sampling method will produce a confidence interval that includes the true population parameter.
+The interpretation of a confidence interval at 95% confidence level is that we have 95% confidence that the parameter will fall within the margin of the interval.
-* Margin of error: range of the values above and below the sample statistic in confidence interval. ''margin of error=critical value*standard deviation of the statistic''.
+* ''Confidence level'': The probability part of a confidence interval. It describes the likelihood that a particular sampling method will produce a confidence interval that includes the true population parameter.
-* Critical value: The central limit theorem states that the sampling distribution of a statistic will be normal or nearly normal and the critical value can be expressed as a t score or as a z score, if ANY of the following conditions apply:
+* ''Margin of error'': Range of the values above and below the sample statistic in confidence interval; ''margin of error $=$ critical value $*$ standard deviation of the statistic''
-**The population distribution is normal;
-**The sampling distribution is symmetric, unimodal, without outliers, and the sample size is 15 or less;
+* ''Critical value'': The central limit theorem states that the sampling distribution of a statistic will be normal or nearly normal, and that the critical value can be expressed as a $t$ score or as a $z$ score provided that ANY of the following conditions apply:
-**The sampling distribution is moderately skewed, unimodal, without outliers, and the sample size is between 16 and 40;
+**The population distribution is normal.
 **The sample size is greater than 40, without outliers.
@@ Line 44: / Line 45: @@
 *To express the critical value as a t score, follow these steps. Find the degree of freedom (DF): when estimating a mean score or a proportion from a single sample, DF is equal to the sample size minus one. For other applications, the degrees of freedom may be calculated differently. We will describe those computations as they come up.
-: The critical t score $(t^*)$ is the t score having degrees of freedom equal to DF and a cumulative probability equal to the critical probability $(p^*)$.
+: The critical $t$ score $(t^*)$ is the $t$ score having degrees of freedom equal to DF and a cumulative probability equal to the critical probability $(p^*)$.
-: Should you express the critical value as a t score or as a z score? As a practical matter, when the sample size is large (greater than 40), it doesn't make much difference. Both approaches yield similar results. Strictly speaking, when the population standard deviation is unknown or when the sample size is small, the t score is preferred. Nevertheless, many introductory statistics texts use the z score exclusively.
+: Should you express the critical value as a t score or as a $z$ score? As a practical matter, when the sample size is large (greater than 40), it doesn't make much difference. Both approaches yield similar results. Strictly speaking, when the population standard deviation is unknown or when the sample size is small, the $t$ score is preferred. Nevertheless, many introductory statistics texts use the $z$ score exclusively.
-* Standard error: an estimate of the standard deviation of a statistic. When the values of population parameters are unknown, it is valuable to compute the standard error as an unbiased estimate of the standard deviation of a statistic. It is computed form known sample statistic. The table below shows how to compute the standard error for simple random samples assuming that the population size is at least 10 times larger than the sample size.
+* ''Standard error'': an estimate of the standard deviation of a statistic. When the values of population parameters are unknown, it is valuable to compute the standard error as an unbiased estimate of the standard deviation of a statistic. It is computed form known sample statistic. The table below shows how to compute the standard error for simple random samples assuming that the population size is at least 10 times larger than the sample size.
 <center>
 {| class="wikitable" style="text-align:center;width:25%"border="1"
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 </center>
-* Degrees of freedom: the number of independent pieces of information on which the estimate is based
+* ''Degrees of freedom'': the number of independent pieces of information on which the estimate is based
 : In general, the degrees of freedom for an estimate is equal to the number of values minus the number of parameters estimated to the estimate in question. Suppose we have sampled 20 data points then our estimate of the variance has 20 – 1 = 19 degree of freedom.

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 ===Motivation===
-To obtain a desired estimator, or estimation, we need to first determine a probability distribution with parameters of interest based on the data. After deciding the probabilistic model, we need to find the theoretically achievable precision available to any estimator based on the model and then develop an estimator based on this model. There are variety of methods and criteria to develop and choose between estimators based on their performance: maximum likelihood estimators, Bayes estimators, method of moments estimators, minimum mean square error estimators, minimum variance unbiased estimator, best linear unbiased estimator, etc. Experiment or simulations can also be run to test estimators’ performance.
+To obtain a desired estimator or estimation, we need to first determine a probability distribution with parameters of interest based on the data. After deciding the probabilistic model, we need to find the theoretically achievable precision available to any estimator based on the model and then develop an estimator based on this model. There is a variety of methods and criteria to develop and choose between estimators based on their performance:
 ===Theory===

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 ===Overview===
 Estimation is an important concept in the field of statistics and application of estimation is widely applied in various areas. It deals with estimating values of parameters of the population based on the sample data. And the parameters describe an underlying physical setting and their value would affect the distribution of the measured data. Two major approaches are commonly used in estimation:
-# the probabilistic approach assumes that the measured data is random with probability distribution dependent on the parameters;
+# The probabilistic approach assumes that the measured data is random with probability distribution dependent on the parameters.
-# the set-membership approach assumes that the measured data vector belongs to a set which depends on the parameter vector.
+# The set-membership approach assumes that the measured data vector belongs to a set which depends on the parameter vector.
-The purpose of estimation is to find an estimator that is interpretable, accurate and exhibits some form of optimality. Indicators like minimum variance unbiased estimator is usually applied to measure estimator optimality, although it is possible that an optimal estimator don’t always exist. Here we present the fundamentals of estimation theory and illustrate how to apply estimation in real studies.
+The purpose of estimation is to find an estimator that can be interpreted, which is accurate and which exhibits some form of optimality. Indicators like minimum variance unbiased estimator is usually applied to measure estimator optimality, although it is possible that an optimal estimator don’t always exist. Here we present the fundamentals of estimation theory and illustrate how to apply estimation in real studies.
 ===Motivation===

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 * Bias: refers to whether an estimator tends to either overestimate or underestimate the parameter. We say an estimator is biased if the mean of the sampling distribution of the statistic is not equal to the parameter. For example, $σ^{2}=\frac{(x-μ)^{2}} {N}$ is a biased estimator of the population variance and sample variance $s^{2}=\frac{(x-\bar{x})^{2}} {N-1}$ is unbiased estimate of the population variance.
-*Sampling variability: refers to how much the estimate varies from sample to sample. It is usually measured by its standard error: the smaller the standard error, the less the sampling variability. For example, the standard error of the mean is $σ_M=σ/√N$. So the larger the sample size $(N)$, the smaller the standard error of the mean, hence the smaller the sample variability.
+*Sampling variability: refers to how much the estimate varies from sample to sample. It is usually measured by its standard error: the smaller the standard error, the less the sampling variability. For example, the standard error of the mean is $σ_M=\frac{σ}{\sqrt{N}}$. So the larger the sample size $(N)$, the smaller the standard error of the mean, hence the smaller the sample variability.
 *Unbiased estimate: $\eta (X_{1},X_{2},…,X_{n})=E[\delta(X_{1},X_{2},…,X_{n})|T]$ then $\delta(X_{1},X_{2},…,X_{n} )$ is unbiased estimate for $g(\theta)$ and $T$ is a complete sufficient statistic for the family of densities.

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 ====Characteristics of Estimators====
-* Bias: refers to whether an estimator tends to either overestimate or underestimate the parameter. We say an estimator is biased if the mean of the sampling distribution of the statistic is not equal to the parameter. For example, $σ^{2}=\frac{(x-μ)^{2}} {N}$ is a biased estimator of the population variance and sample variance $s^{2}=\frac{(x-\overline x ̅ )^{2}} {N-1 }$ is unbiased estimate of the population variance.
+* Bias: refers to whether an estimator tends to either overestimate or underestimate the parameter. We say an estimator is biased if the mean of the sampling distribution of the statistic is not equal to the parameter. For example, $σ^{2}=\frac{(x-μ)^{2}} {N}$ is a biased estimator of the population variance and sample variance $s^{2}=\frac{(x-\bar{x})^{2}} {N-1}$ is unbiased estimate of the population variance.
 *Sampling variability: refers to how much the estimate varies from sample to sample. It is usually measured by its standard error: the smaller the standard error, the less the sampling variability. For example, the standard error of the mean is $σ_M=σ/√N$. So the larger the sample size $(N)$, the smaller the standard error of the mean, hence the smaller the sample variability.