Statistics: Parameter Estimation - Deepstash

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Used in Statistical Models

Used in Statistical Models

Parameters of a probability distribution, such as the mean and standard deviation of a normal distribution

Regression coefficients of a regression model, such as Y=a'X


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Used in Dynamic Models

Used in Dynamic Models

Engineers apply parameter estimation to dynamic models to compute:

Coefficients of transfer functions, including ARX, ARMAX, Box-Jenkins, and output-error models

Entries in state-space matrices

Coefficients of ODEs or well-structured systems with parameter constraints (grey-box system identification).

Regression coefficients, saturation levels, or dead-zone limits for nonlinear dynamic systems, including nonlinear ARX and Hammerstein-Wi.ener


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  • Point estimate 
  • Confidence interval (CI) estimate.

For both continuous variables (e.g., population mean) and dichotomous variables (e.g., population proportion) one first computes the point estimate from a sample.


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Point Estimation

Point Estimation

The process of finding an approximate value of some parameter—such as the mean (average)—of a population from random samples of the population.

It is the value of statistic that estimates the value of a parameter.

The sample standard deviation (s) is a point estimate of the population standard deviation (σ).

The sample mean (̄x) is a point estimate of the population mean, μ.

The sample variance (s2) is a point estimate of the population variance (σ2).


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Confidence Interval

Confidence Interval

Range of values you expect your estimate to fall between if you redo your test, within certain level of confidence.

Confidence-Describes - Probability

Point Estimate  +- Margin of error 

Given : ( α = 0.05 , n ,x̄ , Population standard deviation σ )  


  1. z-test (Population standard is given & n>=30)
  2. t-test (Population standard is not given & n


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Z-Test - z score

Z-Test - z score

Population standard is given & n >= 30.

Point Estimate +- Margin of error 

x̄  +-  z  α/2 (σ/ √ n )

Upper Bound -     x̄ + z  α/2 (σ/ √ n ) = ztable(Answer)

Lower Bound -     x̄ z  α/2 (σ/ √ n )=ztable(Answer)


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T -test - t-table

T -test - t-table

Population standard deviation is not  given & n >≠ 30.

Point Estimate +- Margin of error 

x̄ +- z  α/2 (σ/ √ n )

t = degree of freedom= n-1

Upper Bound -   x̄ + t  α/2 (s/ √ n ) = t table(Answer)

Lower Bound -   x̄  -  t  α/2 (s / √ n )=t table(Answer)

 (s / √ n ) is Standard error



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Standard Error

Standard Error

The standard error is the standard deviation of a sample population. It measures the accuracy with which a sample represents a population.


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Degree of freedom

Degree of freedom

Degrees of freedom, often represented by v or df, is the number of independent pieces of information used to calculate a statistic.

It’s calculated as the sample size minus the number of restrictions.


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Suppose you randomly sample 10 American adults and measure their daily calcium intake. You use a one-sample t test to determine whether the mean daily intake of American adults is equal to the recommended amount of 1000 mg.The test statistic, t, has 9 degrees of freedom:

df = n − 1

df = 10 − 1

df = 9

You calculate a t value of 1.41 for the sample, which corresponds to a p value of .19. You report your results:

“The participants’ mean daily calcium intake did not differ from the recommended amount of 1000 mg, t(9) = 1.41, p = 0.19.”


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Parameter estimation is the process of computing a model’s parameter values from measured data. You can apply parameter estimation to different types of mathematical models, including statistical models, parametric dynamic models, and data-based Simulink® models


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