Estimation of Moments, Method of Moments and Maximum Likelihood Estimation
Statistical parameter estimation is widely used in Civil Engineering. Engineers estimate unknown parameters such as rainfall mean, soil strength distribution, traffic arrival rate, and material reliability using sample data.
Module 1: Estimation of Moments
A moment describes characteristics of a distribution such as mean and variance.
First moment about origin:
$$ \mu_1 = E(X) $$
Second moment about origin:
$$ \mu_2 = E(X^2) $$
Sample moment estimators:
$$ m_1 = \frac{1}{n}\sum x_i $$
$$ m_2 = \frac{1}{n}\sum x_i^2 $$
Worked Example 1 – Rainfall Analysis
Daily rainfall (mm): 20, 25, 30, 35
First moment:
$$ m_1 = \frac{20+25+30+35}{4} $$
$$ m_1 = 27.5 $$
Second moment:
$$ m_2 = \frac{400+625+900+1225}{4} $$
$$ m_2 = 787.5 $$
Worked Example 2 – Concrete Strength
Strength values (MPa): 28, 30, 32
$$ m_1 = \frac{28+30+32}{3} = 30 $$
$$ m_2 = \frac{784+900+1024}{3} $$
$$ m_2 = 902.67 $$
Worked Example 3 – Traffic Flow
Vehicles per minute: 10, 12, 8, 9
$$ m_1 = \frac{10+12+8+9}{4} $$
$$ m_1 = 9.75 $$
$$ m_2 = \frac{100+144+64+81}{4} $$
$$ m_2 = 97.25 $$
Worked Example 4 – Soil Density
Soil density (kN/m³): 18, 19, 20
$$ m_1 = \frac{18+19+20}{3} = 19 $$
$$ m_2 = \frac{324+361+400}{3} $$
$$ m_2 = 361.67 $$
Worked Example 5 – River Discharge
River discharge (m³/s): 150, 160, 170
$$ m_1 = \frac{150+160+170}{3} $$
$$ m_1 = 160 $$
$$ m_2 = \frac{22500+25600+28900}{3} $$
$$ m_2 = 25666.67 $$
Module 2: Method of Moments
The Method of Moments estimates parameters by equating sample moments with theoretical moments.
If population mean is:
$$ \mu = \theta $$
Set sample mean equal to theoretical mean:
$$ \bar{x} = \theta $$
Worked Example 1 – Estimating Mean Rainfall
Rainfall data: 40, 45, 50, 55
Sample moment:
$$ \bar{x} = \frac{40+45+50+55}{4} $$
$$ \bar{x} = 47.5 $$
Estimated rainfall parameter = 47.5 mm
Worked Example 2 – Soil Strength Parameter
Soil bearing capacity (kN/m²): 120, 130, 140
$$ \bar{x} = \frac{120+130+140}{3} $$
$$ \bar{x} = 130 $$
Estimated soil strength parameter = 130 kN/m²
Worked Example 3 – Traffic Arrival Rate
Vehicles per minute: 6, 7, 5, 8
$$ \bar{x} = \frac{6+7+5+8}{4} $$
$$ \bar{x} = 6.5 $$
Estimated arrival rate = 6.5 vehicles/min
Worked Example 4 – Bridge Load Distribution
Load measurements (kN): 100, 105, 110, 95
$$ \bar{x} = \frac{100+105+110+95}{4} $$
$$ \bar{x} = 102.5 $$
Estimated mean load = 102.5 kN
Worked Example 5 – Water Demand
Daily water demand (MLD): 200, 220, 210
$$ \bar{x} = \frac{200+220+210}{3} $$
$$ \bar{x} = 210 $$
Estimated demand parameter = 210 MLD
Module 3: Method of Maximum Likelihood (MLE)
The Maximum Likelihood Method estimates parameters by maximizing the likelihood function.
Likelihood function:
$$ L(\theta) = \prod f(x_i|\theta) $$
For many distributions the MLE of mean equals the sample mean.
$$ \hat{\mu}_{MLE} = \bar{x} $$
Worked Example 1 – Rainfall Mean Estimation
Rainfall data: 30, 35, 40
$$ \bar{x} = \frac{30+35+40}{3} $$
$$ \bar{x} = 35 $$
MLE estimate of mean rainfall = 35 mm
Worked Example 2 – Concrete Strength
Strength data (MPa): 25, 28, 30, 27
$$ \bar{x} = \frac{25+28+30+27}{4} $$
$$ \bar{x} = 27.5 $$
MLE estimate = 27.5 MPa
Worked Example 3 – Traffic Arrival Rate
Vehicles/minute: 4, 5, 6
$$ \bar{x} = \frac{4+5+6}{3} $$
$$ \bar{x} = 5 $$
Estimated arrival rate = 5 vehicles/min
Worked Example 4 – Soil Density
Soil density (kN/m³): 18, 19, 20, 19
$$ \bar{x} = \frac{18+19+20+19}{4} $$
$$ \bar{x} = 19 $$
MLE estimate = 19 kN/m³
Worked Example 5 – River Flow Estimation
River flow (m³/s): 100, 110, 120
$$ \bar{x} = \frac{100+110+120}{3} $$
$$ \bar{x} = 110 $$
Estimated mean river flow = 110 m³/s
Practice Questions
- Calculate the first moment for rainfall data: 10, 15, 20.
- Use Method of Moments to estimate parameter for data: 5, 7, 9.
- Find MLE estimate for mean of values: 12, 15, 18.
- Explain difference between Method of Moments and MLE.
- Give two civil engineering applications of parameter estimation.
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