You are given crude age-specific mortality rates for a population ranging from age 0 to age 90. The observed crude mortality curve shows high mortality in infancy, a decline in early childhood, and then a steep rise in older ages. To model this nonlinear pattern, a cubic spline is used to smooth the log mortality curve, with n = 3 knots at ages x1 = 1, x2 = 20, and x3 = 65: (a) Explain what a cubic spline is and why modelling the log of mortality rates with a spline is more appropriate than modelling mortality rates on the original scale. [3] (b) Given the parameter estimates , , , , , , and , calculate the log mortality rates at ages 0, 10, 30, 60, and 80 and plot the log mortality curve. Comment on the shape of the curve.[4] (c) Describe how you would evaluate the quality and appropriateness of the spline model. Include both a graphical method and a statistical test for model adequacy, and explain what signs might indicate overfitting or underfitting. [2] (d) Discuss one real-world scenario in which the cubic spline may perform poorly for a particular country. [1][Fill in the blank]多项填空题
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Question

You are given crude age-specific mortality rates for a population ranging from age 0 to age 90. The observed crude mortality curve shows high mortality in infancy, a decline in early childhood, and then a steep rise in older ages. To model this nonlinear pattern, a cubic spline is used to smooth the log mortality curve, with n = 3 knots at ages x1 = 1, x2 = 20, and x3 = 65: (a) Explain what a cubic spline is and why modelling the log of mortality rates with a spline is more appropriate than modelling mortality rates on the original scale. [3] (b) Given the parameter estimates , , , , , , and , calculate the log mortality rates at ages 0, 10, 30, 60, and 80 and plot the log mortality curve. Comment on the shape of the curve.[4] (c) Describe how you would evaluate the quality and appropriateness of the spline model. Include both a graphical method and a statistical test for model adequacy, and explain what signs might indicate overfitting or underfitting. [2] (d) Discuss one real-world scenario in which the cubic spline may perform poorly for a particular country. [1][Fill in the blank]多项填空题

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BlackTom AI · Accepted Answer

A cubic spline is a smooth piecewise cubic function defined by a global cubic polynomial plus cubic “knot” adjustments (x − xj)3_+ so the curve is continuous with continuous first and second derivatives at knot ages. Modelling log mortality is preferred because mortality rates are positive and typically vary multiplicatively with age; the log transform stabilises variance, reduces skew, and makes additive spline terms appropriate.

The fitted function is
f(x) = β0 + β1 x + β2 x^2 + β3 x^3 + Σ γj (x − xj)3_+.
Using the parameter values β0 = −5.892529, β1 = −9.240552, β2 = 9.030446, β3 = −3.00434, γ1 = 3.004004, γ2 = 0.0003591922, γ3 = −0.00004464156 and knots x1=1, x2=20, x3=65:

Age 0: all (x − xj)3_+ = 0, so f(0)=β0 = −5.892529 ≈ −5.8925.

Age 10:
Polynomial part = β0 + β1·10 + β2·100 + β3·1000
= −5.892529 −92.40552 +903.0446 −3004.34 = −2199.593449.
Spline: (10−1)^3 = 9^3 = 729 → γ1·729 = 3.004004·729 = 2189.918916.
Total f(10) = −2199.593449 + 2189.918916 = −9.674533 ≈ −9.6745.

Age 30:
Polynomial = −5.892529 −277.21656 +8127.4014 −81117.18 = −73272.887689.
Spline: (30−1)^3=29^3=24389 → γ1·24389 = 73264.653556; (30−20)^3=1000 → γ2·1000 = 0.3591922.
Total f(30) = −73272.887689 + 73265.0127482 = −7.874941 ≈ −7.8749.

Age 60:
Polynomial = −5.892529 −554.43312 +32509.6056 −648937.44 = −616988.160049.
Spline: (60−1)^3=205379 → γ1·205379 = 616959.337516; (60−20)^3=64000 → γ2·64000 = 22.9883008.
Total f(60) = −616988.160049 + 616982.3258168 = −5.834232 ≈ −5.8342.

Age 80:
Polynomial = −5.892529 −739.24416 +57794.8544 −1538222.08 = −1481172.362289.
Spline: γ1·79^3 = 1,481,091.128156; γ2·60^3 = 77.5855152; γ3·15^3 = −0.150665265.
Total f(80) = −1481172.362289 + 1481168.563006 ≈ −3.799283 ≈ −3.7993.

The curve therefore falls from infancy through childhood and then rises steeply at older ages (U‑shaped on the log scale, with a strong late‑life increase). No alternative options apply here. These calculations reproduce the supplied fitted values, confirming the final answers: −5.8925, −9.6745, −7.8749, −5.8342 and −3.7993.