Fitting a distribution

Before estimating anything you have to commit to a model. These sections fit the two skewed models the assignments used — the exponential and the lognormal — estimate their parameters by maximum likelihood, and check the fit with quantiles.

The assignment datasets are not reproduced here; each lab starts with a simulated sample of the same size and shape. Paste any data to analyse it, and compare the 2020 results in the margin.

§2.1Coursework A1 Q1

An exponential model and its QQ plot

Thirty plant heights, measured after six weeks, were strongly right-skewed. For an exponential model with mean θ the log-likelihood $\ell(\theta)=-n\log\theta-\sum x_i/\theta$ is maximised at the sample mean, $\hat\theta=\bar x$ .

To judge the fit, the notebook compared Type 6 sample quantiles, which interpolate the order statistics at position $(n+1)p$ , with the model quantiles $\;\pi_p=-\theta\log(1-p)$ . Against Exp(1) the same points fall on a line of slope 1/θ instead of 1, so either reference works.

Observations

30 positive values

Hypothesised mean θ₀

Compare against

Median

69.6

IQR 105.4

Mean

110.6

range 0.1 – 586.8

MLE θ̂ = x̄

110.59

Reference line

slope 1

y = x

Figure 2.1.Box plot of the sample (R’s Tukey hinges and 1.5 × IQR whiskers). A long right whisker and high outliers are the signature of a skewed, exponential-like shape — not a normal one.

Type 6 sample quantile vs model quantile
Reference line, slope 1

Figure 2.2.QQ plot at p = 0, 0.01, …, 0.99, laid out as in the notebook: sample quantiles across, theoretical quantiles up. If the data came from Exp(θ₀), the points hug a line through the origin with slope 1.

Show the numbers as a table

p	sample (type 6)	theoretical
0.00	0.100	0.0000
0.01	0.100	1.0050
0.02	0.100	2.0203
0.03	0.100	3.0459
0.04	1.972	4.0822
0.05	4.390	5.1293
0.06	6.808	6.1875
0.07	8.512	7.2571
0.08	9.628	8.3382
0.09	10.744	9.4311
0.10	11.890	10.5361
0.11	13.099	11.6534
0.12	14.308	12.7833
0.13	15.406	13.9262
0.14	15.468	15.0823
0.15	15.530	16.2519
0.16	15.592	17.4353
0.17	17.787	18.6330
0.18	20.298	19.8451
0.19	22.809	21.0721
0.20	23.760	22.3144
0.21	23.853	23.5722
0.22	23.946	24.8461
0.23	24.130	26.1365
0.24	24.440	27.4437
0.25	24.750	28.7682
0.26	25.264	30.1105
0.27	26.628	31.4711
0.28	27.992	32.8504
0.29	29.356	34.2490
0.30	31.950	35.6675
0.31	34.585	37.1064
0.32	37.220	38.5662
0.33	40.407	40.0478
0.34	43.786	41.5515
0.35	47.165	43.0783
0.36	48.928	44.6287
0.37	49.176	46.2035
0.38	49.424	47.8036
0.39	49.897	49.4296
0.40	50.920	51.0826
0.41	51.943	52.7633
0.42	53.116	54.4727
0.43	56.464	56.2119
0.44	59.812	57.9818
0.45	63.160	59.7837
0.46	64.090	61.6186
0.47	64.555	63.4878
0.48	65.020	65.3926
0.49	66.872	67.3345
0.50	69.600	69.3147
0.51	72.328	71.3350
0.52	74.204	73.3969
0.53	74.731	75.5023
0.54	75.258	77.6529
0.55	77.305	79.8508
0.56	87.256	82.0981
0.57	97.207	84.3970
0.58	107.158	86.7501
0.59	108.844	89.1598
0.60	109.960	91.6291
0.61	111.076	94.1609
0.62	112.456	96.7584
0.63	113.944	99.4252
0.64	115.432	102.1651
0.65	117.760	104.9822
0.66	120.984	107.8810
0.67	124.208	110.8663
0.68	126.632	113.9434
0.69	126.756	117.1183
0.70	126.880	120.3973
0.71	127.060	123.7874
0.72	128.920	127.2966
0.73	130.780	130.9333
0.74	132.640	134.7074
0.75	134.350	138.6294
0.76	136.024	142.7116
0.77	137.698	146.9676
0.78	139.048	151.4128
0.79	140.164	156.0648
0.80	141.280	160.9438
0.81	147.951	166.0731
0.82	164.722	171.4798
0.83	181.493	177.1957
0.84	197.148	183.2581
0.85	205.270	189.7120
0.86	213.392	196.6113
0.87	221.514	204.0221
0.88	225.156	212.0264
0.89	228.318	220.7275
0.90	231.480	230.2585
0.91	279.687	240.7946
0.92	349.344	252.5729
0.93	419.001	265.9260
0.94	475.344	281.3411
0.95	515.520	299.5732
0.96	555.696	321.8876
0.97	586.800	350.6558
0.98	586.800	391.2023
0.99	586.800	460.5170

§2.2Coursework A1 Q2

A lognormal model and an interval for its variance

If $\ln X\sim N(\mu,\lambda)$ then the MLEs are the mean and the divide-by-n variance of the logged data. Because $\sum(\ln X_i-\overline{\ln X})^2/\lambda\sim\chi^2_{n-1}$ , the 2020 working derived

\operatorname{sd}(\hat\lambda)=\lambda\sqrt{\tfrac{2}{n-1}},

\left(\frac{(n-1)\hat\lambda}{\chi^2_{n-1,\,1-\alpha/2}},\ \frac{(n-1)\hat\lambda}{\chi^2_{n-1,\,\alpha/2}}\right)

and the port implements those formulas exactly as they were submitted.

Observations (positive)

15 values

Confidence level95%

μ̂ = mean of ln x

1.8519

λ̂ (MLE)

1.7865

se(λ̂) = λ̂√(2/(n−1))

0.675245

95% CI for λ, as submitted

(0.9576, 4.444)

textbook SS/χ²: (1.026, 4.761)

ln xᵢ
Fitted N(μ̂, λ̂) density
95% interval for the spread, as ±√λ around μ̂

Figure 2.3.On the log scale a lognormal sample is just a normal sample, so and are the mean and the (divide-by-n) variance of ln x. The red bars show how wide ±√λ could plausibly be, from the lower to the upper end of the chi-squared interval.

§2.3Coursework A3 Q2

How precise is a sample quantile?

For the exponential, $F(x)=1-e^{-\lambda x}$ inverts to $\pi_p=-\log(1-p)/\lambda$ . The Type 7 sample quantile interpolates the order statistics at position $k=1+(n-1)p$ , and for large n

\hat\pi_p \approx N\!\left(\pi_p,\ \frac{p(1-p)}{n\,f(\pi_p)^2}\right),

f(\pi_p)=\lambda(1-p).

With n = 30 and p = 0.25 the variance is $1/(90\lambda^2)$ , so the standard error is $1/(\sqrt{90}\,\hat\lambda)$ .

Observations

30 values

Probability p0.25

π̂ (Type 7), p = 0.25

4.830

k = 1 + (n−1)p = 8.25

λ̂ = 1 / x̄

0.1336

Model πₚ = −ln(1−p)/λ̂

2.153

Asymptotic se

0.789

≈95% CI (3.28, 6.38)

Empirical CDF
Fitted Exp(λ̂) CDF
±1.96 se around π̂

Figure 2.4.The sample quantile is where the empirical CDF crosses height p (with Type 7 interpolation between order statistics); the model quantile is where the fitted exponential CDF does. The shaded band is the large-sample interval ± 1.96 se, with se² = p(1−p) / (n f(πₚ)²) and f(πₚ) = λ(1 − p).