In all likelihood yudi pawitan pdf

Summer 2015 Instructor:

Problem Set 1

( Part 1 - due Sunday, June 14 )

Likelihood construction.

1. (Ex. 2.5, [2], p.49)

The following shows the heart rate (in beats/minute) of a person, measured throughout the day:

73 75 84 76 93 79 85 80 76 78 80

Assume the data are an iid sample from N(µ,σ2), where σ2 is known at the observed sample variance. Denote the ordered values by y(1), . . . ,y(11) . Draw and compare the likelihood of µ if

(a) the whole data y1, . . . ,y11 are reported.

(b) only the sample mean ȳ is reported.

(c) only the sample median y(6) is reported.

(d) only the minimum y(1) and maximum y(11) are reported.

(e) only the lowest two values y(1) and y(2) are reported

2. (Ex. 2.6, [2], p.49)

Given the following data

0.5 -0.32 -0.55 -0.76 -0.07 0.44 -0.48

draw the likelihood of θ based on each of the following models:

(a) The data are an iid sample from a uniform distribution on (θ − 1,θ + 1) .

(b) The data are an iid sample from a uniform distribution on (−θ,θ) .

(c) The data are an iid sample from N(0,θ) .

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Summer 2015 Instructor:

3. (Ex. 2.9, [1], pp.109-110)

The sample Y1, . . . ,Yn is iid with distribution function

FY (y; p0,p1,α,β) = p0 1{0≤y} + (1 −p0 −p1)F(y; α,β) + p1 1{y≥1} ,

where F(y; α,β) is the beta distribution. You may recall that the beta density is positive on 0 < y < 1 so that F(0; α,β) = 0 and F(1; α,β) = 1 , but otherwise you do not need to use or know its form in the following; just use F(y; α,β) or f(y; α,β) where needed. The physical situation relates to test scores standardized to lie in [0, 1] , but where n0 of the sample values are exactly 0 (turned in a blank test), n1 values are 1 (a perfect score), and the rest are in between 0 and 1. Use the 2h method to show that the likelihood is

pn00 p n1 1 (1 −p0 −p1)

n−n0−n1 ∏

0

f(Yi; α,β) .

4. (Ex. 2.11, [1], p.110)

Let Y1, . . . ,Yn be an iid sample, each with distribution function F . We make no restrictions on F . Show that the empirical distribution function is the nonparametric maximum likelihood estimator for the case that there are ties in the data. To fix notation, let nj be the number of sample values at yj , j = 1, . . . ,k,

∑k j=1 nj = n. Then start with the approximate likelihood∏k

j=1 p nj j,h , where pj,h = F(yj +h)−F(yj−h) , and use an argument

similar to that found in ([1], Section 2.2.6, p.45).

5. (Ex. 2.13, [1], p.110)

For the random censoring likelihood, ([1], pp.49-50), we expressed P(Yi ∈ (y − h,y + h],δi = 1) and P(Yi ∈ (y − h,y + h],δi = 0) as double integrals, (with Yi := min(Ti,Ri) and δi := 1{Ti≤Ri}), divided by 2h and took limits. Instead of double integrals, try to find the same expressions by

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Summer 2015 Instructor:

conditioning: e.g., start with

P(Yi ∈ (y −h,y + h],δi = 1) = P(Ti ∈ (y −h,y + h],Ti ≤ Ri) = P(Ti ≤ Ri|Ti ∈ (y −h,y + h]) ×P(Ti ∈ (y −h,y + h]) .

(Hint: bound P(Ti ≤ Ri|Ti ∈ (y−h,y+h]) above by P(y−h ≤ Ri) and below by P(y + h ≤ Ri) .)

6. (Ex. 2.3, [1], p.108)

Recall the zero-inflated Poisson (ZIP) model ([1], p.31)

P(Y = 0) = p + (1 −p) e−λ

P(Y = y) = (1 −p) λy e−λ

y! , y = 1, 2, . . .

(a) Reparameterize the model by defining

π := P(Y = 0) = p + (1 −p) e−λ . Solve for p in terms of λ and π, and then substitute so that the density depends only on λ and π.

(b) For an iid sample of size n, let n0 be the number of zeroes in the sample. Assuming that the complete data is available (no grouping), show that the likelihood factors into two pieces and that π̂ML = n0/n. This illustrates why we obtained exact fits for the 0 cell in ([1], Example 2.1, p.32). Also show that the maximum likelihood estimator for λ is the solution to a simple nonlinear equation involving Ȳ+ (the average of the nonzero values).

(b) Now consider the truncated or conditional sample consisting of the n − n0 nonzero values. Write down the conditional likeli- hood for these values and obtain the same equation for λ̂ML as in (a). (First write down the conditional density of Y given Y > 0 .)

References

[1] Dennis D. Boos & Leonard A. Stefanski (2013). Essential Statistical Inference, Springer Texts in Statistics, Springer.

[2] Yudi Pawitan (2001). In All Likelihood, Oxford University Press.