Assignment #2 STA355H1F
due Monday, October 26, 2015
Problems to hand in:
1. On Blackboard, there is a file containing data on the lengths (in minutes) of 272 eruptions
of the Old Faithful geyser in Yellowstone National Park. Using R and some of the metho

Estimation of the mode
Mean, median . . . mode?
In an elementary statistics course, the mean, median, and mode are often given as descriptive
measures of the centre of the data. Given data x1 , , xn , the sample mode is defined to
be the value or set of v

Likelihood and Bayesian estimation of a population size
Mark/recapture experiments
Mark/recapture (or capture/recapture) experiments are often used to estimate the (unknown) size
of a population. Although originally developed in ecology to measure animal

Bayesian prediction for the Poisson distribution
Introduction
Suppose that (X1 , , Xn , Xn+1 , , Xn+m ) are random variables with probability density or mass
function f (x1 , , xn , xn+1 , , xn+m ; ) for some unknown parameter . Here we will assume that
w

Resampling Data:
Using a Statistical Jackknife
S. Sawyer Washington University March 11, 2005
1. Why Resample? Suppose that we want to estimate a parameter that
depends on a random quantity sample X = (X1 , X2 , . . . , Xn ) in a complicated way. For exam

Likelihood estimation for a non-regular model
The model
Suppose that X1 , , Xn are independent random variables with density
f (x; ) =
|x |1/2
exp(|x |)
2
(1)
where is an unknown real-valued parameter. Given X1 = x1 , , Xn = xn , the loglikelihood functi

Analysis of a two state Markov chain model
Markov chain model
A Markov chain is a stochastic process cfw_Xi where the conditional distribution of Xi+1 given
the past Xi , Xi1 , Xi2 , depends only on Xi . The range of the possible values in the
Markov cha

The Jackknife
Introduction
The jackknife provides a general-purpose approach to estimating the bias and variance (or standard
error) of an estimator. Suppose that b is an estimator of based on independent random variables
X1 , , Xn from an unknown distrib

increases and the variance decreases. In addition, the bias depends on the unknown density,
which makes the choice of the bandwidth more complicated.
(c) Nonparametric regression using kernel (weighted average) smoothing: Similar issues as
for kernel dens

SOLUTIONS: 2013 FINAL EXAM STA355H1S
1. (a) The log-likelihood function is
ln(/\) : nanl) i Aggy,-
and its rst two derivatives are
n 71
$1115) ingfi
d2 72
mm) = v
Setting the rst derivative to O, we obtain in 2 1/? and from the second derivative it
follow

Assignment #1 STA355H1F
due Wednesay, October 5, 2016
Problems to hand in:
1. Suppose that X1 , , Xn are independent Exponential random variables with density
f (x; ) =
exp(
x) for x
0
where > 0 is an unknown parameter.
1
(a) Show that the quantile of the

STA355H1F
Theory of Statistical Practice
Instructor: K. Knight (office: Sidney Smith 5016G; e-mail: keith@utstat.utoronto.ca)
My office is at the west end of the 5th floor.
Office hours: Fridays 10 to noon, or by appointment. Do not hesitate to contact me

Assignment #3 STA355H1F
due Monday November 21, 2016
Problems to hand in:
1. In genetics, the Hardy-Weinberg equilibrium model characterizes the distributions of
genotype frequencies in populations that are not evolving and is a fundamental model of
popul

Goodness of fit using quantile-quantile plots
Normal quantile-quantile plots
Suppose that X1 , , Xn are independent random variables with distribution function F . Given
data x1 , , xn , we would like to assess whether or not F can be taken be a normal di

Distribution-free confidence intervals for quantiles
The Binomial pivot
Suppose that X1 , , Xn are independent random variables with distribution function F and
we want to construct a condence interval for the quantile of a distribution function F ,
that

Some useful results from probability theory
STA355H1S
1
Modes of Convergence
Note: In this section (and subsequently), all random variables are real-valued unless otherwise specified.
p
Convergence in probability. cfw_Xn converges in probability to X (Xn

Assignment #3 STA355H1F
due Wednesday November 25, 2015
Problems to hand in:
1. In genetics, the Hardy-Weinberg equilibrium model characterizes the distributions of
genotype frequencies in populations that are not evolving and is a fundamental model of
po

Assignment #4 STA355H1F
due Monday December 7, 2015
Problems to hand in:
1. In class, we discussed a general class of estimators of g(x) in the non-parametric regression
model
Yi = g(xi ) + i for i = 1, , n.
of the form
gb(x) =
n
X
wi (x)Yi
i=1
where we r

Assignment #1 STA355H1F
due Wednesday, October 7, 2015
Problems to hand in:
1. Suppose that X1 , , Xn are independent Exponential random variables with density
f (x; ) = exp(x) for x 0
where > 0 is an unknown parameter.
(a) Show that the quantile of the E

Assignment #1 STA355H1F
due Wednesay, October 5, 2016
Problems to hand in:
1. Suppose that X1 , , Xn are independent Exponential random variables with density
f (x; ) = exp(x) for x 0
where > 0 is an unknown parameter.
(a) Show that the quantile of the Ex

Assignment #3 STA355H1S
due Friday March 21, 2014
1. Suppose that F is a continuous distribution with density f . The mode of the distribution
is defined to be the maximizer of the density function. In some applications, the centre of the
distribution is

Assignment #2 STA355H1S
due Friday February 14, 2014
1. As mentioned in lecture, the jackknife variance estimator fails as an variance estimator for
sample quantiles. To get a sense of why, consider the sample median when the sample size is even,
i.e. n =

Density estimation
Introduction
Suppose we observe independent random variables X1 , , Xn , which we believe are a random sample from some distribution function F whose form is unknown. Previously, we
dened a simple estimator of F , namely the empirical d

Tips for computing the posterior distribution
The posterior density for is
(|x1 , , xm ) = k(x1 , , xm )()L()
where
Z
k(x1 , , xm ) =
()L() d
0
1
.
However, the integral defining k(x1 , , xm ) may be difficult to evaluate as L() will be
very small for cer

The Newton-Raphson algorithm: Computing the MLE
of the Cauchy distribution
The Newton-Raphson algorithm
The Newton-Raphson algorithm is a general purpose method for solving equations of the
form g(x ) = 0 where g(x) is a (non-linear) differentiable functi

Spacings
From order statistics to spacings
Suppose that X1 , , Xn are independent random variables with a continuous distribution function
F (x) = P (Xi x) and density function f (x). We can also define the order statistics X(1) X(2)
X(n) , which are si

Demonstration of the Central Limit Theorem for sums of
independent but not identically distributed random variables
Introduction
Suppose that X1 , , Xn are independent random variables and S = X1 + + Xn . In
certain cases (for example, when cfw_Xi have n

Computing a posterior distribution
Introduction
Given a prior density (), the posterior density of is
(|x1 , , xn ) = k(x1 , , xn )()L()
where L() is the likelihood function
k(x1 , , xn ) =
1
Z
.
()L() d
Typically, we must evaluate k(x1 , , xn ) numerical