Homework #1 for Stat 610

Due Thursday February 1, in class.

1. Read chapter 6 in Wasserman

2. Models:

 a. Translate into words the following models for a random sample
    of real-valued random variables

    i. scriptF_1 = { F: F is a cdf }
   ii. scriptF_2 = { F in scriptF_1: F(x) = int_{-infty}^x f(y) dy }
  iii. scriptF_3 = { F in scriptF_2: f(x) = f(2 mu - x) for some mu,
                     and for all real x <= mu }
   iv. scriptF_4 = { F in scriptF_2:  h(x) = -log[ f(x) ]  is convex }
    v. scriptF_5 = { F in scriptF_2:  
                f(x;mu,sigma) = (1/sigma) phi[ (x - mu)/sigma ],
		where phi(x) = density of standard normal distribution  }
    Also, show that scriptF_5 is contained in scriptF4, that scriptF_5
    is contained in scriptF_3. What is the relationship between scriptF_3
    and scriptF_4?

 b. A simple Markov chain model for a vector of binary random variables
    X_1, X_2, ..., X_n entails the marginal m(1) = P[X_1=1] and 
    conditionals of the form  
       P[X_{i+1} = x | X_1=x_1, ..., X_i=x_i ] = t(x_i, x)
    for transition probabilities t(,), where x and each x_i are in {0,1}. 
    Argue that the Markov chain model includes the model in which
    X_i's are a random sample of Bernoulli trials.

 c. The univariate logistic regression model may be useful when data are 
    ordered pairs (X_i, Y_i), mutually independent across i=1,2,...,n and when
    Y_i takes only two values. It specifies the conditional distribution of 
    Y_i given X_i=x as Bernoulli[ h(x) ], where
   
      log[ h(x)/{ 1- h(x) } ] =  alpha + beta x
     
    and where (alpha,beta) is a vector of real labels, and
    it leaves unspecified the marginal distribution of X_i.
    The univariate probit model is similar,  Y_i given X_i = x is
    Bernoulli[ g(x) ] where

      g(x) = int_{-infty}^{ a + b x } phi(u) du,

    and where (a,b) are real labels and phi is the standard normal
    density function.

    Describe the intersection of the probit and logit models?

3. Identifiability:
   
   a. The Rasch model is sometimes useful when data
   are arranged as an m x n matrix of Bernoulli trials (X_ij). 
   Specifically, the model asserts mutual independence of
   all the random variables, and further, with h_ij = P[ X_ij = 1 ],
   that

      log[ h_ij/{ 1- h_ij } ] =  mu + alpha_i + beta_j

   where theta = (mu, alpha_1,...,alpha_m, beta_1,...,beta_n) labels
   each distribution in the model, and is a vector of real values.
   Show that without contstraints on elements of theta, this
   parameterization is not identifiable.  Describe an identifiable
   parameterization and confirm this property. 

   b. Consider the mixture of two normals model described in class
      for the real-valued random variable X.  Densities in this
      model have the form

      f(x;theta) = p (1/sigma1) phi[ (x-mu1)/sigma1 ] + 
                     (1-p) (1/sigma2) phi[ (x-mu2)/sigma2 ]
                                
      where theta=(p, mu1, sigma1, mu2, sigma2 ) is a vector
      labeling the density.  To be a valid density, we must
      have p in (0,1) and sigma1,sigma2 > 0.   
      Show that without contstraints on elements of theta, this
      parameterization is not identifiable.  Describe an identifiable
      parameterization and confirm this property. 


4. Limits.
   Consider a random sample X_1, X_2,...,X_n from a normal
   distribution with mean mu and variance sigma^2.  The
   statistic

   T_n = (1/n) sum_{i=1}^n ( X_i -  bar X_n )^2

   estimates sigma^2. Show that T_n -->p sigma^2 as n--> infty