Estimation of Disaggregate Huff and Kaiyu Markov Model: A Lecture Note on Conditional Logit Model

We keep the mathematical level required for this chapter with high school mathematics as much as possible except for the calculus or differentiation and integration of exponential and logarithmic functions and some elementary concepts of probability and random variables.

The expectations of E(U_A) and E(U_B) are calculated as follows.See also Appendix 1, “Expectation” of this chapter for the definition of expectation for discrete and continuous random variables.

The cumulative distribution function F_X(x) for the random variable X is defined as follows F_X(x) = Prob(X ≤ x) for ∀ x ∈ R. See also Appendix 1, “Cumulative Distribution Function” for the definition and properties of the cumulative distribution function.

See Appendix 1, “Independence of Random Variables” for the definition of independence.

See Appendix 1, “Convolution” of this chapter.

The following relations hold. \( {f}_X(x)={\int}_{-\infty}^{+\infty }{f}_{XY}\left(x,s\right) ds \); \( {f}_Y(y)={\int}_{-\infty}^{+\infty }{f}_{XY}\left(t,y\right) dt \). See Appendix 1, “Joint and Marginal Distributions” for some fundamentals.

For a general background of Gumbel distribution, refer to, for example, [3].

For general information about Euler’s constant, see [4].

The problem to find the value of ζ(2) is called the Basel problem, which Euler solved in 1735. To see how Euler solved the problem, refer to, for example, Bonnar [5]. We give another derivation to find the value of ζ(2) based on the Fourier series in Appendix 2, “Fourier Series and the Value of ζ(2): Fourier Series solve the Basel Problem" (cf. Nishimori [6]).

Readers can skip this section without disturbing the understanding of the conditional logit model that follows.

For general information about Gamma function, refer to [7].

See Appendix 2’s Sections, “Proof of the Convergence of e: From Natural Number to Real Number” and “Proof of Related Inequalities of the Convergence of e: From Euler’s Number to Exponential Function” for the proof of the convergence of \( e=\underset{n\to \infty }{\lim }{\left(1+\frac{1}{n}\right)}^n \) and related properties such as \( \underset{n\to \infty }{\lim }{\left(1-\frac{t}{n}\right)}^n={e}^{-t} \).

Exactly speaking, this should be proved by showing that the interchange of the order of the limit and integral operations is possible. The interchangeability of the limit and integral operations in this case is proved by using the Lebesgue dominated convergence theorem. See the following Appendix 2’s Sections, “Lebesgue Convergence Theorem and Proof of Eq. (41): Interchange of Integration and Limit,” “Uniform Convergences of Digamma and Trigamma Functions: Derivatives of Logarithm of Gamma Function,” and “Measure, Lebesgue Integral, and Convergence Theorems: Proof of Lebesgue Dominated Convergence Theorem” for the related concepts, theorems, and proofs.

The sequence of functions f_n(x) defined on (a,b) is said to converge uniformly to the function f(x) on (a,b) if the following holds. For every ε > 0, there exists a positive number N not related to values x in (a,b) such that | f_n(x) − f(x) | < ε for every x in (a,b) and for all n > N.

See Appendix 2, “Uniform Convergences of Digamma and Trigamma Functions: Derivatives of Logarithm of Gamma Function” for the proof.

The characteristic function and the distribution of a random variable have one-to-one correspondence in a sense, which is guaranteed by Levy’s theorem. See Appendix 2’s Section, “Properties of Characteristic Function: Elements of Complex Analysis” for the theorem and its related properties. Also refer to Appendix 2’s Section, “Fourier’s Integral Theorem and Levy’s Inversion Formula: Fourier’s Transform of Distribution Function Gives Reformulated Levy’s Inversion Formula” for the discussion on the relation between Levy’s and Fourier’s inversion formula and the reformulation of Levy’s inversion formula.

The nth moment of a random variable X is defined as \( E\left({X}^n\right)={\int}_{-\infty}^{+\infty }{x}^n{f}_X(x) dx \) . See Appendix 1, “Expectation” of this chapter.

See Appendix 2’s Section, “Properties of Characteristic Function: Elements of Complex Analysis” for several properties of the characteristic function.

See Eq. (97) in Appendix 1’s Section, “Expectation.”

The second-order derivative of the logarithmic function of the Gamma function is called a trigamma function. The interchange of differentiation and limit in the fourth equality of Eq. (64) is permitted by the uniform convergence of the right-hand side of the last equation of Eq. (64). See Appendix 2’s Section, “Uniform Convergences of Digamma and Trigamma Functions: Derivatives of Logarithm of Gamma Function” for the proof.

Here we use the fact that \( \sum \limits_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6} \) without proof. We derive this fact in Appendix 2’s Section, “Fourier Series and the Value of ζ(2): Fourier Series solve the Basel Problem.”

See Appendix 1’s Section, “Newton-Raphson Method” for the outline of the Method.

See Appendix 1’s Section, “Newton-Raphson Algorithm to Find Maximum Likelihood Estimates of Conditional Logit Models: Derivation of Eq. (79).”

Python provides the optimization solver as SciPy package [14]. MatLab has the Optimization Toolbox [15]. SAS on Demand provides a free cloud computing environment that includes IML (Interactive Matrix Language) and many other statistical procedures, such as SAS Studio [16]. CVX is a convex optimization tool utilized with MatLab, and the recent CVXPY can be used with Python [17, 18]. Excel’s Solver can be used as an add-in for Excel, which is introduced on the Microsoft webpage [19].

See Appendix 1, “Python Code”. Scrutinizing the Python code reveals that the code only uses the objective function or the likelihood function in this case. Thus, the package is called a solver in the same way as the Excel solver.

See Appendix 1, “SAS Code”. As for the least square, when data are given as (X,Y), the least square estimate b is obtained by the solution formula, b=(X’X)^-1X’Y. Note that this SAS IML code uses this solution formula to get the least square estimate.

The definition of measurable function is given in Definition 11 below. To understand the definition of measurable function, we need some basic concepts of measure theory. Refer to Definitions 7–10 for these concepts.

We state Fubini’s theorem as Theorem 4 in the section of Appendix 2, “Properties of Characteristic Function: Elements of Complex Analysis”. Fubinit’s theorem specifies the condition for permitting the interchange of the order of integrals. In this case, the fact that the convergence of the integral to the characteristic function is uniform makes it possible to interchange the order of the integrals.

Refer to Appendix 2, “Proofs of \( {\int}_0^{\infty}\frac{\sin x}{x}=\frac{\pi }{2} \) Using Fourier’s Transform and Cauchy’s (Complex) Integral Theorem.”

Refer to Appendix 2’s Sections, “Lebesgue Convergence Theorem and Proof of Eq. (41): Interchange of Integration and Limit” and “Measure, Lebesgue Integral, and Convergence Theorems: Proof of Lebesgue Dominated Convergence Theorem” for the use and the proof of Lebesgue dominated convergence theorem.

In Appendix 2’s Section, “Fourier’s Integral Theorem and Levy’s Inversion Formula: Fourier’s Transform of Distribution Function Gives Reformulated Levy’s Inversion Formula,” Footnote 36 gives a short explanation about the Lebesgue-Stieltjes integral.

The function f(x) defined on interval (a, b) is said piecewise continuous if its discontinuity points are at most finite, and for each discontinuity point x₀, there exist the right limit \( f\left({x}_0+0\right)=\underset{h\to +0}{\lim }f\left({x}_0+h\right) \) and the left limit \( f\left({x}_0-0\right)=\underset{h\to +0}{\lim }f\left({x}_0-h\right) \), and at the endpoints a and b, f(a + 0) and f(b − 0) exist.

The inequality |e^iz − 1| ≤ |z| is derived from the following fact. Considering a circle with a radius of 1 centered on the origin in the complex plane, the length of the chord e^iθ − 1 is shorter than the length of the arc θ.

The function f is said absolutely integrable if it satisfies ∫ ∣ f ∣ dμ < ∞.

To see this, from Euler’s formula, e^iθ =  cos θ + i sin θ, we have

The Lebesgue-Stieltjes integral is defined in the following way. Consider a monotonically increasing function g on ℝ with finite discontinuity points where g has the right and the left limits. Define a function μ of the interval w in ℝ as follows. For an open interval w = (x₁, x₂), μ(w) = g(x₂ − 0) − g(x₁ + 0) and for a point w = [x], μ(w) = g(x + 0) − g(x − 0). While μ is finitely additive, it can be extended to a countably additive measure on Borel field, i.e., the minimum σ algebra constructed by all intervals in ℝ. In general, considering a measure space (X, ℬ, μ), the Lebesgue integral is formulated by first defining the integral of a simple (nonnegative measurable) function \( \varphi ={\sum}_{j=1}^J{\alpha}_j{1}_{A_j}\ \left({\alpha}_j>0,{A}_j\in \mathrm{\mathcal{B}}\right) \) from X to ℝ as \( {\int}_X\varphi d\mu ={\sum}_{j=1}^J{\alpha}_i\mu \left({A}_j\right) \) and next extending this to an arbitrary measurable fuction by approximating the function by the infinite sequence of simple functions. Now, consider the measure space (X, ℬ, μ) where X is ℝ, ℬ is the Borel field, and the measure μ is the above measure. The Lebesgue-Stieltjes integral is defined as the Lebesgue integral applied to the above measure constructed by the monotonically increasing function g and denoted by ∫_Xφdμ = ∫_Xφ(x)dg(x) (cf. Takagi [39], pp. 443–445). As for \( {\int}_{-\infty}^{+\infty }D\left(x,u\right)d{\Phi}_X(u) \), the integrand D(x, u) is a simple function φ which takes the value −1/2 on (x, +∞), 0 at x, and 1/2 on (−∞, x). Thus the Lebesgue-Stieltjes integral becomes

The same formula as the second equality of Eq. (174) was given by Gil-Pelaez [55] without referring to the Fourier transform. (Also refer to Wendel [53].)