In this article, we delve into several real interview questions from these firms, providing detailed explanations and solutions for each. We cover fat tails in linear regression, estimation of uniform distribution bounds, and an advanced combinatorics problem inspired by grid-walking constraints. By understanding both the concepts and the reasoning behind the answers, you can sharpen your preparation for top quant interviews.

Quant Interview Experience from SIG and G-Research

1. Dealing with Fat Tails in Linear Regression (G-Research)

Understanding Linear Regression and Its Assumptions

Linear regression is a foundational technique in statistics and machine learning, used to model the relationship between a dependent variable \( y \) and one or more independent variables \( X \). The classic linear regression model is:

\[ y = X\beta + \epsilon \]

where \( \epsilon \) represents the error term, assumed to be normally distributed with mean zero and constant variance.

What Are Fat Tails?

In probability and statistics, a distribution is said to have "fat tails" if extreme values (outliers) occur more frequently than would be predicted by a normal distribution. In other words, the probability density function of the error term \( \epsilon \) decays more slowly than that of a Gaussian distribution.

Fat tails are common in financial data, where rare but impactful events (such as market crashes or spikes) are more likely than under the standard normal assumption.

Problems Caused by Fat Tails in Linear Regression

• Inference Problems: Standard errors, confidence intervals, and p-values may be unreliable because they rely on the assumption of normality.
• Sensitivity to Outliers: Ordinary Least Squares (OLS) regression minimizes the sum of squared residuals, which can be dominated by outliers associated with fat tails.
• Poor Predictive Performance: Model coefficients may be unstable and generalization may suffer.

How to Deal with Fat Tails in Linear Regression

There are several strategies to address fat tails in regression settings:

1. Use Robust Regression Techniques

• Least Absolute Deviations (LAD) Regression:

  Instead of minimizing the sum of squared errors, minimize the sum of absolute errors: \[ \min_{\beta} \sum_{i} |y_i - X_i\beta| \] This approach is less sensitive to large residuals and is robust to outliers.

• M-estimators:

  Generalize OLS by minimizing a function \(\rho(\cdot)\) of the residuals instead of the squared loss: \[ \min_{\beta} \sum_{i} \rho(y_i - X_i\beta) \] Common choices for \(\rho\) include the Huber loss, which is quadratic for small errors and linear for large errors.

2. Transform the Response Variable or Predictors

Applying transformations such as the logarithm, square root, or Box-Cox transformation to the response or predictors can sometimes reduce the effect of fat tails.

3. Use Heavy-Tailed Distributions for Errors

Explicitly model the error term with a heavy-tailed distribution, such as the Student-t distribution:

\[ \epsilon \sim t_{\nu}(0, \sigma^2) \]

where \(\nu\) is the degrees of freedom, controlling the tail heaviness. As \(\nu \to \infty\), the t-distribution approaches the normal distribution.

The likelihood function for the regression model becomes:

\[ L(\beta, \sigma, \nu) = \prod_{i=1}^n \text{Student-t}(y_i | X_i\beta, \sigma^2, \nu) \]

Maximum likelihood or Bayesian inference can be used to estimate parameters.

4. Winsorizing or Trimming the Data

Winsorizing replaces extreme values with less extreme ones, while trimming removes them entirely. This can reduce the influence of outliers, though it may discard valuable information.

5. Bayesian Regression with Heavy-Tailed Priors

In Bayesian settings, specify priors or likelihoods that are robust to outliers, such as using a Laplace or Student-t likelihood.

import pymc as pm
with pm.Model() as model:
    beta = pm.Normal('beta', mu=0, sigma=10)
    sigma = pm.HalfNormal('sigma', sigma=1)
    nu = pm.Exponential('nu', lam=1/30)
    y_obs = pm.StudentT('y_obs', mu=X.dot(beta), sigma=sigma, nu=nu, observed=y)
    trace = pm.sample()

Summary Table: Approaches to Fat Tails in Regression

2. Uniform Distribution Estimation Problem (G-Research)

Problem Statement

You have \( m \) random numbers taken uniformly in the range between 0 and \( n \). What is the best estimation of \( n \)?

Understanding the Problem

Suppose you observe \( m \) IID samples \( X_1, X_2, ..., X_m \) from the uniform distribution \( U(0, n) \). You wish to estimate the unknown upper bound \( n \).

Maximum Likelihood Estimation (MLE)

The probability density function for \( U(0, n) \) is:

\[ f(x; n) = \begin{cases} \frac{1}{n}, & 0 \leq x \leq n \\ 0, & \text{otherwise} \end{cases} \]

Given observed data \( x_1, ..., x_m \), the likelihood is:

\[ L(n) = \prod_{i=1}^m \frac{1}{n} = n^{-m} \]

But this is only valid if \( n \geq \max_i x_i \), otherwise the likelihood is zero. Thus,

\[ L(n) = \begin{cases} n^{-m}, & n \geq X_{(m)} \\ 0, & n < X_{(m)} \end{cases} \]

where \( X_{(m)} = \max_i x_i \).

Finding the Maximum Likelihood Estimate

To maximize \( L(n) \), we want the smallest possible \( n \) such that \( n \geq X_{(m)} \). Therefore, the MLE is:

\[ \boxed{n_{\text{MLE}} = X_{(m)}} \]

This makes intuitive sense: under the likelihood, any \( n < X_{(m)} \) is impossible, and for \( n \geq X_{(m)} \), larger \( n \) only decrease the likelihood.

Bias of the MLE

However, \( n_{\text{MLE}} \) is a biased estimator: if you take the maximum of \( m \) samples, it will, on average, be less than the true \( n \).

Expected Value of the Maximum

The cumulative distribution function (CDF) of the maximum is:

\[ P(X_{(m)} \leq x) = \left(\frac{x}{n}\right)^m, \quad 0 \leq x \leq n \]

The PDF of \( X_{(m)} \) is then:

\[ f_{X_{(m)}}(x) = \frac{d}{dx} \left(\frac{x}{n}\right)^m = \frac{m}{n^m} x^{m-1}, \quad 0 \leq x \leq n \]

The expected value is:

\[ E[X_{(m)}] = \int_0^n x f_{X_{(m)}}(x) dx = \int_0^n x \frac{m}{n^m} x^{m-1} dx = \frac{m}{n^m} \int_0^n x^m dx = \frac{m}{n^m} \cdot \frac{n^{m+1}}{m+1} = \frac{m}{m+1} n \]

So, on average, the MLE underestimates \( n \).

Unbiased Estimator

To correct for bias, you can use:

\[ n_{\text{unbiased}} = \frac{m+1}{m} X_{(m)} \]

This estimator is unbiased because:

\[ E\left[\frac{m+1}{m} X_{(m)}\right] = \frac{m+1}{m} E[X_{(m)}] = \frac{m+1}{m} \cdot \frac{m}{m+1} n = n \]

Therefore, the best unbiased estimator for \( n \) is:

\[ \boxed{n^* = \frac{m+1}{m} X_{(m)}} \]

where \( X_{(m)} \) is the maximum observed value.

Summary Table: Estimators for Uniform Maximum

Python Code Example

import numpy as np

def estimate_n(samples):
    m = len(samples)
    x_max = np.max(samples)
    n_mle = x_max
    n_unbiased = (m + 1) / m * x_max
    return n_mle, n_unbiased

# Example usage
samples = np.random.uniform(0, 10, size=5)
n_mle, n_unbiased = estimate_n(samples)
print("MLE:", n_mle)
print("Unbiased:", n_unbiased)

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