Nonparametric Curve Estimation

Jaime Frade

Department of Statistics Florida State Univeristy

Victor Patrangenaru STA5334

December 12, 2008

Outline

1 Introduction

Objective

2 Theory

Definitions

Adapative Kernals

3 Financial Application

Financial Definitions Data of Model

4 Simulation Results
5 Conclusion

Introduction

Nonparametric Density Estimation

Probabilty density estimation goes hand in hand with nonparametric estimation of the cumulative distribution. The density function provides a better visual summary of how the random variable is distributed across its support. Skewness, kurtosis, disperseness are just a few properities can ascertiantined from the density plot.

Introduction Objective

\underline{Goal}

Unlike using empirical density function, which places probabilty mass \frac{1}{n} on each observation, the project focuses on the kernel density esimator that more fairly spreads out the probabilty mass of each observation, not arbitrarily in a fixed interval, but smoothly around the observation, typically in a symmetric way.

From there, will illustrate four basic kernals, as well as an adaptive kernal function, to estimate the distribution of returns on a certian asset. Simulation will be completed.

Theory Definitions

With a sample of X_{1}, X_{2}, …, X_{n}, write the density estimator

f̂(x) = ∑_{i=1}^{n}K (\frac{x − x_{i}}{h_{n}} )  (1)

for X_{i} = x_{i}, i = 1, …, n. The kernal function K represents how the probabilty mass is assigned. For example, for the histogram, in any particular interval, K is constant. The smoothing function h_{n} is a positive sequence of bandwidths analoguous to the bin width in a histogram.

From lecture notes, to estimate f , one may use the density of the random variable X̂ + hZ, where X̂ has the distribution (conditionally, given X_{1}, …, X_{n}) and Z is independent of X̂. From (1), the bandwidth satisfies

h ≡ h_{n} ⟶ as n → ∞  (2)

X̂ + hZ has the density.

f̂_{n}(x) = \frac{1}{n} ∑_{i=1}^{n}K_{h}(x − X_{i}),  (3)

where K_{h} is the density of hZ. From (3), using the following definition for K_{h} below, will obtain (1).

K_{h}(y) = \frac{1}{h}K (\frac{y}{h})  (4)

The kernal function K has five important properities K(x) ≥ 0 ∀ x K(x) = K(−x) ∀ x > 0 ∫ K(u)du = 1 ∫ uK(u)du = 0 ∫ u^{2}K(v)dv = σ_{k}^{2} < ∞ Basic idea is that K controls the shape, h_{n} controls the spread of the kernal. The accuracy of a density estimator can be evaluated using he mean intergrated squared error, defined as

MISE = 𝔼 (∫ (f (x) − f̂(x))^{2}dx)
= ∫ Bias^{2}(f̂(x))dx + ∫ 𝕍ar(f̂(x))dx  (5)

To find a density estimator that minimizes tehe MISE under the five constraints, also will assume that f (x) is continuous and twice differentiable, h_{n} → 0 and nh_{n} → ∞ as n → ∞. Under these conditions,

Bias = \frac{σ_{K}^{2}}{2} f′′(x) + O(h_{n}^{4})

Var (f̂(x)) = \frac{f (x)R(K)}{nh_{n}} + O(n^{−1})  (6)

where R(g) = ∫ g(u)^{2}du Determine the minimum MISE by the choice of h_{n}. However, choosing h_{n} to reduce the bias will increase the variance and vice versa, there is a tradeoff. The choice of the bandwidth is important to the construction of f̂(x). If h is chosen to be small, the minor difference in main part of the density will be apparent. If h is choose to be large, the tails of the distribution are better modeled, but fail to see important charateristics of the middle quartiles of the data.

By subsituting the bias and variance into the formula for (5), minimize MISE with

h_{n}^{∗} = (\frac{R(K)}{σ_{K}^{4}R(f′)})^{\frac{1}{5}} n^{− \frac{1}{5}}

From here, still can choose K(x) and insert a representative density for f (x) to solve for the bandwidth.

Epanechnickov (1969) showed that, upon subsituting f (x) = ϕ(x), the kernal that minimizes MISE is

K_{E}(x) = \{ \begin{matrix}−\frac{3}{4}(1 − x^{2}) & |x| ≤ 1 \\ 0 & if |x| > 1 \end{matrix}

The resulting bandwidth becomes h^{∗} ≈ 1.06σ̂n^{− \frac{1}{5}} , where σ̂ is the sample standard deviation. The choice relies on the approximation for σ for f (x), can obtain different answers.

Theory Adapative Kernals

Adapative Kernals were derived to alleviate this problem. If use mroe general smoothing function tied to the density at x_{j}, could generalize the denisty estimator as

f̂(x) = \frac{1}{n} ∑_{i=1}^{n} \frac{1}{h_{n,i}} K (\frac{x − x_{i}}{h_{n,i}} )  (7)

where h_{n} is a decreasing function of n, under |f̂(x) − f (x)| →^{P} 0 In a majority of practical applications the parameter of smoothness is unknown. After all, nonparametric estimation is typically a first glance at the data at hand. Thus the aim of one paper I researched is to discuss estimates that are adaptive to an unknown smoothness, i.e., data-driven. While the MISE of adaptive estimators can attain the minimax rate, that is, it is not necessary to know the parameter, (can visualize in data). See Efromovich (1999) pg 281.

Financial Application

Financial Application Financial Definitions

\underline{Bonds:} \underline{Default:}

Jaime Frade (Florida State Univeristy)  Nonparametric Curve Estimation  December 12, 2008  16 / 18

Financial Application Data of Model

Matlab Programs Simulation Results Conclusion