Estimator Bias, Consistency, Efficiency, and Robustness

/**
 * Fintech Math Bootcamp · Lesson 073 of 120
 * Estimator Bias, Consistency, Efficiency, and Robustness
 * Module 08: Sampling, Estimation, and Statistical Inference
 *
 * Scenario: Comparing estimators on error rather than a fashionable label
 * Rule:     MSE = variance + bias²
 *
 * Try it:   Does shifting every estimate by one change its variance?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/sampling-estimation-and-statistical-inference/estimator-bias-consistency-efficiency-and-robustness/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

export function lesson073() {
  const truth=10, estimates=[9,10,11];
  const avg=(a:number[])=>a.reduce((s,x)=>s+x,0)/a.length;
  const mean=avg(estimates), bias=mean-truth;
  const variance=avg(estimates.map(x=>(x-mean)**2));
  const mse=avg(estimates.map(x=>(x-truth)**2));
  const result={bias,variance,mse};
  return result;
}

export const checkedResult = {"bias":0,"variance":0.6666666666666666,"mse":0.6666666666666666};

// Run this file directly: npx tsx lessons/08-sampling-estimation-and-statistical-inference/073-estimator-bias-consistency-efficiency-and-robustness.ts
if (process.argv[1] && import.meta.url.endsWith(process.argv[1].replace(/\\/g, "/").split("/").pop()!)) {
  console.log(JSON.stringify(lesson073(), null, 2));
}

"""
Fintech Math Bootcamp · Lesson 073 of 120
Estimator Bias, Consistency, Efficiency, and Robustness
Module 08: Sampling, Estimation, and Statistical Inference

Scenario: Comparing estimators on error rather than a fashionable label
Rule:     MSE = variance + bias²

Try it:   Does shifting every estimate by one change its variance?

Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/sampling-estimation-and-statistical-inference/estimator-bias-consistency-efficiency-and-robustness/
Free course:    https://courses.thefintechbuilder.com
Synthetic teaching example, not financial advice or a production library.
"""

import json


def average(values):
    total = 0
    for x in values:
        total += x
    return total / len(values)


def lesson_073():
    truth, estimates = 10, [9, 10, 11]
    mean = average(estimates)
    bias = mean - truth
    variance = average([(x - mean) ** 2 for x in estimates])
    mse = average([(x - truth) ** 2 for x in estimates])
    return {"bias": bias, "variance": variance, "mse": mse}


if __name__ == "__main__":
    print(json.dumps(lesson_073(), indent=2))

// Fintech Math Bootcamp - Lesson 073 of 120
// Estimator Bias, Consistency, Efficiency, and Robustness
// Module 08: Sampling, Estimation, and Statistical Inference
//
// Scenario: Comparing estimators on error rather than a fashionable label
// Rule:     MSE = variance + bias^2
//
// Try it:   Does shifting every estimate by one change its variance?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/sampling-estimation-and-statistical-inference/estimator-bias-consistency-efficiency-and-robustness/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
import java.util.ArrayList;
import java.util.Arrays;
import java.util.LinkedHashMap;
import java.util.List;
import java.util.Map;

public class Main {
    static double average(double[] values) {
        double total = 0;
        for (double x : values) total += x;
        return total / values.length;
    }

    // (x - center)^2 for each value
    static double[] squaredDeviations(double[] values, double center) {
        double[] out = new double[values.length];
        for (int i = 0; i < values.length; i++) {
            out[i] = Math.pow(values[i] - center, 2);
        }
        return out;
    }

    static Map<String, Object> lesson073() {
        double truth = 10;
        double[] estimates = {9, 10, 11};
        double mean = average(estimates), bias = mean - truth;
        Map<String, Object> result = new LinkedHashMap<String, Object>();
        result.put("bias", bias);
        result.put("variance", average(squaredDeviations(estimates, mean)));
        result.put("mse", average(squaredDeviations(estimates, truth)));
        return result;
    }

    public static void main(String[] args) {
        System.out.println(toJson(lesson073(), ""));
    }

    // Minimal JSON writer: two-space indent, whole numbers without a decimal point, NaN as null.
    static String toJson(Object value, String indent) {
        if (value == null) return "null";
        if (value instanceof Boolean) return value.toString();
        if (value instanceof Number) return formatNumber(((Number) value).doubleValue());
        if (value instanceof String) return quote((String) value);
        if (value instanceof double[]) {
            List<Object> boxed = new ArrayList<Object>();
            for (double d : (double[]) value) boxed.add(d);
            return toJson(boxed, indent);
        }
        if (value instanceof Object[]) return toJson(Arrays.asList((Object[]) value), indent);
        String inner = indent + "  ";
        StringBuilder out = new StringBuilder();
        if (value instanceof Map) {
            Map<?, ?> map = (Map<?, ?>) value;
            if (map.isEmpty()) return "{}";
            out.append("{\n");
            int i = 0;
            for (Map.Entry<?, ?> entry : map.entrySet()) {
                out.append(inner).append(quote(entry.getKey().toString())).append(": ")
                   .append(toJson(entry.getValue(), inner));
                out.append(++i < map.size() ? ",\n" : "\n");
            }
            return out.append(indent).append("}").toString();
        }
        List<?> list = (List<?>) value;
        if (list.isEmpty()) return "[]";
        out.append("[\n");
        for (int i = 0; i < list.size(); i++) {
            out.append(inner).append(toJson(list.get(i), inner));
            out.append(i + 1 < list.size() ? ",\n" : "\n");
        }
        return out.append(indent).append("]").toString();
    }

    static String formatNumber(double x) {
        if (Double.isNaN(x) || Double.isInfinite(x)) return "null";
        if (x == Math.rint(x) && Math.abs(x) < 1e15) return Long.toString((long) x);
        return Double.toString(x);
    }

    static String quote(String s) {
        StringBuilder out = new StringBuilder("\"");
        for (char c : s.toCharArray()) {
            if (c == '"' || c == '\\') out.append('\\').append(c);
            else if (c == '\n') out.append("\\n");
            else if (c < 0x20) out.append(String.format("\\u%04x", (int) c));
            else out.append(c);
        }
        return out.append('"').toString();
    }
}

// Fintech Math Bootcamp · Lesson 073 of 120
// Estimator Bias, Consistency, Efficiency, and Robustness
// Module 08: Sampling, Estimation, and Statistical Inference
//
// Scenario: Comparing estimators on error rather than a fashionable label
// Rule:     MSE = variance + bias²
//
// Try it:   Does shifting every estimate by one change its variance?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/sampling-estimation-and-statistical-inference/estimator-bias-consistency-efficiency-and-robustness/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
package main

import (
	"encoding/json"
	"fmt"
	"math"
)

type Lesson073Result struct {
	Bias     float64 `json:"bias"`
	Variance float64 `json:"variance"`
	MSE      float64 `json:"mse"`
}

// average returns the arithmetic mean, summed left to right.
func average(values []float64) float64 {
	total := 0.0
	for _, x := range values {
		total += x
	}
	return total / float64(len(values))
}

// squaredDeviations returns (x - center)^2 for each value.
func squaredDeviations(values []float64, center float64) []float64 {
	out := make([]float64, len(values))
	for i, x := range values {
		out[i] = math.Pow(x-center, 2)
	}
	return out
}

func lesson073() Lesson073Result {
	truth := 10.0
	estimates := []float64{9, 10, 11}
	mean := average(estimates)
	return Lesson073Result{
		Bias:     mean - truth,
		Variance: average(squaredDeviations(estimates, mean)),
		MSE:      average(squaredDeviations(estimates, truth)),
	}
}

func main() {
	out, err := json.MarshalIndent(lesson073(), "", "  ")
	if err != nil {
		panic(err)
	}
	fmt.Println(string(out))
}

// Fintech Math Bootcamp · Lesson 073 of 120
// Estimator Bias, Consistency, Efficiency, and Robustness
// Module 08: Sampling, Estimation, and Statistical Inference
//
// Scenario: Comparing estimators on error rather than a fashionable label
// Rule:     MSE = variance + bias²
//
// Try it:   Does shifting every estimate by one change its variance?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/sampling-estimation-and-statistical-inference/estimator-bias-consistency-efficiency-and-robustness/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <iostream>
#include <optional>
#include <stdexcept>
#include <string>
#include <utility>
#include <vector>

// A minimal JSON value, enough to print this lesson's result.
struct Json {
    enum class Kind { Null, Bool, Number, String, Array, Object };
    Kind kind = Kind::Null;
    bool flag = false;
    double number = 0.0;
    std::string text;
    std::vector<std::string> keys;  // object keys, parallel to items
    std::vector<Json> items;        // array elements or object values

    Json() = default;
    Json(bool value) : kind(Kind::Bool), flag(value) {}
    Json(int value) : kind(Kind::Number), number(value) {}
    Json(double value) : kind(Kind::Number), number(value) {}
    Json(const char* value) : kind(Kind::String), text(value) {}
    Json(const std::string& value) : kind(Kind::String), text(value) {}
    Json(const std::vector<double>& values) : kind(Kind::Array) {
        for (double v : values) items.push_back(Json(v));
    }
};

Json jsonArray(const std::vector<Json>& values) {
    Json array;
    array.kind = Json::Kind::Array;
    array.items = values;
    return array;
}

Json jsonObject(const std::vector<std::pair<std::string, Json>>& fields) {
    Json object;
    object.kind = Json::Kind::Object;
    for (const auto& field : fields) {
        object.keys.push_back(field.first);
        object.items.push_back(field.second);
    }
    return object;
}

// Shortest decimal form that reads back as the same double.
std::string formatNumber(double x) {
    if (!std::isfinite(x)) return "null";
    char buffer[32];
    if (x == std::floor(x) && std::fabs(x) < 1e15) {
        std::snprintf(buffer, sizeof buffer, "%.0f", x);
        return buffer;
    }
    for (int precision = 1; precision <= 17; ++precision) {
        std::snprintf(buffer, sizeof buffer, "%.*g", precision, x);
        if (std::strtod(buffer, nullptr) == x) break;
    }
    return buffer;
}

std::string quote(const std::string& s) {
    std::string out = "\"";
    for (char c : s) {
        if (c == '"' || c == '\\') { out += '\\'; out += c; }
        else if (c == '\n') out += "\\n";
        else out += c;
    }
    return out + "\"";
}

std::string toJson(const Json& value, const std::string& indent = "") {
    switch (value.kind) {
        case Json::Kind::Null: return "null";
        case Json::Kind::Bool: return value.flag ? "true" : "false";
        case Json::Kind::Number: return formatNumber(value.number);
        case Json::Kind::String: return quote(value.text);
        default: break;
    }
    const bool isObject = value.kind == Json::Kind::Object;
    if (value.items.empty()) return isObject ? "{}" : "[]";
    const std::string inner = indent + "  ";
    std::string out = isObject ? "{\n" : "[\n";
    for (std::size_t i = 0; i < value.items.size(); ++i) {
        out += inner;
        if (isObject) out += quote(value.keys[i]) + ": ";
        out += toJson(value.items[i], inner);
        out += i + 1 < value.items.size() ? ",\n" : "\n";
    }
    return out + indent + (isObject ? "}" : "]");
}

double average(const std::vector<double>& values) {
    double total = 0.0;
    for (double x : values) total += x;
    return total / values.size();
}

// (x - center)^2 for each value
std::vector<double> squaredDeviations(const std::vector<double>& values, double center) {
    std::vector<double> out;
    for (double x : values) out.push_back(std::pow(x - center, 2));
    return out;
}

Json lesson073() {
    const double truth = 10;
    const std::vector<double> estimates = {9, 10, 11};
    const double mean = average(estimates), bias = mean - truth;
    return jsonObject({
        {"bias", bias},
        {"variance", average(squaredDeviations(estimates, mean))},
        {"mse", average(squaredDeviations(estimates, truth))},
    });
}

int main() {
    std::cout << toJson(lesson073()) << '\n';
    return 0;
}

// Fintech Math Bootcamp · Lesson 073 of 120
// Estimator Bias, Consistency, Efficiency, and Robustness
// Module 08: Sampling, Estimation, and Statistical Inference
//
// Scenario: Comparing estimators on error rather than a fashionable label
// Rule:     MSE = variance + bias²
//
// Try it:   Does shifting every estimate by one change its variance?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/sampling-estimation-and-statistical-inference/estimator-bias-consistency-efficiency-and-robustness/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.

/// A minimal JSON value, enough to print this lesson's result.
#[allow(dead_code)]
enum Json {
    Null,
    Bool(bool),
    Num(f64),
    Str(String),
    Arr(Vec<Json>),
    Obj(Vec<(String, Json)>),
}

#[allow(dead_code)]
impl Json {
    fn obj(fields: Vec<(&str, Json)>) -> Json {
        Json::Obj(fields.into_iter().map(|(k, v)| (k.to_string(), v)).collect())
    }

    fn nums(values: &[f64]) -> Json {
        Json::Arr(values.iter().map(|&v| Json::Num(v)).collect())
    }

    /// Pretty-prints with two-space indentation.
    fn pretty(&self, indent: &str) -> String {
        let inner = format!("{}  ", indent);
        match self {
            Json::Null => "null".to_string(),
            Json::Bool(b) => b.to_string(),
            Json::Num(x) => format_number(*x),
            Json::Str(s) => quote(s),
            Json::Arr(items) if items.is_empty() => "[]".to_string(),
            Json::Obj(fields) if fields.is_empty() => "{}".to_string(),
            Json::Arr(items) => {
                let body: Vec<String> = items
                    .iter()
                    .map(|v| format!("{}{}", inner, v.pretty(&inner)))
                    .collect();
                format!("[\n{}\n{}]", body.join(",\n"), indent)
            }
            Json::Obj(fields) => {
                let body: Vec<String> = fields
                    .iter()
                    .map(|(k, v)| format!("{}{}: {}", inner, quote(k), v.pretty(&inner)))
                    .collect();
                format!("{{\n{}\n{}}}", body.join(",\n"), indent)
            }
        }
    }
}

fn format_number(x: f64) -> String {
    if !x.is_finite() {
        "null".to_string()
    } else if x == x.trunc() && x.abs() < 1e15 {
        format!("{}", x as i64)
    } else {
        format!("{}", x)
    }
}

fn quote(s: &str) -> String {
    let mut out = String::from("\"");
    for c in s.chars() {
        match c {
            '"' => out.push_str("\\\""),
            '\\' => out.push_str("\\\\"),
            '\n' => out.push_str("\\n"),
            c => out.push(c),
        }
    }
    out.push('"');
    out
}

fn average(values: &[f64]) -> f64 {
    values.iter().fold(0.0_f64, |s, x| s + x) / values.len() as f64
}

/// (x - center)^2 for each value.
fn squared_deviations(values: &[f64], center: f64) -> Vec<f64> {
    values.iter().map(|x| (x - center).powf(2.0)).collect()
}

fn lesson_073() -> Json {
    let truth = 10.0;
    let estimates = [9.0_f64, 10.0, 11.0];
    let mean = average(&estimates);
    let bias = mean - truth;
    Json::obj(vec![
        ("bias", Json::Num(bias)),
        ("variance", Json::Num(average(&squared_deviations(&estimates, mean)))),
        ("mse", Json::Num(average(&squared_deviations(&estimates, truth)))),
    ])
}

fn main() {
    println!("{}", lesson_073().pretty(""));
}

// Fintech Math Bootcamp · Lesson 073 of 120
// Estimator Bias, Consistency, Efficiency, and Robustness
// Module 08: Sampling, Estimation, and Statistical Inference
//
// Scenario: Comparing estimators on error rather than a fashionable label
// Rule:     MSE = variance + bias²
//
// Try it:   Does shifting every estimate by one change its variance?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/sampling-estimation-and-statistical-inference/estimator-bias-consistency-efficiency-and-robustness/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text.Json;

var options = new JsonSerializerOptions { WriteIndented = true };
Console.WriteLine(JsonSerializer.Serialize(Lesson073(), options));

static double Average(IEnumerable<double> values)
{
    double total = 0;
    int count = 0;
    foreach (double x in values)
    {
        total += x;
        count++;
    }
    return total / count;
}

static object Lesson073()
{
    double truth = 10;
    double[] estimates = { 9, 10, 11 };
    double mean = Average(estimates), bias = mean - truth;
    double variance = Average(estimates.Select(x => Math.Pow(x - mean, 2)));
    double mse = Average(estimates.Select(x => Math.Pow(x - truth, 2)));
    return new { bias, variance, mse };
}