Let \(X \sim \text{weibull}(\beta)\), then the PDF is given by: \[ f(x; \beta) = \frac{2x}{\beta^2} \exp\left( -\frac{x^2}{\beta^2} \right), \quad x > 0 \]
# Set seed for reproducibility
set.seed(42)
raw_text = " [1] 2.0506664 2.4418716 1.6895062 3.2531637 2.3198719 4.4698084 1.6939512 2.9590417
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[953] 1.7416125 1.7606352 1.1036271 2.9023427 2.7642357 2.8154811 2.0694465 3.4111210
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[969] 3.6461612 1.5357207 0.9644677 4.4819388 2.2562980 9.2186493 1.0481789 7.8583568
[977] 3.6315160 4.7986213 6.0524314 1.3272624 2.3443622 5.2265570 6.1843015 2.5464572
[985] 3.8249664 5.2177893 3.4499587 1.5294679 4.9628907 5.9890413 1.7007087 1.3118405
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"
clean_text <- gsub("\\[.*?\\]", "", raw_text)
# Step 3: Convert to numeric vector
data <- as.numeric(unlist(strsplit(clean_text, "\\s+")))
data <- data[!is.na(data)]
# Parameters
alpha <- 0.03 # Significance level
n <- 100 # Sample size
# Step 2: Take a random sample
sample <- sample(data, size = n, replace = FALSE)
# Step 3: Calculate test statistic T(x)
Tx <- sum(sample^2)Let \(X_i \sim \text{Weibull}(2, \beta)\). Then define:
\[ Y_i = \left( \frac{X_i}{\beta} \right)^2 \sim \text{Exp}(1) \]
So the sum:
\[ \sum_{i=1}^n Y_i = \sum_{i=1}^n \left( \frac{X_i}{\beta} \right)^2 = \frac{1}{\beta^2} \sum_{i=1}^n X_i^2 \sim \text{Gamma}(n, 1) = \frac{1}{2} \chi^2_{2n} \]
Multiplying both sides by \(\beta^2\):
\[ Critical Value\sum_{i=1}^n X_i^2 \sim \text{Gamma}(n, \theta = \beta^2) \]
Or, in terms of a chi-square distribution:
\[ \sum_{i=1}^n X_i^2 \sim \frac{\beta^2}{2} \cdot \chi^2_{2n} \]
# Confidence level
alpha <- 0.03 # %97
# Sample
n <- 100
sample <- sample(data, size = n, replace = FALSE)
# Test Stat
Tx <- sum(sample^2)
# Chi_square critical value (df = 2n)
chi2_upper <- qchisq(1 - alpha/2, df = 2 * n)/2
chi2_lower <- qchisq(alpha/2, df = 2 * n)/2
# Confidence Interval for beta^2
lower_var <- Tx / chi2_upper
upper_var <- Tx / chi2_lower
# Confidence Interval for beta (sqrt)
lower_beta <- sqrt(lower_var)
upper_beta <- sqrt(upper_var)
cat(sprintf("%d%% Confidence Interval for β: [%.4f, %.4f]\n", (1 - alpha)*100, lower_beta, upper_beta))## 97% Confidence Interval for β: [3.7699, 4.6863]\[ H_0: \beta = \beta_0 \\ H_1: \beta = \beta_1 \quad (\beta_0 \ne \beta_1) \quad \text{(simple vs simple)} \]
Likelihood ratio: \[ \lambda(x) = \frac{L(\beta_1)}{L(\beta_0)} = \left( \frac{\beta_0^2}{\beta_1^2} \right)^n \exp\left[ \left( \frac{1}{2\beta_1^2} - \frac{1}{2\beta_0^2} \right) \sum x_i^2 \right] \]
Let \(T(x) = \sum x_i^2\):
Key Transformation: If \(X \sim \text{Weibull}(alpha=2, \ beta)\), then \(Y = \left(\frac{X}{\beta}\right)^2 \sim \chi^2_2\)
\[ \Rightarrow \frac{X_i^2}{\beta^2/2} \sim \chi^2_2 \Rightarrow \sum \frac{X_i^2}{\beta^2/2} \sim \chi^2_{2n} \quad \text{(under } H_0) \Rightarrow T(x) = \sum X_i^2 \sim \frac{\beta_0^2}{2} \chi^2_{2n} \quad \text{(under } H_0) \]
# --- MP Test (Two-sided) ---
beta0 <- sqrt(mean(sample^2)) # MLE for beta
beta0## [1] 4.179678statistic_mp <- Tx / beta0^2
chi2_crit_low <- qchisq(alpha, df = 2 * n)/2
chi2_crit_high <- qchisq(1 - alpha , df = 2 * n)/2
reject_mp <- (statistic_mp < chi2_crit_low) || (statistic_mp > chi2_crit_high)
cat("MP Test (Two-sided)\n")## MP Test (Two-sided)cat(sprintf("T(x)/σ₀² = %.2f\n", statistic_mp))## T(x)/σ₀² = 100.00cat(sprintf("Critical region: < %.2f or > %.2f\n", chi2_crit_low, chi2_crit_high))## Critical region: < 82.06 or > 119.64cat("Result:", ifelse(reject_mp, "Reject H0", "Do not reject H0"), "\n\n")## Result: Do not reject H0cat("Bounds of theta:") ## Bounds of theta:sqrt((Tx)/chi2_crit_low)## [1] 4.614126sqrt((Tx)/chi2_crit_high)## [1] 3.821321\[ H_0: \beta = \beta_0 \\ H_1: \beta > \beta_0 \text{ or } \beta < \beta_0 \quad \text{(Let it be } \beta_1\text{)} \]
Under \(\beta_0 > \beta_1\): \[ \frac{L(\beta_1)}{L(\beta_0)} = \left(\frac{\beta_0^2}{\beta_1^2}\right)^n \exp\left[\left(\frac{1}{\beta_1^2} - \frac{1}{\beta_0^2}\right)\sum x_i^2\right] \]
Under \(\beta_1 > \beta_0\): \[ \frac{L(\beta_1)}{L(\beta_0)} = \left(\frac{\beta_1^2}{\beta_0^2}\right)^n \exp\left[\left(\frac{1}{\beta_0^2} - \frac{1}{\beta_1^2}\right)\sum x_i^2\right] \]
\(T(x) = \sum x_i^2\), and \(L\) is a non-decreasing function of \(T(x)\), so the MLRS is \(T(x)\).
# --- UMP Test (One-sided: H1: β > β₀) ---
chi2_crit_ump <- qchisq(1 - alpha, df = 2 * n)
reject_ump <- statistic_mp > chi2_crit_ump
cat("UMP Test (H1: β> β₀)\n")## UMP Test (H1: β> β₀)cat(sprintf("T(x)/β₀² = %.2f, Critical value = %.2f\n", statistic_mp, chi2_crit_ump))## T(x)/β₀² = 100.00, Critical value = 239.27cat("Result:", ifelse(reject_ump, "Reject H0", "Do not reject H0"), "\n\n")## Result: Do not reject H0\[ H_0: \beta = \beta_0 \quad \text{vs.} \quad H_1: \beta \ne \beta_0 \]
Parameter spaces: \[ \Omega_0 = \{\beta_0\} \quad \Omega_1 = (0, \beta_0) \cup (\beta_0, \infty) \quad \Omega = \Omega_0 \cup \Omega_1 = (0, \infty) \]
Likelihood: \[ L(\beta) = \prod \frac{2}{\beta^2}x \exp\left(-\frac{x_i^2}{\beta^2}\right) \Rightarrow \ell_n L(\beta) = \sum \ln x_i - 2n \ln \beta + \ln\ 2 - \frac{\sum x_i^2}{\beta^2} \]
Derivative: \[ \frac{\partial \ell_n L(\beta)}{\partial \beta} = -\frac{n}{\beta} + \frac{1}{\beta^3} \sum x_i^2 = 0 \Rightarrow \hat{\beta}_{MLE} = \sqrt{\frac{1}{n} \sum x_i^2} \]
GLR statistic: \[ \lambda = \frac{L(\beta_0)}{L(\hat{\beta})} = \left(\frac{\hat{\beta}^2}{\beta_0^2}\right)^n \exp\left[\sum x_i^2 \left(\frac{1}{\hat{\beta}^2} - \frac{1}{\beta_0^2}\right)\right] \]
This simplifies to: \[ \left(\frac{\sum x_i^2}{n\beta_0^2}\right)^n \exp\left[n - \frac{\sum x_i^2}{\beta_0^2}\right] < \lambda_0 \]
Taking log: \[ \ln \lambda = n \ln \left(\frac{\sum x_i^2}{n\beta_0^2}\right) + n - \frac{\sum x_i^2}{\beta_0^2} < \ln \lambda_0 \]
Define: \[ \ln \left( \frac{\sum x_i^2}{n\beta_0^2} \right) - \frac{\sum x_i^2}{n\beta_0^2} < \frac{\ln \lambda_0}{n} - 1 \]
Let \(Y = \frac{\sum x_i^2}{n \beta_0^2}, Z = \ln Y - Y\), then: \[ P(\text{Reject } H_0 | H_0 \text{ true}) = P(Y < k_1) + P(Y > k_2) = \alpha \]
Since \(Y \cdot n \sim \chi^2_{2n}\), \[ \text{Reject } H_0: \frac{\sum x_i^2}{\beta_0^2} < \chi^2_{2n, 1 - \alpha/2} \text{ or } > \chi^2_{2n, \alpha/2} \]
Final form: \[ -2 \ln \lambda = -2n - \ln \sum x_i^2 + 2 \ln n \beta_0^2 + \frac{2\sum x_i^2}{\beta_0^2} \sim \chi^2_1 \]
Then: \[ \alpha = P(\text{Reject } H_0 | H_0 \text{ TRUE}) = P(\lambda < \lambda_0 | \beta = \beta_0) = P(-2 \ln \lambda > -2 \ln \lambda_0 | \beta_0) \]
Decision rule: \[ \text{Reject } H_0: -2 \ln \lambda > \chi^2_{1, \alpha} \]
# --- GLR Test ---
beta_hat <- sqrt(Tx / n) # MLE for beta
beta_hat## [1] 4.179678lambda_glr <- (beta0 / beta_hat)^(n) * exp(n- Tx/(beta0^2))
test_stat_glr <- -2 * log(lambda_glr)
crit_val_glr <- qchisq(1 - alpha, df = 1)
reject_glr <- test_stat_glr > crit_val_glr
cat("GLR Test\n")## GLR Testcat(sprintf("-2 * ln(λ) = %.2f, Critical value = %.2f\n", test_stat_glr, crit_val_glr))## -2 * ln(λ) = -0.00, Critical value = 4.71cat("Result:", ifelse(reject_glr, "Reject H0", "Do not reject H0"), "\n")## Result: Do not reject H0Let \(X \sim \text{Rayleigh}(\sigma)\), then the PDF is given by: \[ f(x; \sigma) = \frac{x}{\sigma^2} \exp\left( -\frac{x^2}{2\sigma^2} \right), \quad x > 0 \]
# Set seed for reproducibility
set.seed(42)
# Step 1: Read the data from the .txt file
# Set seed for reproducibility
set.seed(42)
# Step 1: Read the data from the .txt file
raw_data = "[1] 3.9838900 3.1404940 8.3858660 2.4450140 16.3581700 4.5726500 11.4172600 5.7863790 7.2289120 12.8344900 7.4979110 2.8077770 7.1752170 9.4481750 9.7778730 6.0916880
[17] 6.3507690 2.8269510 7.0376860 6.1845420 10.9223500 8.1495820 5.2922420 3.2583290 4.0002490 2.2900970 6.7122380 3.1436890 4.1846590 0.2610592 10.5704700 3.6722680
[33] 3.0797980 3.0531820 7.8485750 3.9460790 6.6153560 6.9011300 4.6500100 7.4195490 16.2404200 14.5218000 4.9024380 2.9230470 1.7031220 3.9937600 5.3226830 1.1492960
[49] 3.2206580 8.4931370 6.4404790 4.6833290 4.1367620 3.6528120 6.2377320 6.6108390 3.6836190 6.6583190 1.5467100 3.5003580 8.2520340 6.6173730 4.6525540 9.3933560
[65] 0.6818655 6.4133990 2.7039290 4.7547270 4.9279070 1.8828730 5.2236440 7.7868520 7.9734210 3.1476660 3.2947840 1.7238300 8.5343390 10.7171100 2.3605880 5.8358460
[81] 0.5807991 0.6204383 10.6870200 3.3198640 10.0078400 6.2659670 10.4879000 6.6317080 10.5442300 3.3310990 17.3775000 12.4789700 8.3639790 9.0902200 8.1487030 2.8820280
[97] 5.2786240 6.0293860 8.8291700 0.7051838 4.4633710 3.9695270 3.2523630 7.8681480 4.5520210 3.3227270 15.3077000 1.7804250 5.5542170 5.7294030 0.6140395 3.9069710
[113] 8.2061820 2.4040490 7.0610320 5.7856600 3.4936740 5.3839060 3.6820850 7.6601590 3.3818640 5.3736820 5.9734680 0.7968121 1.8930340 5.2959660 8.4297950 5.9172020
[129] 2.3855760 2.4506050 2.9259330 9.5387880 3.3920040 8.8795570 6.1851380 1.7651920 5.1470710 8.9896480 5.9375090 3.1891030 5.0690550 9.2523730 7.2566890 4.4967990
[145] 7.3534710 9.0676660 3.8657690 2.9331710 9.6418840 9.3680600 6.4217720 14.8365000 2.9197150 6.2208530 1.6540860 13.2191700 10.5846700 0.8499204 1.9056680 5.3731760
[161] 5.7329430 2.0257250 4.0992530 3.8857330 7.5103450 5.7637170 5.8197160 13.1333400 4.1291100 2.5960830 5.1864520 0.2756376 6.0511130 4.3884480 6.8235090 8.0417120
[177] 2.8300500 7.2459060 13.1639700 9.2052400 5.1496650 2.1075970 7.9669070 4.9047050 8.6375220 3.7365050 8.3200310 6.1390200 7.3662720 3.7471090 10.9712300 5.7063830
[193] 15.6062900 3.1501280 5.6347090 4.7673180 16.2381200 9.8650720 2.9560610 11.9607300 6.9830270 2.6991240 9.2950380 4.0704480 6.7950110 2.4819120 2.5125300 5.8371520
[209] 6.7608340 13.9178100 10.1362200 9.6294820 4.0794230 3.7198240 2.1588070 8.5780670 1.9209250 7.5268230 4.8581880 15.7355700 3.4084230 6.5684940 6.1215120 7.0910590
[225] 12.8214400 5.7767850 6.2317370 1.2988770 7.1013670 11.3476600 6.5930870 4.1032830 3.3270110 7.4169700 7.9455280 2.7488570 4.5197340 12.5627300 5.1393280 1.1991310
[241] 5.5249090 5.0297060 6.7942730 8.6721730 4.3388160 2.5118180 5.3362500 6.4785420 2.3675180 4.5844010 7.8593660 9.9053060 1.3619890 2.5198050 4.5373520 6.2917590
[257] 4.8485380 4.5282650 3.2180300 2.7563940 4.4104340 8.4459340 6.2683180 5.0950920 10.4796900 2.9796010 4.4342740 7.6905060 5.1042220 5.8594070 8.3266260 8.4616650
[273] 12.4085300 2.5943430 8.4888630 8.8930230 2.5018990 7.6740380 4.0688100 4.9325660 6.4382630 3.0499290 3.0441860 5.0377080 11.1410500 14.3107800 4.5481160 4.2511880
[289] 3.0301000 5.7340800 4.6443860 3.8799240 5.5591090 4.5230770 7.8546510 6.9069230 7.9239410 5.4828990 6.6512270 12.8824500 9.9696050 7.1532000 9.6106950 3.2986870
[305] 0.9713148 2.9407450 5.3011910 4.8476820 6.6987490 5.3669250 9.0633590 3.2033980 5.3965680 10.6867900 1.2292830 4.6755180 5.3090290 4.1798520 5.6690180 3.1613680
[321] 0.7081749 8.1550920 5.2508730 2.2046240 10.0687300 4.2841340 9.4785310 1.7635080 6.1874520 6.2086060 5.2120090 1.1359310 3.2082110 10.4477800 1.3102420 3.9566350
[337] 2.3274640 8.0675140 4.3895010 7.9644730 7.9695810 5.1003030 5.9628290 1.9647770 8.8331300 12.3460300 9.4898590 0.7470549 6.3507260 1.8179840 1.4783350 8.9999790
[353] 10.5696200 6.7030580 4.6130670 8.5327250 8.2368350 12.1877200 5.4273900 4.4215090 1.0270260 9.5918400 11.7511200 2.7823980 9.7570120 3.3452050 4.4748180 4.5307980
[369] 4.2606920 2.2564310 3.3054160 1.2003270 5.0039800 2.8852510 7.9421540 1.4429100 5.6303750 7.3846790 15.5392100 7.5776860 10.8761100 3.8296360 7.2058840 6.2560630
[385] 9.8327100 4.7973920 3.2663500 5.8165870 8.3082650 1.1218610 8.6503930 1.1999300 2.0819870 7.5811740 6.2289720 7.8197350 8.4259870 4.1498510 2.2658360 2.3327640
[401] 3.1219350 7.1884710 7.1713720 3.4152180 11.4110300 12.5543000 4.8496610 6.9726030 7.4263120 3.4027100 3.3957890 7.2812410 1.3389490 12.2563000 9.8023200 3.2025370
[417] 11.6057300 6.0472180 7.7623700 6.0444000 3.2140510 2.8162000 3.6791370 2.6250130 7.9602660 4.9052000 8.7839020 3.2358140 5.6206050 1.4955380 3.8346160 3.4296340
[433] 3.0369820 2.9299190 5.4387770 4.8192580 7.8179530 19.1919800 4.8850110 7.1767720 5.9766480 3.9424150 5.5975760 8.4060210 10.5328300 5.3249490 7.0098450 6.0914440
[449] 2.0260950 8.3268350 1.6619170 3.6253140 11.2295300 5.7285140 3.6330740 5.6252450 4.7315610 4.0663350 6.0822710 3.2273960 7.0308460 5.9843320 4.8524610 4.4386710
[465] 6.2281500 2.5408820 10.3117000 12.2002900 7.8582290 2.4574290 2.7266070 2.1117050 1.5954500 3.4911910 10.1328500 10.5851500 4.8508480 1.4243130 3.9475660 5.6425810
[481] 2.9949380 2.1924070 5.6830960 4.9269460 2.3895540 9.1129830 4.2448170 5.7651690 4.9589070 3.8984430 3.7061170 2.5149590 2.4359060 8.1020280 1.9474210 7.2182310
[497] 7.7114230 5.7243430 10.6804400 4.8687990 6.8599620 5.6379220 9.1748270 7.8755680 7.8668670 4.8299800 6.2278900 6.9550230 5.5830290 5.8482740 7.4660070 8.5145830
[513] 5.2997480 5.4024960 10.9953500 5.3271070 0.6354547 3.2230590 15.1330700 7.5575630 4.5493970 10.6114800 3.5540560 1.1785930 5.8818530 3.4622010 7.3917880 9.6250130
[529] 8.3451320 6.7625450 2.2492910 5.2748160 8.6380020 10.1220600 5.4623060 12.8996400 3.6128170 6.0745190 1.7208500 6.0230460 4.7678930 3.0109880 2.8139250 6.4232580
[545] 8.9648620 7.3310480 2.8729200 5.8590210 14.4235800 10.5755000 1.8026360 6.3093270 0.8229984 2.7603070 6.3599530 7.1602360 17.1972500 7.1521460 1.0369610 6.6623640
[561] 5.9339020 1.9074400 6.3262120 4.1776490 2.4134310 2.9423390 2.0548470 1.3999330 1.1825300 6.2174670 6.0591380 0.7694072 9.2190370 1.8305160 5.2355590 12.0092100
[577] 7.8248590 7.5811710 6.1865320 8.4177610 0.9024701 6.3392500 3.1846940 7.6687450 10.0680600 7.8838880 5.3911380 10.6018800 0.4612450 4.7307750 7.5877080 2.0864070
[593] 4.7364420 3.3728130 2.0404840 9.8735380 3.0252760 4.2881050 8.4457740 4.4418810 21.9031900 9.8580880 6.2168050 3.7202480 10.5606500 8.3880070 1.8988150 4.1126160
[609] 6.6634630 6.1666570 4.0363040 6.9417680 10.4917000 6.3523300 2.5389390 12.9966000 8.3515620 6.3341980 6.2427490 3.7284680 5.7858820 7.3312460 5.7471080 4.7215840
[625] 0.3232862 2.2295680 6.4720420 10.1448500 4.7652580 9.0542960 5.8213720 3.8533600 3.8516770 0.9014661 12.3822600 15.3987100 5.7437960 8.3215330 7.0808530 2.8626570
[641] 7.8787130 6.0928630 9.6514170 5.3891270 9.2580180 14.4705900 11.1949400 4.8889160 6.7607360 8.9786280 3.8696510 7.1548400 7.5123520 8.4970990 10.9137000 10.3576200
[657] 7.0148670 4.8798580 2.7623380 8.8472160 5.8006890 2.6616600 4.6394580 3.9274900 5.8560830 2.9523860 9.1644470 9.0284200 9.3739520 13.3400600 6.9562900 7.9342790
[673] 2.5808390 3.4844690 6.0054520 9.8737080 4.4517870 11.8641300 4.4970330 2.6901500 8.4752500 4.9102760 2.6422190 3.6741440 7.8789440 10.2552300 9.7744090 2.9287570
[689] 6.0597390 3.2128490 1.3130840 7.6576340 3.1768000 10.8426300 2.7127420 3.5357880 3.0074570 3.2078330 3.5601630 4.7624730 2.7386140 2.0800870 3.7558060 5.5783250
[705] 1.2118010 5.3047110 6.1017680 3.3427430 10.8294700 14.0404200 0.5472427 4.7611210 4.9637060 1.5422050 8.4173950 6.0039740 12.5841500 2.0085480 9.1374980 8.2591940
[721] 8.3341100 3.5564430 5.5375320 9.8135620 0.7643371 3.6504350 1.9288630 3.4997950 7.8515830 10.6468900 7.0985670 1.5676410 2.1949330 9.8883420 2.5421720 9.2275870
[737] 8.1539920 9.3640090 2.8430920 9.3250400 5.5053930 0.8852075 4.5874320 9.6686950 8.9375760 6.6866790 3.9373710 7.4415240 4.8848920 5.0542580 7.3125510 14.9774800
[753] 8.3857360 11.6007000 10.0136800 9.5500700 11.0458600 5.1229550 7.4299060 2.9637200 8.1329520 3.7930610 6.4015370 7.3230160 7.0517910 4.1722190 4.6853410 4.8548890
[769] 3.9830300 5.8955900 5.0169290 6.3502270 2.4569130 4.9072590 14.1636000 10.4447800 7.3318000 5.1942340 9.6513600 4.9132420 3.1074760 7.8369910 5.2095800 2.0690980
[785] 3.1856500 4.7570450 2.2833860 4.2403510 6.7507820 7.2918020 11.3946800 7.2865530 1.3772880 7.6252460 5.2958130 5.4419540 8.9756390 7.3369790 2.5680960 10.7005900
[801] 2.3204810 11.7230900 2.0315730 7.0616240 6.0922970 8.2313760 2.9610640 4.1960350 6.3820200 6.8895930 7.6263220 2.8901690 8.9506250 5.3272100 3.4865750 4.9247120
[817] 6.3666470 4.9159900 8.6534780 1.4692790 3.4400730 3.4032250 2.7425830 4.8465380 3.8920220 9.3887740 1.6113090 5.5031310 5.4605610 5.3611640 2.5297860 3.7952530
[833] 8.3436670 6.3954260 1.8315830 7.2516760 8.9483500 7.4906500 3.7936040 6.7497230 8.0043010 5.5203330 3.0177480 6.4759660 3.8792150 2.2023780 9.7212040 1.2040290
[849] 4.2304170 9.5626610 11.6837600 2.0858190 6.8264370 0.9451860 7.0763980 4.7903860 11.6017800 5.3220170 7.4672480 9.0792040 9.2744670 4.9235990 4.1115280 5.4290820
[865] 8.3356250 4.5502910 7.7697420 1.4312350 7.2065080 6.8749240 4.7840020 11.9162600 14.8588000 10.0330500 4.5668850 3.8934410 6.4980470 8.4863340 4.8690590 2.7392270
[881] 1.6678930 6.3996410 1.7797260 7.5037560 7.9055430 5.7540680 3.7835080 4.2239310 12.1340500 7.3003820 5.3704150 10.3611700 6.0755230 0.9583251 10.4790200 7.0802120
[897] 1.2646880 2.5932480 2.2096290 6.8151250 6.4335190 3.3863190 5.8850080 5.5411560 2.9565890 9.3398130 7.6772830 5.3521090 8.3566160 6.6129880 3.5404610 9.2564500
[913] 3.5719500 4.0317170 7.1027260 4.1380320 8.2373370 7.3940420 6.7660740 6.0773340 6.3459200 10.8210700 2.4245490 3.7337100 5.3854140 4.1715400 0.7638671 16.3828000
[929] 6.9185910 6.4656970 11.3714900 4.6914530 8.8113410 7.8090360 3.2030260 8.8390010 3.9628650 3.5520210 3.7899050 6.0960610 2.0444220 9.4524710 5.2632890 4.5744820
[945] 7.0847110 10.4211200 9.6488570 9.9366710 3.8221930 4.7300950 4.2333230 6.9340400 2.5873230 1.9215430 5.2166780 6.1254370 3.9267530 1.8597140 3.8524370 5.7101480
[961] 6.0830770 8.4537570 3.3432040 8.1227380 4.7244950 11.2103600 9.2503010 4.4366040 4.8862580 3.1781180 3.3945190 4.6508030 4.0153680 7.1731530 4.0363600 8.0688120
[977] 3.6558360 3.5959990 2.2335050 11.0316800 3.4343430 8.3616230 6.2100880 4.0419980 8.6536260 3.6709230 7.1269540 5.0199440 7.7640050 1.9930760 6.7938030 11.3330500
[993] 1.9096390 3.0946710 7.5686540 4.2500460 10.5878100 4.9576460 7.6567110 7.8919400
"
# Process the data
clean_text <- gsub("\\[.*?\\]", "", raw_data)
data <- as.numeric(unlist(strsplit(clean_text, "\\s+")))
data <- data[!is.na(data)]
# Parameters
alpha <- 0.03 # Significance level
n <- 100 # Sample size
# Step 2: Take a random sample
sample <- sample(data, size = n, replace = FALSE)
# Step 3: Calculate test statistic T(x)
Tx <- sum(sample^2)Key Transformation: If \(X \sim \text{Rayleigh}(\sigma)\), then \(Y = \left(\frac{X}{\sigma}\right)^2 \sim \chi^2_2\)
\[ \Rightarrow \frac{X_i^2}{\sigma^2} \sim \chi^2_2 \Rightarrow \sum \frac{X_i^2}{\sigma^2} \sim \chi^2_{2n} \quad \text{(under } H_0) \Rightarrow T(x) = \sum X_i^2 \sim \sigma_0^2 \chi^2_{2n} \quad \text{(under } H_0) \]
\[\frac{\sum X_i^2}{\sigma^2} \sim \chi^2_{2n}\]
\[ P\left( \frac{\sum X_i^2}{\chi^2_{\alpha/2,\; 2n}} < \sigma^2 < \frac{\sum X_i^2}{\chi^2_{1 - \alpha/2,\; 2n}} \right) = 1 - \alpha \]
\[ \Rightarrow \left[ \frac{\sum X_i^2}{\chi^2_{\alpha/2,\; 2n}},\; \frac{\sum X_i^2}{\chi^2_{1 - \alpha/2,\; 2n}} \right] \]
\[ \Rightarrow \left[ \sqrt{ \frac{\sum X_i^2}{\chi^2_{\alpha/2,\; 2n}} },\; \sqrt{ \frac{\sum X_i^2}{\chi^2_{1 - \alpha/2,\; 2n}} } \right] \quad \text{(for } \sigma \text{)} \]
# Confidence level
alpha <- 0.03 # %97
# Sample
n <- 100
sample <- sample(data, size = n, replace = FALSE)
# Test Stat
Tx <- sum(sample^2)
# Chi_square critical value (df = 2n)
chi2_upper <- qchisq(1 - alpha/2, df = 2 * n)
chi2_lower <- qchisq(alpha/2, df = 2 * n)
# Confidence Interval for sigma^2
lower_var <- Tx / chi2_upper
upper_var <- Tx / chi2_lower
# Confidence Interval for sigma (sqrt)
lower_sigma <- sqrt(lower_var)
upper_sigma <- sqrt(upper_var)
cat(sprintf("%d%% Confidence Interval for σ: [%.4f, %.4f]\n", (1 - alpha)*100, lower_sigma, upper_sigma))## 97% Confidence Interval for σ: [4.6011, 5.7195]\[ H_0: \sigma = \sigma_0 \\ H_1: \sigma = \sigma_1 \quad (\sigma_0 \ne \sigma_1) \quad \text{(simple vs simple)} \]
Likelihood ratio: \[ \lambda(x) = \frac{L(\sigma_1)}{L(\sigma_0)} = \left( \frac{\sigma_0^2}{\sigma_1^2} \right)^n \exp\left[ \left( \frac{1}{2\sigma_1^2} - \frac{1}{2\sigma_0^2} \right) \sum x_i^2 \right] \]
Let \(T(x) = \sum x_i^2\):
Key Transformation: If \(X \sim \text{Rayleigh}(\sigma)\), then \(Y = \left(\frac{X}{\sigma}\right)^2 \sim \chi^2_2\)
\[ \Rightarrow \frac{X_i^2}{\sigma^2} \sim \chi^2_2 \Rightarrow \sum \frac{X_i^2}{\sigma^2} \sim \chi^2_{2n} \quad \text{(under } H_0) \Rightarrow T(x) = \sum X_i^2 \sim \sigma_0^2 \chi^2_{2n} \quad \text{(under } H_0) \]
# --- MP Test (Two-sided) ---
sigma0 <- sqrt(Tx / (2 * n)) # MLE for sigma # Null hypothesis value for sigma
statistic_mp <- Tx / sigma0^2
chi2_crit_low <- qchisq(alpha /2, df = 2 * n)
chi2_crit_high <- qchisq(1 - alpha /2 , df = 2 * n)
reject_mp <- (statistic_mp < chi2_crit_low) || (statistic_mp > chi2_crit_high)
cat("MP Test (Two-sided)\n")## MP Test (Two-sided)cat(sprintf("T(x)/σ₀² = %.2f\n", statistic_mp))## T(x)/σ₀² = 200.00cat(sprintf("Critical region: < %.2f or > %.2f\n", chi2_crit_low, chi2_crit_high))## Critical region: < 159.10 or > 245.85cat("Result:", ifelse(reject_mp, "Reject H0", "Do not reject H0"), "\n\n")## Result: Do not reject H0lower_sigma2 <- Tx / chi2_crit_high
upper_sigma2 <- Tx / chi2_crit_low
lower_sigma <- sqrt(lower_sigma2)
upper_sigma <- sqrt(upper_sigma2)
cat("Critical values for σ")## Critical values for σcat("=", "c1:", lower_sigma, "c2:" ,upper_sigma)## = c1: 4.60109 c2: 5.719541\[ H_0: \sigma = \sigma_0 \\ H_1: \sigma > \sigma_0 \text{ or } \sigma < \sigma_0 \quad \text{(Let it be } \sigma_1\text{)} \]
Under \(\sigma_0 > \sigma_1\): \[ \frac{L(\sigma_1)}{L(\sigma_0)} = \left(\frac{\sigma_0^2}{\sigma_1^2}\right)^n \exp\left[\left(\frac{1}{2\sigma_1^2} - \frac{1}{2\sigma_0^2}\right)\sum x_i^2\right] \]
Under \(\sigma_1 > \sigma_0\): \[ \frac{L(\sigma_1)}{L(\sigma_0)} = \left(\frac{\sigma_1^2}{\sigma_0^2}\right)^n \exp\left[\left(\frac{1}{2\sigma_0^2} - \frac{1}{2\sigma_1^2}\right)\sum x_i^2\right] \]
\(T(x) = \sum x_i^2\), and \(L\) is a non-decreasing function of \(T(x)\), so the MLRS is \(T(x)\).
# --- UMP Test (One-sided: H1: σ > σ₀) ---
chi2_crit_ump <- qchisq(1 - alpha, df = 2 * n)
reject_ump <- statistic_mp > chi2_crit_ump
cat("UMP Test (H1: σ > σ₀)\n")## UMP Test (H1: σ > σ₀)cat(sprintf("T(x)/σ₀² = %.2f, Critical value = %.2f\n", statistic_mp, chi2_crit_ump))## T(x)/σ₀² = 200.00, Critical value = 239.27cat("Result:", ifelse(reject_ump, "Reject H0", "Do not reject H0"), "\n\n")## Result: Do not reject H0\[ H_0: \sigma = \sigma_0 \quad \text{vs.} \quad H_1: \sigma \ne \sigma_0 \]
Parameter spaces: \[ \Omega_0 = \{\sigma_0\} \quad \Omega_1 = (0, \sigma_0) \cup (\sigma_0, \infty) \quad \Omega = \Omega_0 \cup \Omega_1 = (0, \infty) \]
Likelihood: \[ L(\sigma) = \prod \frac{x_i}{\sigma^2} \exp\left(-\frac{x_i^2}{2\sigma^2}\right) \Rightarrow \ell_n L(\sigma) = \sum \ln x_i - 2n \ln \sigma - \frac{\sum x_i^2}{2\sigma^2} \]
Derivative: \[ \frac{\partial \ell_n L(\sigma)}{\partial \sigma} = -\frac{2n}{\sigma} + \frac{1}{\sigma^3} \sum x_i^2 = 0 \Rightarrow \hat{\sigma}_{MLE} = \sqrt{\frac{1}{2n} \sum x_i^2} \]
GLR statistic: \[ \lambda = \frac{L(\sigma_0)}{L(\hat{\sigma})} = \left(\frac{\hat{\sigma}^2}{\sigma_0^2}\right)^n \exp\left[\sum x_i^2 \left(\frac{1}{2\hat{\sigma}^2} - \frac{1}{2\sigma_0^2}\right)\right] \]
This simplifies to: \[ \left(\frac{\sum x_i^2}{2n\sigma_0^2}\right)^n \exp\left[n - \frac{\sum x_i^2}{2\sigma_0^2}\right] < \lambda_0 \]
Taking log: \[ \ln \lambda = n \ln \left(\frac{\sum x_i^2}{2n\sigma_0^2}\right) + n - \frac{\sum x_i^2}{2\sigma_0^2} < \ln \lambda_0 \]
Define: \[ \ln \left( \frac{\sum x_i^2}{2n\sigma_0^2} \right) - \frac{\sum x_i^2}{2n\sigma_0^2} < \frac{\ln \lambda_0}{n} - 1 \]
Let \(Y = \frac{\sum x_i^2}{2n \sigma_0^2}, Z = \ln Y - Y\), then: \[ P(\text{Reject } H_0 | H_0 \text{ true}) = P(Y < k_1) + P(Y > k_2) = \alpha \]
Since \(Y \cdot 2n \sim \chi^2_{2n}\), \[ \text{Reject } H_0: \frac{\sum x_i^2}{\sigma_0^2} < \chi^2_{2n, 1 - \alpha/2} \text{ or } > \chi^2_{2n, \alpha/2} \]
Final form: \[ -2 \ln \lambda = -2n - \ln \sum x_i^2 + 2 \ln 2n \sigma_0^2 + \frac{\sum x_i^2}{\sigma_0^2} \sim \chi^2_1 \]
Then: \[ \alpha = P(\text{Reject } H_0 | H_0 \text{ TRUE}) = P(\lambda < \lambda_0 | \sigma = \sigma_0) = P(-2 \ln \lambda > -2 \ln \lambda_0 | \sigma_0) \]
Decision rule: \[ \text{Reject } H_0: -2 \ln \lambda > \chi^2_{1, \alpha} \]
# --- GLR Test ---
sigma_hat <- sqrt(Tx / (2 * n)) # MLE for sigma
lambda_glr <- (sigma0 / sigma_hat)^(2 * n) * exp((1 - (sigma0^2 / sigma_hat^2)) * Tx / (2 * sigma0^2))
test_stat_glr <- -2 * log(lambda_glr)
crit_val_glr <- qchisq(1 - alpha, df = 1)
reject_glr <- test_stat_glr > crit_val_glr
cat("GLR Test\n")## GLR Testcat(sprintf("-2 * ln(λ) = %.2f, Critical value = %.2f\n", test_stat_glr, crit_val_glr))## -2 * ln(λ) = -0.00, Critical value = 4.71cat("Result:", ifelse(reject_glr, "Reject H0", "Fail to reject H0"), "\n")## Result: Fail to reject H0# Test statistic
test_stat <- sum(sample^2) / sigma0^2
cat("Test Statistic :" ,test_stat ,"\n")## Test Statistic : 200# Critical values
upper_crit <- qchisq(1- alpha / 2, df = n*2)
cat("Upper Critical Value:" ,upper_crit ,"\n")## Upper Critical Value: 245.8451lower_crit <- qchisq(alpha / 2, df = n*2)
cat("Lower Critical Value:" ,lower_crit ,"\n")## Lower Critical Value: 159.0965# Decision
if (test_stat < lower_crit || test_stat > upper_crit) {
cat("Reject H0.\n")
} else {
cat("Fail to reject H0.\n")
}## Fail to reject H0.Let \(X \sim \text{Uniform}(a, b)\), where \(a = 2\). The PDF is:
\[ f(x; b) = \begin{cases} \frac{1}{b - 2}, & 2 \le x \le b \\ 0, & \text{otherwise} \end{cases} \]
The CDF is:
\[ F(x; b) = \begin{cases} \frac{x - 2}{b - 2}, & 2 \le x \le b \\ 0, & \text{otherwise} \end{cases} \]
set.seed(42)
raw_text = " [1] 2.663621 3.719691 2.017035 2.164160 3.678541 3.378565 2.117577 2.202661 2.275739
[10] 3.674937 2.897971 2.905268 2.059480 2.867312 2.162826 3.274812 3.713663 2.086917
[19] 2.531441 2.549707 2.699910 2.376508 3.135720 3.205476 2.827482 3.235600 2.795574
[28] 3.675428 2.192815 2.354276 3.090142 3.016486 2.912484 2.277020 2.067384 3.810705
[37] 3.216785 2.479067 2.467721 3.465114 2.993363 3.658151 3.641604 2.519516 2.906663
[46] 3.724780 2.570132 3.039599 3.463408 2.754499 2.828312 3.569176 3.008819 3.463273
[55] 3.651341 3.465800 3.486750 3.556692 3.079041 2.096812 2.111632 3.484208 3.603337
[64] 2.899568 2.057530 2.856121 2.976772 2.851672 3.904215 2.721099 2.906733 3.631381
[73] 3.158846 3.622023 2.020901 2.964106 2.597594 2.083036 2.754207 2.239494 3.113160
[82] 2.362410 3.780881 2.869399 3.525334 3.069901 3.345015 3.365031 2.989089 3.795102
[91] 2.723967 3.350508 3.188972 2.095746 2.668814 3.670119 3.700714 3.261475 2.231906
[100] 2.963503 3.575509 3.081385 2.632728 2.773702 3.711460 3.922208 2.080387 2.906779
[109] 2.711601 3.401000 2.047337 2.263007 2.533348 2.331349 2.544480 2.398135 2.802361
[118] 2.393106 2.996377 3.283875 3.987811 3.638404 2.439576 2.117281 3.705830 2.195954
[127] 2.175716 3.202325 2.478414 2.275158 3.850307 3.939562 2.796597 2.103550 2.309119
[136] 2.975220 2.956471 2.597938 2.781806 3.656677 2.122953 2.496162 2.790332 3.908046
[145] 3.257915 2.143870 2.621087 3.240883 2.637803 3.713815 3.957882 2.587821 3.261055
[154] 3.047381 2.260638 2.517469 3.355109 2.129659 2.623983 2.465968 3.310790 3.320290
[163] 2.840429 3.642476 3.767820 2.083566 3.095760 2.119193 3.314760 3.146432 2.463800
[172] 2.687378 3.645619 2.904312 2.426555 2.177044 2.540458 3.859999 3.230073 3.717644
[181] 2.313047 2.533545 2.092364 2.028786 2.601089 3.411821 2.328651 3.449246 2.207781
[190] 3.636488 2.122207 2.814976 3.644852 2.358352 2.054776 3.510887 3.809067 3.073782
[199] 2.544307 3.117265 2.797197 2.673574 3.400855 3.113256 2.320663 3.144448 3.929980
[208] 3.715611 3.946040 3.990222 3.581176 2.991170 3.797662 3.469349 3.588758 3.483175
[217] 2.205566 2.600155 3.561080 2.106795 3.681636 3.411598 2.421913 3.086589 3.504433
[226] 3.845067 2.946498 2.753043 3.119020 3.951742 2.137848 3.906378 3.846519 3.890667
[235] 2.696465 3.807961 3.906170 2.077847 2.646424 2.617993 3.780063 3.506835 3.047141
[244] 3.388590 3.844747 2.111301 2.593678 2.443815 2.838243 3.929339 3.190321 3.627884
[253] 2.471517 3.163949 2.112584 2.555148 2.621020 3.933363 2.163520 2.030342 3.660950
[262] 2.191912 2.418312 2.795992 2.776286 3.132091 3.752974 3.841899 2.600588 2.373443
[271] 2.989930 3.359068 2.000142 2.686738 3.073891 3.414127 2.576576 2.852739 3.648324
[280] 2.884435 2.065092 2.244715 2.003479 2.947998 3.187076 2.359932 2.048152 3.405613
[289] 2.094115 2.670801 3.672257 2.829378 2.622770 2.755119 3.949487 3.191897 3.403655
[298] 2.822619 2.849621 3.232000 2.239239 2.031237 3.709551 3.794981 2.964269 2.337420
[307] 3.427339 2.344190 3.362000 3.912588 3.420951 2.947922 3.184728 2.173259 3.271149
[316] 2.926523 2.883571 2.898221 3.631705 3.385374 2.083779 3.624535 3.774494 3.452251
[325] 3.922635 3.277668 3.996363 2.173574 2.031969 3.122913 2.864853 2.493077 2.204682
[334] 2.797468 2.919548 2.664990 3.991716 2.244455 2.231676 3.548801 3.925715 2.295617
[343] 3.350612 2.180727 2.880788 3.088446 2.905243 2.216631 3.654952 2.305906 3.507629
[352] 2.575571 2.562985 2.710327 2.788357 3.766948 2.173219 2.956216 3.546882 2.738688
[361] 2.603296 3.507473 3.929698 2.731363 3.025692 3.026686 2.621363 2.857946 3.667466
[370] 3.948852 2.697422 3.979490 2.473204 2.392021 2.987477 3.469977 3.176836 2.222916
[379] 3.551299 3.256679 3.541934 3.051749 2.391193 2.640895 3.465495 2.924979 2.217012
[388] 2.977353 3.407280 2.603286 3.778744 2.413910 3.606457 2.756039 3.808171 3.274811
[397] 3.907092 3.858524 2.085288 2.482097 2.573389 3.312762 2.971628 3.486146 3.293284
[406] 3.132844 2.303481 2.771884 3.015393 3.277188 2.941679 2.481365 2.193130 2.318667
[415] 2.820162 3.631782 3.135979 3.769292 3.759790 2.190188 3.789205 2.537375 2.490709
[424] 2.373372 3.433263 3.452605 3.188092 2.695100 2.878318 2.308094 2.128544 3.055415
[433] 3.258182 2.748111 3.741216 2.623718 3.134335 3.619225 3.206428 3.771856 2.121059
[442] 2.736096 2.581522 3.722913 3.405482 3.108372 2.264826 3.197247 2.890230 2.744098
[451] 3.865511 3.790807 3.608629 2.560825 3.731265 2.025492 2.899368 3.195058 2.471034
[460] 3.911730 2.731687 3.593911 2.702266 2.654844 2.606919 2.046090 2.559190 3.445816
[469] 3.217578 3.218765 3.705715 2.067280 3.754891 2.249893 3.466682 2.919506 3.854194
[478] 3.407955 3.433376 2.077168 3.719660 2.064889 3.096523 3.972399 2.856720 3.903453
[487] 3.142344 2.620556 2.806127 2.611029 2.767564 3.824011 2.936287 2.009581 3.024101
[496] 2.257305 2.833677 3.462514 3.199104 2.040301 2.363172 3.551644 3.612526 2.611930
[505] 3.156146 2.098094 3.879702 3.939784 2.569947 3.983748 2.612695 2.252280 2.374017
[514] 2.927339 2.099046 2.754386 3.394857 2.894617 2.107886 2.747617 3.220423 2.566434
[523] 3.240223 3.372539 3.150674 3.682545 3.050635 2.104853 3.639177 2.964202 2.955960
[532] 2.587896 2.893608 2.182181 3.952425 2.624209 2.631968 3.804051 3.828991 3.296230
[541] 3.890326 3.692126 2.468652 2.480452 2.129814 2.727162 3.311260 2.698144 3.044802
[550] 2.556288 2.973136 3.124453 3.950354 2.635932 3.942196 2.952486 2.052242 2.507722
[559] 3.825739 3.924928 3.184960 2.085447 2.754121 3.709696 3.352922 3.241102 3.030106
[568] 2.616632 2.426163 3.031755 2.927097 3.106903 2.619579 3.529550 2.259768 3.360798
[577] 3.420122 3.292172 2.507914 2.592781 2.762888 3.298596 3.241231 3.071329 2.477496
[586] 2.117015 3.324431 2.526936 3.528335 3.710070 3.268832 2.419218 3.059785 2.026687
[595] 2.996845 2.287508 3.522110 2.536549 3.637732 3.009265 3.323508 2.322827 3.219932
[604] 2.913942 2.194856 3.316801 2.408771 3.975937 2.424495 2.372593 3.936162 2.348808
[613] 2.487431 2.115482 2.174412 2.751236 2.335225 2.513667 3.334714 3.577065 2.431711
[622] 3.807677 3.721653 3.297705 2.968679 3.943218 2.777172 2.020256 3.919411 2.025720
[631] 2.231937 2.322247 2.178401 3.862908 2.599086 2.742460 3.613763 2.341518 2.154332
[640] 2.108338 3.957688 2.312757 2.435865 2.777558 2.424704 2.197730 2.648260 3.930130
[649] 3.298536 3.520007 3.820596 2.911657 2.919349 2.369423 2.614106 3.462244 2.533629
[658] 2.060231 2.248170 2.794997 2.032308 2.427261 2.898747 2.544656 3.282118 3.473619
[667] 3.408229 2.432161 2.826300 3.740850 3.988504 2.899310 3.369924 2.698727 3.955615
[676] 2.941528 2.388658 2.270501 2.595392 3.542870 3.607551 2.290938 3.597335 3.891193
[685] 3.406865 2.254553 2.429042 3.117934 3.537524 2.150339 3.668073 2.021571 3.838646
[694] 3.528124 2.600114 3.044467 2.538443 3.963835 3.343779 3.973384 3.259711 2.196727
[703] 3.407215 3.287100 2.659139 3.120794 3.123574 2.027499 2.154761 2.932777 3.714275
[712] 2.115376 2.511614 2.435213 3.630536 2.524426 3.268395 2.454139 2.206882 2.677176
[721] 2.303091 3.946637 2.642464 2.817032 2.103334 3.055083 2.624042 2.466906 3.594984
[730] 3.622354 2.457629 2.556736 2.008185 3.907153 2.379803 3.130619 2.577052 3.527095
[739] 3.364947 2.326627 2.190322 3.026715 3.749403 2.086865 2.322406 2.083619 2.997685
[748] 3.534270 3.222302 2.798390 3.731822 3.145327 3.724936 2.682607 3.330709 3.803648
[757] 3.598026 2.042255 2.621577 3.417986 2.090147 2.429382 3.080479 2.931406 3.666316
[766] 3.687842 3.794752 2.612679 2.434277 2.211080 3.588855 3.287702 3.820640 3.329308
[775] 3.811281 2.895364 2.923476 3.711414 3.408864 2.356313 3.790532 2.671648 2.860639
[784] 3.001603 2.501607 2.272224 2.519897 3.700921 3.002709 3.211645 2.268247 3.244728
[793] 2.970524 3.086987 3.486535 3.188821 2.937428 2.112923 2.742558 2.883942 2.204031
[802] 3.858861 3.690473 3.677587 2.910144 2.422590 3.249226 3.174314 2.577616 3.325207
[811] 3.576910 2.934369 3.812576 2.853111 2.631798 2.393923 3.977034 2.613453 2.905863
[820] 3.147328 3.885572 3.333348 3.884459 3.015449 3.853529 3.246359 2.547410 2.504726
[829] 3.341873 2.141223 3.923367 3.225120 2.879114 2.585154 2.168133 3.361611 3.413252
[838] 3.359384 2.824799 2.303277 2.537249 2.340689 3.939308 3.198269 2.067624 3.584507
[847] 3.568521 2.857178 2.947313 3.005177 2.211407 3.765567 3.900815 2.633975 3.866715
[856] 2.935907 3.680821 3.864761 3.370870 3.059530 3.573389 2.311139 2.372755 3.491877
[865] 3.395655 2.305772 3.289605 2.959034 3.220917 2.711930 3.236365 3.947636 2.734149
[874] 3.765392 2.443012 2.216904 3.593100 3.338195 3.189790 2.973668 2.476051 3.606860
[883] 3.431178 2.553765 3.435443 2.391645 2.814854 2.869893 2.152242 2.434799 2.108694
[892] 3.959800 3.568305 2.839860 2.655718 2.701637 2.643903 3.445083 3.196835 2.759733
[901] 2.207070 3.662692 3.359245 2.501345 3.769289 2.571965 3.371700 2.390206 3.336850
[910] 2.197374 2.725372 2.888732 2.044582 2.482600 2.334211 2.641208 2.088972 3.262926
[919] 3.895987 2.816219 2.877834 2.702481 3.389631 3.313000 3.917481 2.959654 2.580787
[928] 3.979322 3.642783 3.678813 2.542172 3.566290 3.162492 2.547413 3.209201 2.268793
[937] 2.001616 2.942817 3.202387 2.687960 2.859732 3.916186 3.409755 3.026521 3.523265
[946] 3.023305 2.329919 2.012723 3.939088 3.176342 3.043906 2.251255 2.477426 2.292342
[955] 2.634596 3.338837 2.065543 2.129009 2.700892 2.564697 3.456502 2.695993 2.428529
[964] 3.059795 3.707981 2.764431 2.135229 3.368364 3.842271 3.237610 2.259229 3.836547
[973] 2.447999 2.519087 3.465969 2.006225 2.578073 3.869987 3.056396 3.020982 2.742596
[982] 2.938689 3.175106 2.034125 2.520023 2.721818 3.412443 2.985365 2.811582 2.944410
[991] 2.578869 3.559594 2.781051 3.421633 3.053965 3.377630 2.574170 2.928157 2.912884
[1000] 2.715684
"
clean_text <- gsub("\\[.*?\\]", "", raw_text)
# Step 3: Convert to numeric vector
data <- as.numeric(unlist(strsplit(clean_text, "\\s+")))
data <- data[!is.na(data)]# Parameters
alpha <- 0.03 # Significance level
n <- 100 # Sample size
# Step 2: Take a random sample
sample <- sample(data, size = n, replace = FALSE)
# Parameters
a <- 2
b_true <- 3.99
n <- 100
alpha <- 0.03Let \(X_{(n)}\) denote the maximum of an i.i.d. sample of size \(n\) and it is our sufficent statistic and MLE. Then:
\[ F_{X_{(n)}}(x) = \left( \frac{x - 2}{b - 2} \right)^n, \quad 2 \le x \le b \]
Define:
\[ Y = \frac{X_{(n)} - 2}{b - 2} \sim \text{Beta}(n, 1) \]
Then:
\[ P(Y \le y) = y^n \Rightarrow P(X_{(n)} \le 2 + y(b - 2)) = y^n \]
Solving for \(b\), we get a \((1 - \alpha)\) upper confidence bound:
\[ P\left(b \ge 2 + \frac{X_{(n)} - 2}{(1 - \alpha)^{1/n}}\right) = 1 - \alpha \]
Thus, a two-sided \((1 - \alpha)\) confidence interval is:
\[ \left[ 2 + \frac{X_{(n)} - 2}{(1 + \alpha/2)^{1/n}}, \; 2 + \frac{X_{(n)} - 2}{(1 - \alpha/2)^{1/n}} \right] \]
x_max <- max(sample)
# Compute confidence interval
lower_bound <- 2 + (x_max - 2) / (1 + alpha/2)^(1/n)
upper_bound <- 2 + (x_max - 2) / (1 - alpha/2)^(1/n)
ci <- c(lower_bound, upper_bound)
ci## [1] 3.979195 3.979789We want to test:
\[ H_0: b = b_0, \quad H_1: b = b_1 \quad (b_0 \ne b_1) \text{(simple vs simple)} \]
Let’s assume \(b_1 >b_0\) Then: \[ H_0: b = b_0, \quad X_{(n)}\le b_0, \quad \text{and} \quad H_1: b = b_1,\quad X_{(n)}\le b_1 \] So any observation X >\(b_0\) is impossible under \(H_0\), but possible under \(H_1\) , so we should reject \(H_0\) if \(X_{(n)}\)> \(c\) , for some critical value \(c\). Since under \(H_1\), \(X_{(n)} \le b_1\), we should reject \(H_0\) if \(X_{(n)} \le c\), for some critical value \(c \in (b_0, b_1)\).
Find \(c \in [2, b_0] \cup (b_0,
b_1]\) such that
\[
P(X_{(n)} > c \mid H_0 \text{ TRUE}) = \alpha
\]
\[ \Rightarrow \quad P(X_{(n)} \leq c) = 1 - \alpha \]
\[ \Rightarrow \quad \left( \frac{c - 2}{b_0 - 2} \right)^n = 1 - \alpha \quad \Rightarrow \quad c = 2 + (1 - \alpha)^{1/n} (b_0 - 2) \]
\[ P(X_{(n)} \le c \mid H_0) = 1 - \alpha \]
We have:
\[ P\left( \frac{X_{(n)} - 2}{b_0 - 2} \le y \right) = y^n \Rightarrow y = \left(1 - \alpha\right)^{1/n} \]
Then:
\[ \frac{X_{(n)} - 2}{b_0 - 2} \le \left(1 - \alpha\right)^{1/n} \Rightarrow c = 2 + (b_0 - 2) \cdot \left(1 - \alpha\right)^{1/n} \]
# Step 2: Compute the critical value c
c <- a + (b_true - a) * (1 - alpha)^(1 / n)
# Step 3: Make the decision
reject_H0 <- (x_max > c)
# Step 4: Print results
cat("Maximum statistic X(n):", x_max, "\n")## Maximum statistic X(n): 3.97949cat("Critical value c:", c, "\n")## Critical value c: 3.989394cat("Reject H0?", reject_H0, "\n")## Reject H0? FALSE# Optional: wrap in a list if you'd like to store
results <- list(
x_max = x_max,
critical_value = c,
reject_H0 = reject_H0
)Hypotheses:
\[ H_0: b = b_0 \quad \text{vs} \quad H_1: b < b_0 \quad (\text{or } b > b_0) \]
This is most natural because for Uniform distributions, we typically test the endpoint based on the maximum order statistic.
To construct a test, we use the likelihood ratio:
\[ \lambda = \frac{\sup_{b < b_0} L(b)}{\sup_{b \ge b_0} L(b)} = \frac{L(b)}{L(X_{(n)})}, \quad \text{if } X_{(n)} \le b_0 \quad (H_1: b < b_0 \text{ (1st scenario)}) \]
Since \(L(b)\) is decreasing in \(b\), for a fixed sample, the MLE of \(b\) is:
\[ \hat{b}_{MLE} = X_{(n)} = \max X_i = T(X) \]
So:
\[ \lambda = \left( \frac{X_{(n)} - 2}{b_0 - 2} \right)^n, \quad \text{for } X_{(n)} \le b_0 \]
Reject \(H_0\) if \(\lambda < c \iff X_{(n)} < c'\) for some threshold \(c'\).
So the UMP Test is:
For a test of size \(\alpha\), we find \(c_1\) such that:
\[ P(X_{(n)} < c_1 \mid H_0 \text{ TRUE}) = \alpha \]
Set:
\[ \left( \frac{c_1 - 2}{b_0 - 2} \right)^n = \alpha \Rightarrow c_1 = (b_0 - 2) \alpha^{1/n} + 2 \]
2nd Scenario:
For a test of size \(\alpha\), we find \(c_2\) such that:
\[ P(X_{(n)} > c_1 \mid H_0 \text{ TRUE}) = \alpha \Rightarrow P(X_{(n)} < c_2) = 1 - \alpha \]
Set:
\[ \left( \frac{c_2 - 2}{b_0 - 2} \right)^n = 1 - \alpha \Rightarrow c_2 = (b_0 - 2)(1 - \alpha)^{1/n} + 2 \]
# Function to compute UMP test threshold and decision
ump_test <- function(data, b_0, alpha = 0.03, alternative = c("less", "greater")) {
n <- length(data)
x_max <- max(data)
alternative <- match.arg(alternative)
if (alternative == "less") {
# H1: b < b0 --> reject if X(n) < c1
c1 <- (b_0 - 2) * alpha^(1/n) + 2
reject <- x_max < c1
cat("Alternative: H1: b < b0\n")
cat(sprintf("Critical value c1 = %.4f\n", c1))
} else if (alternative == "greater") {
# H1: b > b0 --> reject if X(n) > c2
c2 <- (b_0 - 2) * (1 - alpha)^(1/n) + 2
reject <- x_max > c2
cat("Alternative: H1: b > b0\n")
cat(sprintf("Critical value c2 = %.4f\n", c2))
}
cat(sprintf("X(n) = %.4f\n", x_max))
if (reject) {
cat("Result: Reject H0\n")
} else {
cat("Result: Fail to reject H0\n")
}
}
ump_test(data = sample, b_0 = 3.99, alpha = 0.03, alternative = "less")## Alternative: H1: b < b0
## Critical value c1 = 3.9214
## X(n) = 3.9795
## Result: Fail to reject H0ump_test(data = sample, b_0 = 3.99, alpha = 0.03, alternative = "greater")## Alternative: H1: b > b0
## Critical value c2 = 3.9894
## X(n) = 3.9795
## Result: Fail to reject H0Let: - \(X_1, \ldots, X_n \overset{i.i.d.}{\sim} \text{Uniform}(2, b)\) - We test the hypotheses:
\[ H_0: b = b_0 \quad \text{vs} \quad H_1: b \ne b_0 \]
The maximum likelihood estimator is \(\hat{b} = X_{(n)}\). The likelihood ratio statistic is:
\[ \Lambda(x) = \frac{L(b_0)}{L(\hat{b})} = \begin{cases} \left(\frac{X_{(n)} - 2}{b_0 - 2}\right)^n, & X_{(n)} \le b_0 \\ 0, & \text{otherwise} \end{cases} \]
We reject \(H_0\) when:
\[ X_{(n)} < c_1 \quad \text{or} \quad X_{(n)} > c_2 \]
Critical values:
\[ c_1 = 2 + (b_0 - 2) \cdot \left( \frac{\alpha}{2} \right)^{1/n}, \quad c_2 = 2 + (b_0 - 2) \cdot \left(1 - \frac{\alpha}{2} \right)^{1/n} \]
x_max <- max(sample)
c1 <- a + (b_true - a) * (alpha / 2)^(1/n)
c2 <- a + (b_true - a) * (1 - alpha / 2)^(1/n)
reject_H0 <- (x_max < c1) | (x_max > c2)
list(x_max = x_max, c1 = c1, c2 = c2, reject_H0 = reject_H0)## $x_max
## [1] 3.97949
##
## $c1
## [1] 3.908156
##
## $c2
## [1] 3.989699
##
## $reject_H0
## [1] FALSE# Example: Assume binomial_data is already loaded as a numeric vector of 0s and 1s
# Uncomment the following line if reading from a CSV file:
# binomial_data <- read.csv("your_data.csv")$your_column_name
raw_text= " [1] 38 40 35 42 34 39 35 36 37 39 37 36 39 37 38 38 34 40 43 40 40 33 37 41 40 38 37 38 38
[30] 41 37 37 40 39 34 38 41 33 36 40 36 34 38 37 34 33 41 37 36 39 36 31 38 38 35 42 33 35
[59] 40 38 38 37 38 36 37 37 37 39 42 39 40 33 41 38 34 32 37 36 37 30 34 39 43 31 43 37 39
[88] 39 35 36 33 42 39 33 40 41 37 35 39 42 44 32 38 36 38 42 38 36 38 37 30 34 39 32 34 34
[117] 34 36 38 39 36 36 35 33 43 37 36 32 36 37 38 41 37 40 34 37 36 38 35 38 37 37 35 33 41
[146] 42 38 38 38 35 41 35 43 44 35 35 39 34 40 38 37 39 38 39 36 32 39 40 39 41 38 36 33 41
[175] 27 46 40 42 40 39 39 36 36 39 34 37 32 34 44 40 36 33 41 40 39 41 31 38 36 35 38 41 40
[204] 33 36 36 37 31 41 41 38 35 39 39 36 31 36 38 34 38 38 34 35 39 34 38 39 37 38 30 35 38
[233] 37 33 41 42 38 40 33 39 32 35 39 41 37 36 35 31 29 40 34 40 36 41 41 38 34 36 38 35 33
[262] 37 37 36 38 42 41 38 41 33 37 38 37 38 35 33 39 43 39 39 40 35 32 39 40 36 39 44 43 34
[291] 38 36 37 39 41 31 38 34 37 35 35 33 34 37 41 44 40 37 40 34 34 37 38 34 40 39 38 38 37
[320] 39 39 36 43 31 37 32 38 41 38 39 41 35 40 42 37 42 39 37 37 33 41 36 42 35 40 35 38 34
[349] 38 33 32 36 31 38 40 36 38 39 39 36 33 40 32 43 38 38 38 38 37 39 36 39 39 37 39 34 35
[378] 36 35 37 38 34 40 40 42 37 41 36 31 36 38 37 40 34 32 36 39 39 39 37 39 36 40 42 34 29
[407] 33 38 38 35 38 43 36 39 35 40 43 36 39 33 39 36 40 35 42 35 41 41 37 38 34 38 38 35 39
[436] 37 38 33 36 38 36 37 40 34 38 40 37 39 36 33 37 34 37 36 35 42 36 39 40 37 41 38 40 42
[465] 40 35 33 38 36 35 35 41 40 40 38 36 31 35 40 38 34 33 36 38 37 36 39 34 37 33 33 33 37
[494] 40 41 27 35 42 39 34 38 33 37 35 40 42 38 31 38 40 33 37 38 39 39 39 36 43 40 36 39 38
[523] 35 36 40 35 42 37 31 35 37 35 32 42 39 35 38 38 39 44 38 40 43 38 37 36 39 39 36 39 39
[552] 39 38 33 41 40 43 38 35 41 37 29 36 41 36 30 38 32 42 37 37 38 43 40 41 39 37 35 32 39
[581] 45 36 39 41 40 37 40 41 40 39 41 39 37 37 40 39 40 39 35 40 39 33 39 32 34 40 41 37 36
[610] 36 35 36 40 40 38 35 33 32 35 40 36 34 37 38 37 41 41 34 34 37 45 37 37 35 35 39 36 35
[639] 40 39 38 34 37 40 39 41 31 35 33 34 32 41 40 38 39 34 35 36 38 44 38 37 41 39 41 35 39
[668] 39 37 34 41 31 41 35 37 39 33 36 32 40 40 36 41 36 36 35 36 39 37 35 38 40 41 39 38 33
[697] 38 37 37 32 38 33 38 38 34 39 40 36 38 35 41 36 35 35 41 40 38 34 35 31 34 36 38 43 33
[726] 38 38 36 34 38 41 34 41 38 40 36 37 36 38 28 35 39 40 38 37 31 34 40 36 37 39 38 40 37
[755] 36 43 42 41 42 35 37 39 40 34 36 39 35 34 35 40 39 39 39 39 36 37 40 40 35 40 38 36 41
[784] 38 37 38 40 39 35 42 41 35 34 38 38 38 38 41 32 36 34 44 37 41 39 38 40 35 43 38 38 35
[813] 34 36 29 37 38 33 36 41 38 37 38 41 35 37 35 40 40 38 36 39 41 39 40 34 30 36 37 26 38
[842] 35 36 33 34 39 39 40 40 39 38 39 40 39 37 40 40 37 38 34 37 35 43 38 37 42 38 40 36 41
[871] 30 42 36 34 35 39 28 34 33 43 37 35 31 42 37 34 41 40 39 40 33 35 39 37 34 38 39 41 39
[900] 38 36 41 38 39 38 38 39 36 32 36 38 43 41 35 35 39 38 37 34 41 38 39 44 36 39 42 37 37
[929] 39 40 39 37 35 42 36 38 40 35 35 38 37 38 37 43 36 40 34 43 38 36 40 43 39 40 35 38 40
[958] 34 32 40 42 34 39 43 39 39 37 39 38 39 38 34 38 42 41 38 34 38 35 42 35 42 36 29 35 35
[987] 36 36 34 34 37 33 38 38 40 38 41 44 32 43"
clean_text <- gsub("\\[.*?\\]", "", raw_text)
binomial_data <- as.numeric(unlist(strsplit(clean_text, "\\s+")))
binomial_data <- binomial_data[!is.na(binomial_data)]
# Set seed for reproducibility
set.seed(123)
# Step 1: Sample 100 observations from your binomial_data (size 1000)
sample_data <- sample(binomial_data, size = 100, replace = FALSE)The \(100(1 - \alpha)\%\) confidence interval for the success probability \(p\) of a Binomial\((50, p)\) distribution, based on the normal approximation, is given by:
\[ \hat{p} \pm z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n \cdot 50}} \]
Where: - \(\hat{p} = \frac{\bar{x}}{50}\) is the estimator of \(p\) - \(z_{\alpha/2}\) is the critical value from the standard normal distribution - \(n\) is the number of Binomial observations (sample size) - Each observation is based on 50 trials
n <- length(sample_data) # Number of observations
trials <- 50 # Binomial trials per observation
xbar <- mean(sample_data) # Sample mean
p_hat <- xbar / trials # Estimated p
# Standard error for p_hat
SE <- sqrt(p_hat * (1 - p_hat) / (n * trials))
# 97% confidence interval
z <- qnorm(0.985)
lower<- p_hat - z * SE
upper <- p_hat + z * SE
cat("Estimated p:", p_hat, "\n")## Estimated p: 0.7458cat("97% CI for p: [", lower, ",", upper, "]\n")## 97% CI for p: [ 0.7324374 , 0.7591626 ]Let:
n <- 100
p <- 0.75
alpha <- 0.03
for (c in 0:n) {
prob <- pbinom(c, n, p)
if (prob > alpha) {
cat("Critical Value c =", c - 1, "\n")
cat("P(X <= c) =", pbinom(c - 1, n, p), "\n")
break
}
}## Critical Value c = 66
## P(X <= c) = 0.02759456pbinom(66, size = 100, prob = 0.75)## [1] 0.02759456pbinom(67, size = 100, prob = 0.75)## [1] 0.04459633dbinom(67, size = 100, prob = 0.75)## [1] 0.01700176We are testing:
\[ H_0: X \sim \text{Bin}(100, 0.75) \quad \text{vs.} \quad H_1: X \sim \text{Bin}(100, 0.5) \]
Let \(Y = \sum X_i\), then \(Y \sim \text{Bin}(100, 0.75)\) under \(H_0\).
We want a randomized test such that:
\[ \alpha = P(Y \leq c) + k \cdot P(Y = c+1) \]
From R calculations:
So to find \(k\):
\[ 0.03 = 0.02759456 + k \cdot 0.01700176 \]
Solving:
\[ k = \frac{0.03 - 0.02759456}{0.01700176} = 0.14148182 \]
The test function \(\phi(y)\) is:
\[ \phi(y) = \begin{cases} 1, & y < 67 \\ 0.1415, & y = 67 \\ 0, & y > 67 \end{cases} \]
# Set seed for reproducibility
set.seed(123)
# Step 1: Sample 100 observations from your binomial_data (size 1000)
sample_data <- sample(binomial_data, size = 100, replace = FALSE)
# Sum of observed successes
Y_obs <- sum(sample_data)/50
cat("Observed Y =", Y_obs, "\n")## Observed Y = 74.58# Randomized test decision function
phi <- function(y) {
if (y < 67) return(1)
else if (y == 67) return(0.1415)
else return(0)
}
# Compute decision
decision <- phi(Y_obs)
cat("Test function φ(Y) =", decision, "\n")## Test function φ(Y) = 0# Interpretation
if (decision == 1) {
cat("Reject H0 with probability 1.\n")
} else if (decision == 0) {
cat("Do not reject H0.\n")
} else {
cat("Reject H0 with probability", decision, "(randomized).\n")
}## Do not reject H0.We compare the 97% confidence intervals and two-sided hypothesis tests for three distributions: Weibull, Rayleigh, and Uniform.