Evaluating the accuracy of time phasing in missile and munition development using continuous distribution functions

In defense acquisition, one of the most difficult and consequential tasks is the early estimation of program development costs and how those costs are planned throughout the program lifecycle. This challenge is especially pronounced during the Research, Development, Test, and Evaluation (RDT&E) phase, where spending patterns often shift in response to technical risk, schedule changes, and evolving system requirements. This process, known as time phasing, supports financial planning, congressional budget requests, contract design, and risk management.

Misaligned phasing—whether too aggressive or too conservative—can trigger a series of problems. Premature funding may lead to inefficient portfolio execution and insufficient resources may delay design reviews or test events. Stakeholders may misinterpret program progress. Inaccurate phasing is more than a budget issue; it can undermine program credibility, execution, and strategic value.

To mitigate these risks early in the RDT&E phase, the Department of Defense (DoD) and its service branches often use a set of heuristics to guide early planning. One of the most common heuristics is the “60/40 rule,” which assumes that 60% of development costs will occur by the midpoint of the program timeline. While this rule offers simplicity, it is not empirically grounded and may mislead when applied across diverse system types. Missile and munition programs, in particular, often diverge from this pattern.

Missile and munition programs differ from other commodities in specific ways. They tend to follow shorter development timelines, involve fewer configuration changes, and rely on mission-specific performance goals that shape testing schedules. Some are heavily front-loaded due to early prototyping, while others are less aggressive due to costly integration or range testing. Applying a standard S-curve without accounting for these patterns introduces risk into budget formulation and contract structure.

Despite these distinctions, few studies have focused on the time phasing behavior of missile development. Most prior research emphasizes broader categories, such as aircraft (Brown et al., 2015), space systems (Burgess, 2006), or general R&D programs (Brown et al., 2002; Bills, 2023). These studies test continuous distributions—including Rayleigh, Weibull, and Beta curves—as alternatives to heuristics. However, little work has examined how well these models fit the cost progression unique to missile programs.

This study analyzes normalized monthly cost data from 21 DoD missile and munition development programs, drawn from the Cost Assessment Data Enterprise (CADE) and the Earned Value Management Central Repository (EVM-CR). The data are refined to isolate non-recurring engineering (NRE) costs in the RDT&E phase. The analysis proceeds in three stages: first, testing the empirical validity of the “60/40 rule” for missile and munition systems; second, comparing the fit of Rayleigh, Weibull, and Beta cumulative distribution functions to observed expenditure profiles; and third, developing regression models that estimate distribution parameters from basic program traits such as weapon type, acquisition category, development duration, and service branch. These tools enable early-stage cost analysts to generate data-informed time phasing curves prior to contract award or detailed schedule definition.

The findings support a shift from generalized heuristics to tailored, statistically grounded models. By improving phasing accuracy, these methods enhance budget credibility, reduce systemic error, and improve alignment between financial planning and technical execution. More broadly, the work contributes to the growing integration of quantitative methods into defense cost analysis.

Accurate time phasing of RDT&E expenditures is essential for effective defense acquisition, particularly as the Department of Defense (DoD) adopts more agile procurement models and shorter development timelines. Precision in early budget planning now carries heightened importance, as phasing profiles influence Program Objective Memorandum (POM) formulation, annual budget submissions, congressional justification materials, and contract funding strategies. When these projections are misaligned—especially early in a program—they can trigger cascading disruptions in execution, oversight, and financial control.

The Air Force Cost Analysis Handbook (AFCAH, 2007) outlines three dominant approaches to time phasing: milestone-based (schedule-driven), analogy-based, and S-curve methods. Each of these strategies has its own strengths and applicability depending on program maturity and data availability.

The milestone-based method aligns cost distribution with planned program events such as Preliminary Design Review (PDR), Critical Design Review (CDR), and key test activities. When schedules are well-defined, this method offers high fidelity and traceability. However, in early-stage programs—prior to Milestone B or contract award—milestone sequencing is often tentative, making this approach impractical.

The analogy method relies on historical data from similar completed programs. Analysts scale the cost profile of a past program to fit the cost and duration of the current effort. While expedient, this method is subject to several limitations: program uniqueness, technology maturity differences, and classification discrepancies can undermine analogy validity, leading to unreliable forecasts.

The third and most flexible method is the use of S-curve models, also known as cumulative distribution functions (CDFs). These mathematical functions express cumulative expenditures as a function of normalized time, where both axes range from 0 to 1. S-curve models assume that project execution follows a nonlinear progression—typically starting slowly, accelerating toward the middle, and tapering toward the end. These curves are typically parameterized to allow flexibility in shape and skew, enabling better alignment with empirical program data even in the absence of detailed schedule milestones.

A widely applied heuristic in early-stage time phasing is the “60/40 rule,” which assumes that 60% of program expenditures are incurred by the 50% schedule mark. This rule emerged from simplified approximations of the Rayleigh curve and has since become institutionalized in DoD planning, often serving as a default when no detailed information is available (Lee et al., 1997). However, its empirical validity across different program types has rarely been scrutinized, and recent research has shown significant deviations. This warrants specific examination in missile and munition programs where development patterns differ structurally from more complex platforms like aircraft or satellites.

Further, the 60/40 rule is not always consistently applied, with variations in interpretation across different organizations and phases. In some cases, the 60% is applied to 50% of the schedule, while in others, it is interpreted in terms of work completion—a distinction that can cause significant variance in funding allocations.

S-curve modeling typically relies on one of three continuous probability distributions: Rayleigh, Weibull, or Beta. Each model defines a cumulative distribution function F (t) that describes the fraction of total program cost expected to be expended by time t, where t is normalized between 0 and 1.

This distribution models gradual initial expenditures that increase toward a peak and then taper off — producing a characteristic S-shaped cumulative cost curve. Crucially, the distribution is defined by a scale parameter, β, that determines the rate at which costs accumulate over time: smaller values front-load spending, while larger values delay it. For example, achieving 60% of total expenditure by the program’s midpoint (t = 0.5) requires selecting a β such that the distribution reaches 97% completion by t = 1 (F(1) = 0.97). While the Rayleigh model captures many realistic expenditure profiles it is limited by its fixed form making it less flexible for modeling strongly front- or back-loaded programs (Lee et al., 1997).

Although simple to implement, the Rayleigh model lacks the flexibility needed for modern missile programs, where asymmetric cost phasing is common due to variable testing sequences, integration milestones, and procurement cycles. Because the Rayleigh curve’s shape is fixed by a single scale parameter and assumes a smooth, gradually accelerating expenditure profile, it cannot accommodate highly front-loaded or back-loaded spending patterns. As a result, it may misrepresent actual cost dynamics in programs with irregular or event-driven funding profiles.

The Rayleigh distribution is a special case of the Weibull distribution with a fixed shape. The Weibull distribution generalizes the Rayleigh function by introducing a flexible shape parameter α, in addition to the scale parameter β, which allows for curve skewness:

When α is fixed at a constant equal to 2, the Weibull distribution reduces to the Rayleigh form, which produces the classic 60/40 expenditure rule when satisfying the assumptions discussed previously. When α is less than 2, the curve becomes increasingly front-loaded, concentrating more cost early in the schedule, compared to the Rayleigh. When α exceeds 2, the curve is backloaded, delaying cost accumulation into later phases, compared to the Rayleigh. This added flexibility makes the Weibull distribution well-suited to modeling the idiosyncrasies of missile development, including programs with delayed testing or late-stage system integration. Figures 1 and 2 demonstrate the flexibility of the general Weibull distribution when adjusting the parameters α and β. Note the dotted line in Figure 1 reflects the special case Rayleigh distribution approximating the 60/40 rule.

Brown et al. (2002) and Unger (2001) found that Weibull distributions offer a better empirical fit for RDT&E programs than Rayleigh models, particularly in scenarios involving significant cost skew. These findings have been echoed in more recent studies (e.g. Brown et al., 2015), although missile-specific validation has not yet been demonstrated.

The Beta distribution is the most flexible of the three models, possessing two shape parameters α and β, unlike the Weibull distribution’s use of one shape parameter and one scale parameter. While both influence the curve’s shape, the Weibull’s scale parameter stretches the distribution across a wider range, whereas the Beta’s two shape parameters control its behavior entirely within the 0 to 1 interval. Because of this, F(1) always equals 1 in a Beta distribution, whereas the Weibull distribution only approaches 1 asymptotically—unless the scale parameter is exactly zero (which is not meaningful in practice), F(1) is always less than 1. The Beta distribution can approximate a wide variety of curves depending on the relative magnitudes of its shape parameters. When α = β, the curve is symmetric. When α > β, the profile is skewed left (back-loaded), and when α < β, it is skewed right (front-loaded). The flexibility of the Beta distribution can be demonstrated by the extreme case where α = β = 1, where the distribution collapses to a straight line with an unchanging slope of 1 throughout the CDF. Figures 3 and 4 graphically demonstrate the flexibility of the beta distribution across various values of α and β for both symmetric and asymmetric scenarios.

Despite its flexibility, the Beta distribution often presents estimation challenges. It is more computationally demanding and sensitive to sparse or irregular data. As Burgess (2006) and Brown et al. (2015) observed, while Beta models may provide excellent fits in some cases, their parameters can be unstable or non-convergent—especially when used for regression-based forecasting in early acquisition stages.

Historical applications of these S-curve models have largely centered on aircraft (Brown et al., 2015), software systems (Putnam, 1978), and satellite programs (Burgess, 2006). Brown et al. (2002) derived regression models for Weibull parameters using program characteristics like duration and service branch from 128 various research and development programs, enhancing early-stage applicability, however the analysis lacked commodity specificity. Bills (2023) extended this work by using the Comprehensive Cost and Requirements enterprise data to validate S-curve models across 89 Air Force RDT&E programs. Despite the rich history of S-curve analysis in DoD acquisitions, few studies have rigorously tested these distributions on missile and munition data, and no study exclusively.

This omission is critical, given that missile development programs are inherently different in lifecycle, technology readiness levels, and procurement pathways. As early as Weida (1977), researchers hypothesized that using commodity-specific data would improve curve fit and reduce confidence interval width. Yet until now, there has been no systematic effort to isolate missile and munition programs and evaluate which distribution best describes their time phasing behavior—or to estimate model parameters using missile-relevant features.

This section details the empirical methodology used to evaluate the applicability of continuous distribution functions in modeling RDT&E expenditure profiles for missile and munition programs. The approach is grounded in statistical modeling and regression-based parameter estimation, using actual program cost data to test the validity of heuristic rules and the accuracy of fit time phasing curves. The methodological framework is structured in sequential stages: data collection and cleaning, normalization and adjustment, heuristic evaluation, model fitting, and regression-based parameter estimation.

We derive our dataset from two primary repositories maintained by the Department of Defense:

1. The Cost Assessment Data Enterprise (CADE), which includes DD Form 1921 Cost Data Summary Reports (CDSRs), and

2. The Earned Value Management Central Repository (EVM-CR), which houses Format 1 reports detailing monthly actual costs.

The analysis targets missile and munition development programs that meet several criteria:

1. The contract was identified as RDT&E non-recurring effort,

2. The program was at least 92.5% complete, based on the findings of Tracy and White (2011) which finds no statistical difference in reports past the 92.5% completion point, and

3. Monthly Actual Cost of Work Performed (ACWP) data was available in a continuous, granular format.

We retained 21 unique program data points from an initial pool of 51 missile and munition programs after applying our filters and conducting data quality checks. Several programs contained multiple RDT&E efforts that were reported as distinct line items but represented continuous development sequences. These were consolidated where appropriate. We excluded programs with disjointed or production-centric profiles. Table 1 lists the missile and munition programs used in this study. Table 2 details aggregate program characteristics. Characteristics marked with * indicate that multiple data points were extracted from a unique program. For example, the Patriot Missile System officially entered the Army inventory in 1981 and provides several efforts over its 40+ year history.

We extracted and aligned the Actual Cost of Work Performed (ACWP) values from Format 1 reports with reporting periods. Programs frequently exhibited anomalies in early months, such as abnormally high or zero expenditures, due to cost loading issues, delayed obligation recognition, or data entry errors. We corrected these anomalies through linear interpolation or backward smoothing where justified by adjacent period trends and informed by comparison across similar program types. Next, we calculated nonrecurring engineering (NRE) cost weights using 1921 report data for each WBS line item and reporting period, and then applied to monthly ACWP figures to isolate development-only costs. When multiple 1921 submissions existed for a program, the NRE ratios were linearly interpolated or assumed constant between reporting intervals.

We then normalized the final monthly ACWP values, adjusted for NRE. Time was normalized between 0 and 1 by dividing each month’s elapsed time since the start by total program duration. Similarly, we divided cumulative costs by the program’s final cumulative total to yield a normalized expenditure trajectory. This process facilitated direct comparison across programs of different durations and magnitudes.

To assess the empirical accuracy of the 60/40 heuristic, we calculated interpolated expenditure values at the 50% schedule mark for each program. Because most programs did not report a data point exactly at 50%, we estimated mid-schedule expenditures using linear interpolation between adjacent months. We then averaged these values across the sample and grouped them by weapon type and acquisition characteristics to identify trends or systematic deviations.

We further evaluated the accuracy of the heuristic by fitting a Rayleigh distribution parameterized to reflect the 60/40 rule per Lee et al. (1997) by fixing the general Weibull distribution with α = 2 and truncating at 97% by setting β = $13.5$ ≈ 0.535. We compared the fit curve against actual program profiles using Mean Absolute Percentage Error (MAPE) and Root Mean Squared Error (RMSE) as our primary performance metrics.

We fit each program’s normalized cost curve to three cumulative distribution functions: Rayleigh, Weibull, and Beta. We implemented these fits in Python using SciPy’s optimize.minimize function with specified bounds and convergence criteria. Our objective function minimized the Sum of Squared Errors (SSE) between the actual cumulative expenditure curve and the model-predicted curve.

The Rayleigh and Weibull models are expressed as:

For Rayleigh, α is fixed at 2, reducing the function to a single-parameter model with scale β. For Weibull, both parameters α (shape) and β (scale) are estimated.

The Beta CDF is calculated via the regularized incomplete beta function:

Where B(α, β) is the beta function, and x is a dummy variable for integration. Due to the nonlinearity of this function and potential convergence issues, parameter estimation for the Beta model was conducted using constrained bounded optimization with parameter ranges α, β > 0.1.

To enhance the utility of these models for forward-looking estimation, we employed Ordinary Least Squares (OLS) regression to derive empirical relationships between program characteristics and the optimized distribution parameters.

The general regression model form is:

Where:

1. θ: either the shape or scale parameter for Weibull or Beta distributions,

2. γ: intercept term,

3. X_i: program characteristic variables (e.g. ACAT level, service branch, weapon type, duration),

4. δ_i: estimated coefficients.

We conducted separate regressions for each parameter of the Weibull and Beta distributions independently. Due to the small sample size, we limited models to one or two predictors and interaction terms. We evaluated all models for influential observations using Cook’s Distance, applying a conservative threshold of 0.5. No observations exceeded this threshold—the highest Cook’s D value was 0.21—and we retained the full sample for all final regressions. We confirmed that residuals exhibited no significant departures from normality or homoscedasticity, and that standard OLS assumptions were met. These procedures yielded parsimonious and generalizable specifications with acceptable predictive performance and low parameter variance.

The final stage of this research involves generating predicted S-curves using the regression-derived parameter estimates for each program, rather than direct optimization. These regression-generated curves can be compared against actual cost profiles using MAPE to determine predictive accuracy. We compare performance across the three models to identify the most stable and accurate distribution for early-phase use.

We organize results into three core segments: (1) evaluation of the 60/40 heuristic, (2) performance of Rayleigh, Weibull, and Beta distribution fits, and (3) regression-based estimation of model parameters. We assess the accuracy of each approach using the interpolated program data, and we report the performance metrics to support comparative analysis across models and estimation methods.

A key initial objective was to test whether the widely adopted 60/40 rule, asserting that 60% of development costs are incurred by 50% of the schedule, holds true for missile and munition programs. We interpolated the expenditure percentage at 50% normalized schedule completion for each of the 21 programs using linear interpolation between the closest adjacent months. The average interpolated value across the dataset is approximately 60.5%, appearing at first glance to support the rule’s broad applicability.

However, further inspection revealed substantial variation. The range of expenditures at 50% schedule extended from as low as 41.0% to as high as 90.1%, with a standard deviation exceeding 13% points. This high degree of variability undermines the predictive reliability of the 60/40 rule when applied to individual missile programs. Table 3 displays the descriptive analysis of the 60/40 rule in the data set.

When stratified by commodity type, distinct patterns emerge. Table 4 shows that Air Force programs load fewer expenditures in the early phases than the 60/40 heuristic suggests, while Navy programs generally align with it. Army and Missile Defense Agency (MDA) programs exceed the heuristic by roughly 10% points, indicating more front-loaded spending.

These service-level differences reflect underlying variation in commodity types. Surface-to-Air programs—common in Army and MDA portfolios—exhibit the most frontloading. In contrast, Air-to-Air and Air-to-Surface programs, which dominate the Air Force sample, spend less aggressively early in the schedule. This pattern suggests that frontloading behavior stems not only from service practices but also from the nature of the weapon system.

ACAT level and program origin (upgrade vs. new start) show smaller effects, with both categories trending slightly more front-loaded than the 60/40 rule. The program start year does not reveal a consistent pattern, indicating that reforms such as the 2009 WSARA may not significantly change expenditure timing in missile and munition programs. However, given the limited sample size and the absence of a direct test for WSARA effects, this question warrants further investigation.

These conditional variations confirm that while the 60/40 rule approximates the mean behavior, it lacks generalizability as a deterministic forecasting tool for missile program budgeting. To further quantify its limitations, Figure 5 displays a Rayleigh curve configured to represent the 60/40 profile, as per Lee et al. (1997), is compared to actual program data using MAPE and RMSE (reported as a percentage point error, as the dependent variable itself is a percentage).

The 60/40 Rayleigh model produces an average MAPE of 22.17% and an RMSE of 9.17%, indicating reasonable predictive accuracy across the sample (Lewis, 1982). These error levels suggest that while the curve captures general trends in expenditure acceleration, it lacks the flexibility to reflect program-specific cost dynamics. Given these limitations, we believe more adaptive or data-driven models can offer improved fidelity.

To explore alternatives to heuristic modeling, we fit Rayleigh, Weibull, and Beta distributions to the full set of normalized cumulative cost profiles. Using SciPy’s “optimize.minimize” function, we applied non-linear least squares optimization to estimate a single set of parameters for each distribution that minimized the sum of squared errors across all programs. This approach focuses on identifying general distributional shapes that best approximate overall expenditure behavior, without incorporating the regression-based parameter tuning based on programmatic characteristics that we explore later.

We assess fit quality using MAPE and RMSE, which capture proportional and absolute deviations between the fit curve and the aggregated empirical data and also provide Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) calculations to evaluate distributional fit. Table 5 summarizes these results.

When fit globally to the full dataset, all three distributions—Rayleigh, Weibull, and Beta—outperform the 60/40 heuristic. While the Beta distribution achieves slightly better RMSE (7.54%) and marginally lower AIC and BIC values than the Weibull, the Weibull distribution delivers the lowest MAPE (19.49%), indicating superior proportional accuracy in predicting real expenditure behavior. Further, the Beta distribution’s MAPE (21.75%) was the worst of the three, and only slightly outperformed the 60/40 heuristic.

Given its strong overall performance, interpretability, and computational simplicity, the Weibull distribution remains the preferred choice. Its shape (α = 0.47) and scale (β = 1.65) parameters are easily applied and provide a more accurate and defensible planning heuristic in missile and munition programs than the traditional 60/40 rule, which trails with a higher MAPE of 22.17% and weaker model selection statistics. Even modestly tuned, historically informed distributions offer significant improvements over legacy heuristics in modeling cost phasing.

After establishing each distribution’s baseline performance when globally fit to the aggregated expenditure profile, we explored whether model accuracy could be improved by adjusting distribution parameters based on observable program characteristics. This approach builds on prior research demonstrating the value of linking S-curve parameters to programmatic factors (Brown et al., 2002, 2015; Burgess, 2006; Bills, 2023).

We began by applying non-linear least squares optimization—again using SciPy’s “optimize.minimize”—to estimate the best-fitting parameters for each individual program across the Rayleigh, Weibull, and Beta distributions. These program-level parameter estimates served as dependent variables in a series of Ordinary Least Squares (OLS) regressions, in which we tested whether characteristics such as commodity class (e.g. air-to-air, surface-to-air), service affiliation, ACAT level, and upgrade status significantly explained variation in shape and scale parameters.

To ensure model parsimony and mitigate overfitting given our limited sample size (n = 21), we restricted each regression to a maximum of two predictors and their interaction term, retaining variables only if they achieved significance at the 10% level. We also conducted standard diagnostic tests—including the Anderson-Darling test for residual normality and the Breusch-Pagan test for homoscedasticity—to validate regression assumptions and confirm model reliability.

This parameter adjustment process enabled us to construct functional forms for each distribution, where key coefficients were tied directly to observable program attributes. The resulting equations are provided in the Appendix. We then applied these adjusted models across the dataset and evaluated their performance using MAPE, RMSE, AIC, and BIC. Table 6 summarizes these results.

The Rayleigh distribution, constrained by a fixed shape parameter (α = 2), showed comparatively weaker performance than the more flexible Weibull and Beta models. While it delivered consistent results across programs, its structural rigidity limited its ability to capture the diversity of expenditure profiles observed in missile and munition development. Rayleigh produced a mean MAPE of 18.63% and an RMSE of 7.07%, underperforming the Weibull model on both metrics. Although the Beta distribution exhibited a slightly higher MAPE, Rayleigh’s limited adaptability makes it less suitable in contexts requiring tailored or highly asymmetric phasing behavior.

The Beta distribution demonstrated strong local fit in select programs, achieving the lowest overall RMSE (6.60%) and marginally more favorable AIC and BIC values compared to the Weibull model. However, these advantages were modest and did not offset its significantly higher mean MAPE of 19.11%, the worst among the three distributions tested. Given that MAPE is the most interpretable metric for proportional forecast accuracy in this context, the Beta distribution’s performance suggests that it often failed to track the actual shape of cumulative expenditures, despite achieving lower absolute error in a few cases. Furthermore, its practical application was limited by sensitivity to program variability and less stable performance across the dataset. While the Beta distribution offers greater theoretical flexibility through its two shape parameters, this flexibility did not translate into consistent predictive value, making it a less reliable choice than the Weibull model for general use in missile and munition RDT&E phasing.

The Weibull distribution delivered the strongest overall performance, achieving the lowest average MAPE (17.45%) and a competitive RMSE of 6.88%—only marginally higher than the Beta distribution’s 6.60%. Its two-parameter structure allowed it to capture a broad spectrum of phasing behaviors while maintaining consistent accuracy across the full set of 21 programs. Unlike the Rayleigh model, it provided sufficient flexibility to accommodate asymmetric expenditure profiles, and unlike the Beta distribution, it did so without sacrificing proportional accuracy or model stability. These results reinforce prior findings in the literature and establish the Weibull distribution as the most reliable and broadly applicable time-phasing model for missile and munition RDT&E programs under current data constraints.

Across all three distributions, fitting parameters globally to the aggregated dataset yielded meaningful improvements over the traditional 60/40 heuristic. However, precision improved further when we tailored model parameters to specific program characteristics. This approach—first explored by Brown et al. (2002) and later refined in Brown et al. (2015), Burgess (2006), and Bills (2023)—demonstrated that phasing accuracy can be enhanced by linking distribution parameters to observable program traits. We build directly on that foundation by extending the methodology to missile and munition development programs, a category that prior research had either subsumed within broader RDT&E datasets or excluded entirely.

Our results confirm the value of this refinement. For example, the Weibull distribution achieved a MAPE of 19.49% when fit globally, but this fell to 17.45% after incorporating regression-based parameter adjustments—a relative improvement of approximately 10.5%. Similar gains were observed across the other distributions, underscoring the utility of commodity-specific tuning. By isolating missile and munition programs as a distinct analytical class and demonstrating measurable benefits from tailored parameterization, this study expands the existing body of research and strengthens the empirical foundation for early-stage cost estimation in defense acquisition.

In this study, we set out to improve how cost analysts model time phasing in missile and munition development programs. Our findings confirm that generalized heuristics like the 60/40 rule fall short when applied to individual programs and that continuous distribution models—particularly the Weibull distribution—offer a more accurate and defensible alternative. We also show that by linking distribution parameters to observable program characteristics, we can generate practical forecasting tools for early-stage budgeting.

We found that the 60/40 rule, while close to the average across our dataset, masked substantial variability in actual cost progression. In some programs, just over 40% of costs had accrued by the midpoint; in others, more than 90% had. This wide spread makes clear that a one-size-fits-all heuristic cannot reliably guide funding allocation for missile or munition programs.

When we examined this variation more closely, we observed that it followed consistent patterns. Surface-to-air systems, often developed by the Army or Missile Defense Agency, tended to front-load expenditures likely due to programmatic events such as early testing and integration demands. Air-to-air programs, by contrast, delayed major costs until later development phases. These differences suggest that time phasing is shaped by program structure and mission profile, not randomness—further reinforcing the need for tailored models.

We fit Rayleigh, Weibull, and Beta distributions to normalized cost data and found that all three outperformed the 60/40 benchmark. However, only the Weibull distribution consistently delivered high accuracy while remaining computationally tractable and broadly applicable. The Rayleigh model, while simple, lacked the flexibility to represent heavily skewed expenditure patterns. The Beta distribution showed strong absolute fit in a few cases but suffered from apparent parameter instability and elevated MAPE on average—findings consistent with earlier studies by Burgess (2006) and Brown et al. (2015).

The Weibull distribution stood out because it consistently matched observed behavior across programs, even when we used the same parameters across the dataset. Its two-parameter form let us model both front- and back-loaded profiles without sacrificing stability or interpretability. To our knowledge, this is the first time Weibull’s advantages have been validated specifically in missile and munition development.

To move from retrospective curve fitting to predictive modeling, we developed regression equations that estimate Weibull parameters based on program characteristics. We used variables like weapon type, service branch, and development duration, and tested all models for normality, variance stability, and outlier sensitivity. The results held up well across these diagnostics.

This approach builds directly on the work of Brown et al. (2002) and addresses the longstanding need that Unger (2001) identified—namely, the lack of a method for estimating S-curve parameters before cost data become available. By using regression to derive those parameters from observable inputs, we enable analysts to generate credible phasing curves much earlier in the acquisition lifecycle.

While our results are promising, we recognize several limitations. Our dataset includes only 21 missile and munition programs, which constrains statistical generalizability. We excluded canceled or restructured efforts, which may exhibit different expenditure dynamics. We also inferred start and end points using EVM-CR timelines rather than SAR-defined milestones, introducing some ambiguity into schedule normalization. Finally, although our regression models performed well in-sample, they require further testing on broader and more diverse portfolios to validate out-of-sample accuracy.

Despite these limitations, our findings offer immediate, actionable value for the defense acquisition community. Using regression-informed Weibull models, analysts can generate realistic time-phasing curves before detailed schedules or analog programs are available. This enables more accurate cost projections in Cost Analysis Requirements Descriptions, improves milestone planning, and supports better alignment of funds across the Program Objective Memorandum. Program offices can integrate these models into Excel tools or automated scripts to produce defensible S-curves with minimal input—transforming early-phase planning from rule-of-thumb to data-driven estimation. In our view, this transition marks a meaningful step toward more agile, transparent, and analytically grounded cost analysis in the Department of Defense.

In preparing this manuscript, we employed OpenAI’s ChatGPT (GPT-4) solely for language refinement and copyediting. Specifically, ChatGPT assisted with phrasing, grammar polishing, and stylistic consistency. All substantive content—including data analyses, theoretical framing, results, interpretations, and conclusions—was generated independently by the authors. We reviewed and approved all edits produced by the AI tool to ensure accuracy, coherence, and scholarly integrity. No AI tools were used to create or alter figures, tables, or data interpretations, and authors retain full responsibility for the scientific content.

The views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the United States Government. This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

To improve distributional fit and account for program-level variation, we used OLS regression to link shape and scale parameters to observable program characteristics. The following equations define the final parameterized forms used to generate adjusted S-curves for each distribution.

Let:

1. Army, Upgrade, Surface-to-Air, Air Force, and Navy be binary indicator variables.

2. The dependent variable is normalized cumulative expenditure as a function of percent schedule.

Rayleigh is a special case of the Weibull distribution where α = 2. The fitted scale parameter β is defined as:

Weibull is defined by two tunable parameters—shape (α) and scale (β):

Beta is defined by two shape parameters (α, β) with both adjusted based on service affiliation:

All regressions met standard OLS assumptions. Residuals exhibited no significant deviations from normality or homoscedasticity, and no individual observations were found to unduly influence parameter estimates.