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Statistical parameter

In statistics, a statistical parameter is a fixed numerical value that summarizes a characteristic of an entire population, such as the population mean (μ) or proportion (p), and is typically unknown because it requires data from every member of the population.^[1]^[2] Unlike a sample statistic, which is computed from a subset of the population and serves as an estimate of the parameter, a parameter remains constant for a given population and forms the basis for probabilistic models and inference.^[3]^[4] Statistical parameters play a central role in statistical inference, where they represent true but often unobservable features of a population that analysts aim to estimate or test using sample data.^[2] Common examples include the population mean (μ), which measures the central tendency; the population variance (σ²) or standard deviation (σ), which quantify dispersion; and the population proportion (p), which indicates the fraction of the population with a specific trait.^[1]^[4] In probability theory, parameters also define the shape and properties of probability distributions, such as μ and σ for the normal distribution or λ for the Poisson distribution, enabling the modeling of random phenomena.^[5]^[6] Estimating parameters involves methods like the method of moments or maximum likelihood estimation, which use sample statistics to approximate the true values and assess uncertainty through confidence intervals or hypothesis tests.^[7] These techniques are foundational in fields like data science, economics, and medicine, where parameters inform decisions under uncertainty by bridging observed samples to broader populations.^[8] Parameters are distinguished from hyperparameters in advanced modeling, but in core statistics, they remain essential descriptors fixed by the population's underlying distribution.^[9]

Fundamentals

Definition and Scope

A statistical parameter is a numerical quantity that characterizes a specific feature of a population distribution, serving as a fixed value that summarizes an inherent property of the entire population.^[1] Examples include the population mean, denoted as \mu, which represents the average value across all members of the population, or the population variance, denoted as \sigma^2, which measures the spread of values around that mean.^[2] These parameters are typically unknown in practice, as they pertain to the complete population rather than observable data, forming the foundation for inferential statistics.^[10] The scope of statistical parameters extends to both finite and infinite populations, where a finite population consists of a concrete, countable set of units (such as all registered voters in a country at a given time), allowing parameters to be theoretically computable if full data are available, though often impractical.^[10] In contrast, infinite or conceptual populations treat the data-generating process as ongoing, making parameters abstract theoretical constructs that describe long-run behavior, such as the expected value in repeated trials.^[11] This distinction highlights true parameters as fixed, population-level truths versus empirical approximations derived from observed data, which serve as estimates rather than the parameters themselves.^[1] The concept of statistical parameters was formally introduced by Ronald A. Fisher in the early 1920s as a core element of the parametric inference framework, emphasizing models where distributions are specified up to a finite set of adjustable values.^[12] In his seminal 1922 paper, Fisher outlined the mathematical foundations of theoretical statistics, using parameters to bridge probability theory and data analysis, a development that revolutionized statistical methodology. This framework assumed that the form of the population distribution is known, with parameters tuning its specifics, laying groundwork for modern estimation techniques. Understanding statistical parameters requires familiarity with foundational probability concepts, such as random variables—quantities that assume numerical values based on chance—and probability distributions, which assign probabilities to possible outcomes of those variables.^[1] These prerequisites enable the conceptualization of parameters as deterministic features embedded within probabilistic structures, distinct from the variability observed in samples.^[10]

Distinction from Sample Statistics

In statistics, a population parameter is a fixed numerical value that summarizes a characteristic of the entire population, such as the true mean \mu of a distribution, whereas a sample statistic is a value computed from a subset of the population, such as the sample mean \bar{x}, which serves as an estimator of the parameter.^[13]^[3] Parameters are inherently constant because they describe the complete population without variability, while sample statistics exhibit sampling variability due to the random selection process involved in drawing samples from the population.^[2] This variability means that different samples from the same population will yield different statistics, reflecting the inherent randomness in sampling, whereas the parameter remains unchanged across all possible samples.^[13] The primary goal of inferential statistics is to use sample statistics as estimators to infer the unknown population parameters, treating the parameters as fixed but typically inaccessible targets for estimation.^[3] An estimator \hat{\theta} of a parameter \theta is generally expressed as a function of the sample data, \hat{\theta} = g(X_1, \dots, X_n), where X_1, \dots, X_n are independent and identically distributed random variables drawn from the population.^[14] For unbiasedness, the expected value of the estimator must equal the true parameter, \mathbb{E}[\hat{\theta}] = \theta, ensuring that, on average over repeated samples, the estimator does not systematically deviate from the parameter due to bias.^[15] Beyond unbiasedness, consistency addresses the long-run behavior of estimators as the sample size n increases, requiring that \hat{\theta} converges in probability to \theta as n \to \infty, meaning the probability of the estimator being arbitrarily close to the true parameter approaches 1.^[16] This property ensures that larger samples provide estimators that approximate the fixed population parameter more reliably, without systematic error, thereby justifying parameters as the ultimate targets of statistical inference.^[17]

Role in Probability and Distributions

Parameters in Parametric Distributions

Parametric distributions form a class of probability distributions where the entire form is defined by a finite-dimensional parameter vector, distinguishing them from non-parametric distributions that do not impose such a restrictive structure and may require an infinite number of parameters to fully describe the data-generating process.^[18] This finite parameterization enables concise modeling of complex phenomena under specific assumptions about the underlying population.^[19] In these distributions, the parameters directly influence key characteristics such as location, scale, and shape, fully determining the probability density or mass function. For instance, the normal distribution is parameterized by the mean \mu and variance \sigma^2, which control its central tendency and spread, respectively, as denoted by N(\mu, \sigma^2).^[20] Similarly, the Bernoulli distribution, a foundational discrete case, is governed by a single parameter p \in [0,1], representing the probability of success in a binary trial.^[21] The general form of a parametric probability density function (for continuous cases) or mass function (for discrete cases) is expressed as f(x \mid \theta), where x is the random variable and \theta is the parameter vector that indexes the family of distributions.^[22] A core assumption of parametric distributions is that the parameter vector \theta fully specifies the distribution, meaning the family encompasses all necessary flexibility without redundancy or insufficiency. Under-parameterization occurs when the chosen family is too restrictive, failing to capture the true data distribution and often leading to biased inferences.^[23] Conversely, over-parameterization introduces redundant parameters, resulting in non-identifiability where multiple \theta values yield the same distribution, complicating unique recovery of parameters.^[24] This requirement ensures that parameters like shape descriptors remain meaningful within the model's structure.^[25]

Functional Forms and Identifiability

In statistical models, parameters often take functional forms that relate them directly to key characteristics of the probability distribution, such as moments or quantiles, facilitating both theoretical analysis and estimation. For instance, the mean parameter \mu in a normal distribution is expressed as the first moment, \mu = \mathbb{E}[X], while higher-order parameters like variance capture second moments. This moment-based parameterization underpins the method of moments estimation, where population parameters are solved as functions of theoretical moments equated to their sample counterparts. Similarly, quantile-parameterized distributions express parameters in terms of specific percentiles, such as the median or interquartile range, offering robustness to outliers in elicitation or modeling tasks.^[26]^[27] Reparameterization involves transforming the parameter space, such as substituting \mu with \log(\mu) for positive constraints, without altering the underlying probability model or its implications for inference. This change re-expresses the likelihood or posterior in terms of new parameters, potentially simplifying optimization or sampling in Bayesian or maximum likelihood frameworks, as the Jacobian adjustment ensures equivalence in the parameter space. For example, in multilevel survival models, reparameterizing variance components from standard deviations to correlations can enhance MCMC convergence rates while preserving the model's probabilistic structure. Such transformations are particularly useful in hierarchical models, where they mitigate identifiability issues arising from overparameterization without affecting the validity of downstream inferences.^[28] A critical property of these functional forms is identifiability, which ensures that distinct parameter values \theta produce distinct distributions, enabling unique recovery from observed data. Formally, a parameter \theta is identifiable if the mapping from \theta to the induced probability measure P_\theta is injective, meaning \theta_1 \neq \theta_2 implies P_{\theta_1} \neq P_{\theta_2}. This condition guarantees that the likelihood function separates different parameter values, supporting consistent estimation. Non-identifiability arises in models like finite mixtures, where label switching among component parameters yields equivalent distributions, complicating inference unless constraints like ordering are imposed. In such cases, the mapping becomes non-injective, leading to multiple \theta values corresponding to the same P_\theta, as analyzed in early mixture model theory.^[29] The injectivity requirement can be expressed mathematically as the parameter-to-distribution map being one-to-one:

\theta \mapsto P_\theta \quad \text{is injective.}

This structural property must hold for the model to permit precise parameter recovery, distinguishing identifiable formulations from those requiring auxiliary constraints.

Types and Classification

Location Parameters

A location parameter is a scalar value in a probability distribution that specifies its position along the real line, effectively translating the entire distribution horizontally without affecting its shape or variability. This parameter determines the central tendency of the distribution, such as its mean or median, and is fundamental in location families of distributions where varying the parameter shifts the probability density function (PDF). For instance, in a pure location family, the PDF is given by f(x \mid \mu) = g(x - \mu), where \mu is the location parameter and g is the PDF of a fixed standard distribution.^[30] In broader location-scale families, the location parameter \mu combines with a scale parameter \eta > 0 to form the PDF f(x \mid \mu, \eta) = \frac{1}{\eta} g\left( \frac{x - \mu}{\eta} \right), where the term (x - \mu)/\eta standardizes the variable relative to the location shift \mu. Properties of location parameters include their role as measures of central tendency; for symmetric distributions, \mu often coincides with both the mean and median. These parameters exhibit equivariance under affine transformations of the random variable—for example, if Y = aX + b with a > 0, the location of Y becomes a\mu + b, preserving the family's structure.^[31]^[32] Prominent theoretical examples illustrate this concept. In the normal distribution N([\mu](/page/MU), \sigma^2), [\mu](/page/MU) serves as the location parameter, shifting the symmetric bell curve while \sigma^2 controls spread.^[30] Similarly, the Cauchy distribution's location parameter \mu translates its heavy-tailed, symmetric PDF, which lacks a defined mean but has \mu as its median.^[30] The uniform distribution on interval (a, b) can be parameterized with location \mu = (a + b)/2 and scale (b - a)/2, where \mu centers the flat density.^[33] In the logistic distribution, \mu acts as the location parameter, coinciding with the mean and median, and shifting the S-shaped cumulative distribution function.^[34]

Scale and Shape Parameters

Scale parameters characterize the spread or dispersion of a probability distribution by multiplicatively stretching or compressing its scale, thereby affecting the variability of the random variable without altering its fundamental shape.^[35] In many parametric families, the scale parameter appears in the density function as a factor that normalizes the distribution after transformation, ensuring it integrates to unity. For instance, in the normal distribution, the standard deviation σ serves as the scale parameter, controlling the width of the bell-shaped curve. Similarly, in the exponential distribution, the parameter β (the mean, equivalent to the inverse of the rate λ) acts as a scale parameter, where larger values of β expand the distribution's tail, representing longer expected waiting times.^[36] A canonical form for scale families, often in conjunction with a location parameter μ, is given by the probability density function:

f(x \mid \mu, \sigma) = \frac{1}{\sigma} g\left( \frac{x - \mu}{\sigma} \right),

where g is the base density (e.g., standard normal), σ > 0 is the scale parameter, and the transformation ensures scale invariance: if X has scale σ, then cX (c > 0) has scale cσ.^[30] This multiplicative property implies that scale parameters transform proportionally under linear scaling of the variable, preserving the relative spread.^[37] Shape parameters, in contrast, modify the underlying form of the distribution, influencing aspects such as asymmetry, peakedness, or the heaviness of tails, which in turn affect higher-order moments like skewness and kurtosis.^[31] For example, in the beta distribution defined on [0, 1], the parameters α > 0 and β > 0 are shape parameters that determine the distribution's skewness: when α > β, the density skews right; when β > α, it skews left; and symmetry occurs when α = β.^[38] In the kappa distribution, the shape parameter κ controls the tail behavior and boundedness, with negative κ yielding unbounded support and heavy tails suitable for modeling extreme events in hydrology or finance.^[39] An illustrative case is the gamma distribution, with density

f(x \mid \alpha, \beta) = \frac{1}{\beta^\alpha \Gamma(\alpha)} x^{\alpha-1} e^{-x/\beta}, \quad x > 0,

where α > 0 is the shape parameter influencing skewness (decreasing as α increases toward normality) and β > 0 is the scale parameter.^[35] Shape parameters thus enable flexible modeling of non-standard forms within parametric families, distinct from mere rescaling.^[40]

Estimation Methods

Point Estimation Techniques

Point estimation techniques seek to produce a single numerical value, \hat{\theta}, as an approximation to an unknown population parameter \theta based on observed sample data from a random sample.^[41] This approach contrasts with methods that incorporate uncertainty through ranges, focusing instead on a direct, data-driven guess for \theta.^[42] The goal is to select \hat{\theta} such that it closely mirrors the true parameter in expectation or through optimization criteria, balancing factors like bias and variance in the estimation process.^[43] One classical method is the method of moments, pioneered by Karl Pearson in the late 19th century.^[44] It involves solving a system of equations where the first k sample moments are set equal to the first k theoretical population moments, with k matching the number of parameters to estimate.^[26] For instance, in estimating the mean \mu and variance \sigma^2 of a distribution, the first sample moment (the sample mean \bar{x}) equals \mu, and the second central sample moment equals \sigma^2, yielding the method of moments estimator for variance as

\hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2,

where n is the sample size.^[26] This technique is computationally simple and does not require assuming a specific distributional form beyond the moments, though it may yield less efficient estimators compared to other methods in finite samples.^[45] A more widely adopted technique is maximum likelihood estimation (MLE), formalized by Ronald A. Fisher in 1922.^[46] Given independent and identically distributed observations x_1, \dots, x_n from a probability density (or mass) function f(x_i \mid \theta), MLE defines the likelihood function as

L(\theta) = \prod_{i=1}^n f(x_i \mid \theta)

and selects the estimator \hat{\theta} that maximizes L(\theta), often by maximizing the log-likelihood \ell(\theta) = \log L(\theta) for computational convenience.^[46] Equivalently,

\hat{\theta} = \arg\max_{\theta} \ell(\theta).

^[47] Under standard regularity conditions—such as the existence of the support not depending on \theta and differentiability of the log-likelihood—MLE estimators are consistent, meaning \hat{\theta} \to_p \theta as n \to \infty, and asymptotically normal, with

\sqrt{n} (\hat{\theta} - \theta) \xrightarrow{d} \mathcal{N}\left(0, \mathcal{I}(\theta)^{-1}\right),

where \mathcal{I}(\theta) is the Fisher information matrix.^[47] These properties establish MLE as efficient in large samples, though individual estimators may exhibit bias, prompting a tradeoff between bias reduction and variance minimization in practical applications.^[43]

Interval Estimation and Confidence

Interval estimation extends point estimation by providing a range of plausible values for an unknown statistical parameter θ, rather than a single value, to account for sampling variability. A confidence interval (CI) is constructed as an interval [L, U] such that the probability P(L ≤ θ ≤ U) = 1 - α holds asymptotically, where α is the significance level, meaning that in repeated sampling, the interval will contain the true parameter with frequency 1 - α as the sample size increases. This approach, formalized by Jerzy Neyman, emphasizes the interval's behavior over hypothetical repetitions of the experiment rather than a direct probabilistic statement about θ given a fixed sample.^[48] Confidence intervals are typically constructed using the sampling distribution of a point estimator or a related pivotal quantity. For instance, when estimating the mean μ of a normal distribution with known standard deviation σ based on a sample of size n, the CI takes the form \bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}, where \bar{x} is the sample mean and z_{\alpha/2} is the (1 - α/2) quantile of the standard normal distribution; this derives from the fact that \sqrt{n} (\bar{x} - μ)/σ follows a standard normal distribution. More generally, a pivotal quantity Q(X, θ) with a known distribution independent of θ allows construction of a (1 - α) CI as the set of θ values for which α/2 ≤ F(Q(X, θ)) ≤ 1 - α/2, where F is the cumulative distribution function of Q.^[49] The coverage probability of a CI is interpreted as the long-run proportion of intervals that contain the true θ across repeated samples from the population, not as the probability that a specific observed interval contains θ given the data. This frequentist perspective avoids assigning probability to fixed parameters and focuses on the procedure's reliability. For non-parametric cases where the sampling distribution is unknown or complex, the bootstrap method resamples the data with replacement to approximate the distribution of the estimator, enabling percentile or bias-corrected CIs; this technique, introduced by Bradley Efron, provides robust interval estimates without assuming a parametric form for the underlying distribution.^[50]

Applications and Examples

Univariate Case Studies

In the univariate case, the normal distribution provides a foundational example of statistical parameters, featuring a location parameter \mu, which determines the center of the distribution, and a scale parameter \sigma^2, which governs its spread or variance.^[51] These parameters fully characterize the distribution, with the probability density function given by f(x; \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right).^[52] Estimation of \mu is typically achieved using the sample mean \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i, which serves as the maximum likelihood estimator (MLE), while \sigma^2 is estimated by the sample variance s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2.^[53] For discrete univariate distributions, the Bernoulli distribution illustrates a shape parameter p, representing the probability of success in a single trial, where the distribution takes values 0 or 1 with probabilities $1-p and p, respectively.^[54] This extends to the binomial distribution for n independent trials, where p remains the key parameter governing the proportion of successes. The MLE for p in a sample of n Bernoulli trials with k successes is the sample proportion \hat{p} = \frac{k}{n}, which unbiasedly estimates the true parameter and achieves the Cramér-Rao lower bound for efficiency. The exponential distribution offers another univariate example, parameterized by a scale parameter \lambda > 0, often interpreted as the rate of events in a Poisson process, with the probability density function f(x; \lambda) = \lambda e^{-\lambda x} for x \geq 0.^[36] The mean of the distribution is $1/\lambda, directly linking the parameter to the expected waiting time between events, while the variance equals $1/\lambda^2.^[55] This parameterization highlights \lambda's role in scaling the distribution's tail behavior. To illustrate estimation in practice, consider a hypothetical dataset from a normal distribution: observations x = \{2.1, 1.9, 2.3, 2.0, 2.2\} with assumed true \mu = 2. The sample mean is \bar{x} = \frac{2.1 + 1.9 + 2.3 + 2.0 + 2.2}{5} = 2.1, providing an estimate \hat{\mu} = 2.1 that is close to the true value, demonstrating the consistency of the estimator for moderate sample sizes. The sample variance is s^2 = \frac{1}{4} \sum (x_i - 2.1)^2 = 0.025, yielding \hat{\sigma} \approx 0.158, which captures the data's low variability.

Multivariate Extensions

In multivariate statistical models, parameters generalize from scalar values to vectors and matrices to capture joint behaviors across multiple dimensions. A prominent example is the multivariate normal distribution, where the primary parameters are the mean vector \mu \in \mathbb{R}^k, which acts as the location parameter indicating the center of the distribution in k-dimensional space, and the covariance matrix \Sigma \in \mathbb{R}^{k \times k}, a symmetric positive definite matrix that describes both the dispersion and interdependencies among variables. These parameters fully characterize the distribution, enabling analysis of correlated data in fields such as finance, genetics, and signal processing. The mean vector \mu extends the univariate location parameter, such as the mean, to specify the expected value in each dimension, with marginal means aligning with its components. The covariance matrix \Sigma combines scale and shape aspects: its diagonal elements represent variances (scale in individual dimensions), off-diagonal elements capture covariances (linear dependencies), eigenvalues quantify scale along principal axes, and eigenvectors determine the orientation or shape of the elliptical contours of constant density. The determinant |\Sigma| provides a measure of overall multivariate scale, reflecting the generalized volume of the distribution, while the trace \operatorname{tr}(\Sigma) sums the variances for total dispersion. This structure allows \Sigma to model both isotropic (spherical) and anisotropic (elongated or rotated) spreads, distinguishing it from univariate scale parameters like standard deviation. Estimation of these parameters typically employs maximum likelihood methods for a sample \mathbf{x}_1, \dots, \mathbf{x}_n drawn from a k-variate normal distribution. The maximum likelihood estimator (MLE) for the mean vector is the sample mean vector \hat{\mu} = \bar{\mathbf{x}} = \frac{1}{n} \sum_{i=1}^n \mathbf{x}_i, which is unbiased and minimum-variance under normality.^[56] For the covariance matrix, the MLE is \hat{\Sigma} = \frac{1}{n} \sum_{i=1}^n (\mathbf{x}_i - \bar{\mathbf{x}})(\mathbf{x}_i - \bar{\mathbf{x}})^T, a biased but consistent estimator that converges to the true \Sigma as n increases; an unbiased alternative divides by n-1.^[56] These estimators arise from maximizing the log-likelihood function derived from the multivariate normal density. The probability density function for a single observation \mathbf{x} from the k-variate normal distribution is given by

f(\mathbf{x} \mid \mu, \Sigma) = (2\pi)^{-k/2} |\Sigma|^{-1/2} \exp\left( -\frac{1}{2} (\mathbf{x} - \mu)^T \Sigma^{-1} (\mathbf{x} - \mu) \right),

where |\Sigma| is the determinant and \Sigma^{-1} is the inverse covariance matrix (precision matrix). For n independent observations, the likelihood is the product of these densities, and taking the logarithm yields a function quadratic in the parameters, leading directly to the MLEs above upon differentiation and setting to zero.^[56] This framework underpins inference in multivariate settings, such as hypothesis testing for \mu or \Sigma, and extends to more complex models like factor analysis or Gaussian processes.

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Outline of a Theory of Statistical Estimation Based on the Classical ...
1937Outline of a Theory of Statistical Estimation Based on the Classical ... Mathematical notations produced through Infty OCR. DOWNLOAD PDF. Figures ...
[49]
The Normal Distribution - Utah State University
A normal distribution has two parameters, the mean μ μ , and the variance σ2 σ 2 . The mean can be any real number and the variance can be any non-negative ...Missing: parameterization source
[50]
Week 4: The Normal Distribution - NTNU
The estimate $\hat{\mu}$ is just the sample mean, and $\hat{\sigma}^2$ is the sample variance. These two statistics can be used to summarise the whole ...
[51]
9.4 Estimating the parameters of a Normal distribution
This chapter will introduce two models for inferring the parameters of a (single) normal distribution, both of which are set-up in such a way that it is ...Missing: parameterization | Show results with:parameterization
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1.2 - Maximum Likelihood Estimation | STAT 415
The first example on this page involved a joint probability mass function that depends on only one parameter, namely p , the proportion of successes. Now ...
[53]
15.1 - Exponential Distributions | STAT 414 - STAT ONLINE
If λ (the Greek letter "lambda") equals the mean number of events in an ... θ = 1 λ and λ = 1 θ. For example, suppose the mean number of customers to ...
[54]
Maximum-likelihood estimation of the parameters of a multivariate ...
This paper provides an exposition of alternative approaches for obtaining maximum- likelihood estimators (MLE) for the parameters of a multivariate normal ...